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NONLINEAR PROBLEMS: PRESENT A N D FUTURE

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NORTH-HOLLAND MATHEMATICS STUDIES

61

Nonlinear Problems: Present and Future Proceedings of the First Los Alamos Conference on Nonlinear Problems, Los Ala mos, NM, U.S.A., March 2-6,1981

Edited by

ALAN BISHOP DAVID CAMPBELL BASIL NICOLAENKO Los Alarnos National Laboratory LosAlarnos, New Mexico, U.S.A.

1982 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

lil North- Holland

Publishing Company 1982

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted. in any form or by any means, electronic, mechanical, photocopying. recording o r otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 86395 8

Publish ers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data

Los Alamos Conference on N o n l i n e a r Problems (1st : 1981) N o n l i n e a r problems. (North-Holland mathematics s t u d i e s ; 61) 1. N o n l i n e a r t h e o r i e s - - C o n g r e s s e s . 2 . MatheI. B i s h o p , A.R. m a t i c a l hysics--Congresses. (Alan R.f 11. Campbell, David, 1944 J u l y 23111. Nicolaenko. B a s i l . 1947IV. T i t l e . V. S e r i e s . 530.1'5 82 -3504 QC20.7.N6L67 1981 ISBN 0-444-86395-8 AACF2

.

PRINTED IN T H E NETHERLANDS

V

PREFACE

Recognizing the growing awareness o f common problems and answers i n t h e nonlinear sciences, t h e Los Alamos Center f o r Nonlinear Studies (CNLS) was established by the Laboratory D i r e c t o r i n October 1980, t o coordinate i n t e r d i s c i p l i n a r y studies, t o strengthen t i e s among Laboratory researchers and w i t h the academic community, and t o i n t e r f a c e between basic and a p p l i e d areas. The breadth o f research problems a t Los Alamos and the v a r i e t y o f technical e x p e r t i s e r e f l e c t e d i n i t s s t a f f o f f e r s a f e r t i l e environment f o r the CNLS. But from i t s inception, the Center was intended n o t f o r Los Alamos alone, b u t a l s o as an i n t e r n a t i o n a l resource f o r the e n t i r e nonl inear communi ty. To r e a l i z e t h i s i n t e n t i o n , the CNLS, through i t s Chairman, Alwyn C. Scott, has developed several continuing modes o f operation: *

i d e n t i f y i n g broad themes f o r i n t e n s i v e i n t e r d i s c i p l i n a r y research ( c u r r e n t l y , these themes are nonlinear phenomena i n r e a c t i v e flows and chaos and coherence i n physical systems); coordinating an a c t i v e v i s i t o r program, i n c l u d i n g b o t h guest l e c t u r e series and longterm c o l l a b o r a t i v e v i s i t s ;

*

-

sponsoring technical workshops aimed a t b r i n g i n g experts together t o expound recent r e s u l t s and d e l i n e a t e f u t u r e d i r e c t i o n s i n s p e c i f i c nonlinear problems (recent workshops have included "Adaptive G r i d Methods,'' "Coupled Nonlinear Osci 1 l a t o r s ,I' and "Sol itons and t h e Bethe Ansatz") ; and hosting an annual i n t e r n a t i o n a l conference on some t o p i c a l area o f nonlinear science.

I n the l a s t category, the major event f o r the CNLS i n i t s f i r s t year was an i n t e r n a t i o n a l conference on 'Nonlinear Problems: Present and Future' h e l d a t t h e Los Alamos National Security and Resources Study Center, Los Alamos, March 2-6, 1981, chaired by Mark Kac and Stanislav Ulam. This volume contains the e d i t e d proceedings o f t h a t conference. We are honored t o dedicate t h e proceedings t o Fermi, Pasta, and Ulam, distinguished Los Alamos alumni who have s e t a t r a d i t i o n o f excellence t o which t h e CNLS must aspire. [We have a l l been saddened by t h e death o f John Pasta since the Conference took place and hope t h a t t h i s volume w i l l be accepted as our small t r i b u t e t o h i s imaginative career.] As b e f i t s an inaugral conference, a very wide spectrum o f t o p i c s were represented ranging from pure mathematics, through numerical methods, t o sophist i c a t e d experiments on f l u i d s and s o l i d s , More s p e c i a l i z e d meetings are a n t i c ipated i n f u t u r e years i n c l u d i n g some o f t h e many t o p i c s t h a t i t was impossible t o cover i n 1981. However, the d e l i b e r a t e l y i n t e r d i s c i p l i n a r y atmosphere o f t h e inaugral meeting t r u l y r e f l e c t e d an e x c i t i n g stage o f development o f nonlinear science as a u n i f i e d subject. The provocative t i t l e a t t r a c t e d w e l l over two hundred p a r t i c i p a n t s and a distinguished l i s t o f speakers, many o f whom succeeded admirably i n overcoming s c i e n t i f i c language b a r r i e r s and generating a broad i n t e r e s t i n t h e i r fundamental problems. Four major t o p i c s were reperesented through survey l e c t u r e s and workshop a c t i v i t i e s : turbulence i n plasmas and f l u i d s (both the onset and fully-developed turbulence); n o n l i n e a r i t y i n f i e l d theory and i n low-dimensional s o l i d s ; reaction- d i f f u s i o n processes; and new methods i n nonl i n e a r mathematics. As described i n d e t a i l i n t h e Contents, we have preserved these d i v i s i o n s i n the Proceedings, supplementing i n v i t e d papers w i t h a small number o f r e l e v a n t c o n t r i b u t e d ones. I t i s our f i r m impression t h a t these art i c l e s , through survey and o r i g i n a l work, represent the c u t t i n g edges o f several important areas o f n o n l i n e a r i t y . We hope they w i l l be valuable reading f o r novi c e s and experts a l i k e .

vi

PREFACE

We t h i n k t h a t a l l those who attended t h e Conference w i l l remember i t f o r i t s stimulation unobtrusive organization. T h i s c r u c i a l combination c o u l d n o t have been achieved w i t h o u t the advice and support o f our colleagues i n t h e CNLS. The D i r e c t o r and h i s e f f i c i e n t s t a f f provided every conceivable help i n c o o r d i n a t i n g the splendid Laboratory f a c i l i t i e s . Mark Kac and S t a n i s l a v Ulam were supportive conference chairman and Stanislav Ulam g r a c i o u s l y agreed t o g i v e a n o s t a l g i c a f t e r - d i n n e r speech on the FPU problem which was admired by a l l . No conference i s b e t t e r than i t s secretary. I n Janet Gerwin we had a secretary whose competence was apparent a t every stage before, during, and a f t e r t h e conference. A l l t h r e e organizers are immeasurably indebted t o her f o r her s k i l l i n t h e face o f t h e continual crises! We are a l s o happy t o thank Janet, Chris Davis, Frankie Gomez, Mary Plehn, agd Kate Procknow f o r t h e i r s k i l l f u l assistance i n preparing these proceedings. Last b u t n o t l e a s t we must thank our p u b l i s h e r s f o r t h e i r e x c e l l e n t cooperation and every conference attendee f o r j o i n i n g us i n t h i s c e l e b r a t i o n o f nonlinear physics. Los Alamos

A. R. Bishop D. K. Campbell B. Nicolaenko

vii

DEDICATION

It started with an experiment--the new kind in which the same instrument, the new computer, both creates and probes an idealization of the real world. The purpose, quite modest, was to test what seemed beyond doubt, namely, that in a nonlinear discretized string the energy initially concentrated in one vibrational mode ultimately distributes itself among all modes. The result, first announced in a Los Alamos report, was however, startlingly different: after an initial tendency toward equipartition, the energy flowed back to the initial mode. Thus a new chapter of nonlinear science began and much of its spectacular growth in the past quarter of a century is directly or indirectly traceable to the pioneering experiment with the nonlinear string. It is therefore fitting that this Conference marking the creation of the Los Alamos Center for Nonlinear Studies be dedicated to the authors of that historic 1955 report: Fermi, Pasta, and Ulam.

Los Alamos

M. Kac N. Metropol is

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ix

TABLE OF CONTENTS

PREFACE

V

vii

DEDICATION PART I :

NEW METHODS AND RESULTS I N NONLINEAR ANALYSIS

Optimal C o n t r o l o f Non-Well-Posed D i s t r i b u t e d Systems and R e l a t e d N o n l i n e a r P a r t i a l D i f f e r e n t i a l Equations J.-L. LIONS

3

Compactness and T o p o l o g i c a l Methods f o r Some N o n l i n e a r V a r i a t i o n a l Problems o f Mathematical Physics P.L. LIONS

17

On Yang-Mills F i e l d s I . M. SINGER

35

Gauge Theories f o r S o l i t o n Problems D. H SATTI NGER

51

The I n v e r s e Monodromy Transform i s a Canonical T r a n s f o r m a t i o n H. FLASCHKA and A.C. NEWELL

65

Numerical Methods f o r N o n l i n e a r D i f f e r e n t i a l Equations J.M. HYMAN

91

.

L i m i t A n a l y s i s o f P h y s i c a l Models M. FREMOND, A. FRIAA, and M. RAUPP

109

V i s c o s i t y S o l u t i o n s o f Hamil ton-Jacobi Equations M.G. CRANDALL

117

B i f u r c a t i o n o f S t a t i o n a r y Vortex C o n f i g u r a t i o n s J.I. PALMORE

127

Exact I n v a r i a n t s f o r Time-Dependent N o n l i n e a r Hami 1t o n i a n S y s t e m H.R. LEWIS and P.G.L. LEACH

133

I s o l a t i n g I n t e g r a l s i n G a l a c t i c Dynamics and t h e C h a r a c t e r o f S t e l 1a r O r b i t s W . I . NEWMAN

147

PART I 1

:

NONLINEARITY I N FIELD THEORIES AND LOW DIMENSIONAL SOLIDS

Physics i n Few Dimensions V.J. EMERY

159

S o l u t i o n o f t h e Kondo Problem N. ANDRE1

169

Kinks o f F r a c t i o n a l Charge i n Quasi-One Dimensional Systems J.R. SCHRIEFEER

179

TABLE OF CONTENTS

X

T h e o r e t i c a l l y Predicted Drude Absorption by a Conducting Charged S o l i t o n i n Doped-Polyacetylene M.J. R I C E

189

Pol arons in Polyacetylene A.R. BISHOP and D.K. CAMPBELL

195

S. ETEMAD and A.J.

L i g h t S c a t t e r i n g and Absorption i n Polyacetylene HEEGER

209

Quasi-Solitons: A Case Study o f the Double Sine-Gordon Equation P. KUMAR gnd R.R. HOLLAND

229

Classical F i e l d Theory w i t h Z(3) Symmetry H.M. RUCK

237

PART 111 :

PART

REACTION-DIFFUSION PROCESSES

Some C h a r a c t e r i s t i c N o n l i n e a r i t i e s o f Chemical Reaction Engineering R. A R I S

247

Propagating Fronts i n Reactive Media P.C. FIFE

267

IV :

NONLINEAR PHENOMENA I N FLUIDS AND PLASMAS

Regularity Results f o r t h e Equations o f Incompressible F l u i d s Mechanic a t the B r i n k o f Turbulence C. BARDOS

289

F i n i t e Parameter Approximative S t r u c t u r e o f Actual Flows C. FOIAS and R. TEMAM

317

The Role o f C h a r a c t e r i s t i c Boundaries i n the Asymptotic S o l u t i o n o f the Navier-Stokes Equations F.A. HOWES

329

Some Approaches t o the Turbulence Problem J. MATHIEU, 0. JEANDEL, and C.M. BRAUNER

335

I n s t a b i l i t y o f Pipe A.T. PATERA and S.A.

Flow ORSZAG

367

Some Formalism and Predictions o f the Period-Doubling Onset o f Chaos M.J. FEIGENBAUM

379

T r i c r i t i c a l Points and B i f u r c a t i o n s i n a Q u a r t i c Map S.-J. CHANG, M. WORTIS, and J.A. WRIGHT

395

Recent Experiments on Convective Turbulence J.P. GOLLUB

403

Experimental Observations o f Complex Dynamics i n a Chemical Reaction J.C. ROUX, J.S. TURNER, W.D. McCORMICK, and H.L. SWINNEY

409

Nonl inear P1asma Dynamics Be1ow the Cyclotron Frequency J.M. GREENE

423

TABLE OF CONTENTS

xi

Turbulence and S e l f - c o n s i s t e n t F i e l d s i n Plasmas D. PESME and D.F. DUBOIS

435

S e l f - F o c u s i n g Tendencies o f t h e N o n l i n e a r S c h r o d i n g e r and Zakharov Equations H.A. ROSE and D.F. DUBOIS

465

C h a o t i c O s c i l l a t i o n s i n a S i m p l i f i e d Model f o r Langmuir Waves P.K.C. WANG

479

INDEX OF CONTRIBUTORS

483

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PART I New Methodsand ResuIts in Nonlinear Analysis

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.1 0 North-Holland Publishing Company, 1982

3

OPTIMAL CONTROL OF NON WELL POSED DISTRIBUTED SYSTEMS AND RELATED NON LINEAR PARTIAL DIFFERENTIAL EQUATIONS Jacques-Louis LIONS College de France and I N R I A

Various p r a c t i c a l problems l e a d t o the question o f optimal cont r o l o f non w e l l posed systems such as non l i n e a r unstable systems. We begin w i t h simple l i n e a r quadratic examples o f non..well posed e v o l u t i o n systems ; the o p t i m a l i t y system leads t o new non l i n e a r P a r t i a l D i f f e r e n t i a l Equations o f R i c a t t i ' s type. We study then non l i n e a r parabolic unstable systems, w i t h a d i s t r i b u t e d o r a boundary c o n t r o l , w i t h o r without const r a i n t s . The o p t i m a l i t y system i s given.

-

-

INTRODUCTION

1. L e t d be a p a r t i a l d i f f e r e n t i a l operator, l i n e a r o r not, o f e v o l u t i o n o r o f s t a t i o n a r y type. I n the usual theory o f optimal c o n t r o l o f d i s t r i b u t e d systems, the s t a t e equation i s given, i n a formal manner, by

(1)

ffy = B v

where v denotes the c o n t r o l v a r i a b l e ( o r f u n c t i o n ) and is an operator which can be thought o f as g i v i n g boundary c o n d i t i o m ( 1 ) one has t o add i n i t i a l condit i o n s i f ff i s o f e v o l u t i o n type. I n the usual theory one assumes t h a t , given v i n a s u i t a b l e Banach s ace U , equation (1) subject t o appropriate boundary and i n i t i a d i t s a u n i ue s o l u t i o n denoted by y ( v ) ; y ( v ) i s t h e s t a t e o f t h e system ; y ( v ) m o n g s t o a space w en v spans U.

+

Then the cost f u n c t i o n i s given by J(v)

Cp(Y(V))

+

uJ( I I V l l " )

(2)

where Cp i s a continuous f u n c t i o n a l from Y * R , where l l v l l v i n U and where X + $ ( A ) i s continuous f o r A t 0, $(O)

=\,

denotes the norm o f $ ( A ) * +EC as X + i-.

IfUad denotes a ( s u i t a b l e ) subset o f U, then the problem i s t o f i n d

i n f J(v), v

E

uad.

(3)

If( 3 ) admits a s o l u t i o n u, one o f the main questions i s then t o f i n d a s e t o f necessary ( o r necessary and s u f f i c i e n t ) conditions f o r c h a r a c t e r i z i n g u, i.e. t o f i n d the o p t i m a l i t y system. For these questions we r e f e r t o J.L. LIONS [1][2][3] and t o the Bibliography therein. 2. A s l i g h t l y d i f f e r e n t s i t u a t i o n can occur i f the f u n c t i o n a l 4 i n ( 2 ) i s not def i n e d on t h e whole space Y . Then one has t o introduce new f u n c t i o n a l spaces.

4

J.-L. LIONS

L e t us g i v e an example. Let R be a bounded open s e t o f R3 w i t h boundary consider the s t a t e equation

a t y

-

By = v ( t ) b ( x - b ) i n Q = R

x

r

; we

IO,TC

0 on C = T x 10,TC,

(4)

Y(XY0) = 0 where 6(x-b) denotes the Dirac measure a t p o i n t b c R . Given V E LL(O,T), probl m (4) admits a unique weak s o l u t i o n ( c f . J.L. E. MAGENES C11) y ( v ) E L ( Q ) .

5

1

LIONS and

L e t us consider now the c o s t f u n c t i o n J(v) = [y(X,T;V)-Zd12 dx + N V2 d t ,

IT

R

(5)

0

2 where Zd i s given i n L ~ ( R )and N > 0. I n general, f o r v E L (o,T), y ( * ,T;v) i s d e f i n d b u t as an element o f H-l(Q) (Sobolev space o f order -1) and n o t an e l e m $R). Therefore ( 5 ) does n o t make sense f o r v E Lz(0,T). One h a s X e n t o r e s t r i c t J t o those v ' s such t h a t V E

L2 (0,T) and y(*,T;v)

E

L2(Q).

(6)

This defines a H i l b e r t space (when provided w i t h the norm ( say U, and i f

we consider again problem ( 3 ) . I n order t o proceed i t i s necessary t o study U, n o t only t o make things more precise b u t a l s o because the dual U ' o f U i s needed f o r w r i t i n g the o p t i m a l i t y system. One v e r i f i e s t h a t U coincides w i t h the s e t o f those v ' s i n LZ(0,T) such t h a t

l',:

( 2 T - ( t t ~ ) ) - ~ " v ( t ) v ( x ) d t ds <

(8)

m.

For questions o f t h i s type we r e f e r t o J.L. LIONS C4lC51, J. SIMON C11. 3. A t h i r d s i t u a t i o n can occur when (1) i s n o t a w e l l posed problem. Equations(1) which have a physical i n t e r e s t and which l e a d t o non w e l l s e t problems a r i s e i n unstable phenomena, i n s i t u a t i o n s where we have b i f u r c a t i o n s c f . J.P. KERNEVEZ J.L. LIONS and D. THOMAS C l l . One has then t o change s i g n i f i c a n t l y t h e p o i n t o f view. One considers the s e t o f v and z such t h a t

-

VE

u,

Z E

Y

(9)

.

nz =bv

(10)

Then one considers the cost f u n c t i o n J(V¶Z)

=

4(z)

+

+(

IIVII" 1

and one looks f o r i n f J(v,z),

v,z subject t o ( 9 ) (10)

w i t h the possible added c o n s t r a i n t

OPTIMAL CONTROL OF NON-WELLPOSED DISTRIBUTED SYSTEMS

VE

uad

5

9

As an example (without physical i n t e r e s t ) we consider

with the conditions

z(x,O) = 0 , z = 0 on

C

(15)

(one can prove t h a t conditions (15) do make sense ; c f . Section 1 below). Let the cost function be given by

and l e t Uad be a closed convex subset of L2 (9) such t h a t the s e t of those v , z ' s such that V E uad and (14) (15) hold t r u e is not empty. Then inf J(v,z)

,

V E

uad

, v,z

.

s a t i s f y (14) (15)

admits a unique solution {u,yl.

(17)

Returning t o the general case, we want t o find an optimality system f o r these problems of optimal control. 4. We consider i n t h i s paper three ( o f the many) situations of such problems. I n Section 1 we consider a system of type (14) b u t which i s also non well posed f o r t < T, namely

x+ m(t)

~z = v ,

(18)

w i t h m > O (resp. < O ) near 0 (resp. near T ) . Decou l i n the optimality system leads t o apparently new nonlinear Partial Diffed u a t i o n s . In Section 2 we consider unstable systems governed by

(or with boundary control). Other situations are indicated in J.L. LIONS C51, such as the case of e l l i p t i c systems which can be controlled by Cauchy data on part of the boundary. Problems of optimum design where again the s t a t e equation i s not well s e t will be studied elsewhere. 5 . Existence problems f o r not necessarily well s e t problems (such as Navier-Stokes equations in space dimension equal 3) have been studied by A.V. FOURSIKOV C11 ; t h i s author does not consider the optimality system. The optimality system for problem ( 1 7 ) involves new functional spaces (of distributions of i n f i n i t e order) ; we refer t o P . RIVERA C11.

6

J.-L. LIONS

1. New non l i n e a r P a r t i a l D i f f e r e n t i a l equation o f R i c c a t i ' s type. 1.1. S e t t i n g o f the problem. We consider couples I v , z l such t h a t 2 V,ZE L ( Q ) x L 2 ( Q ) , Q = RxlO,TC, and

at

t

m ( t ) z = v i n Q,

z(x,O)

z

= 0 i n R,

0 on C = rxl0,TC.

I n (1.2) m denotes a continuous f o n c t i o n w i t h a graph as represented on F i g . 1. Conditions ( 1 . 3 ) (1.4) make sense. L e t us check i t f o r (1.3)llows (1.1) (1.2) t h a t

r"

from

(1.5)

(where H-2(R)

= dual o f

2 2 Ho(R), Ho(R)

=

{$I$,

i t f o l l o w s from (1.5) and from standerd r e s u l t s t h a t z i s continuous form C O Y T I * H- (Q) SO t h a t (1.3) makes sense. One checks by s i m i l a r techniques t h a t (1.4) makes sense.

Fig. 1

O f course (1.2) (1.3) (1.4) i s a non w e l l posed problem. Moreover the system i s a l s o non w e l l posed i n the backward time d i r e c t i o n ; i f we replace (1.3) by z(x,T) = 0 i n R , the corresponding problem i s a l s o non w e l l posed. The cost function t h a t we consider i s given by

n

zd given i n L L ( Q ) , N>O. We consider the "no-constraints" problem, namely i n f J(v,z)

, v,z

subject t o (1.1) ...( 1.4).

1.2. O p t i m a l i t y system. I t i s a simple matter t o check t h a t problem (1.7) admits a unique s o l u t i o n u,y. We want t o characterize u,y ; t h i s c h a r a c t e r i z a t i o n i s given by the f o l l o w i n g r e s u l t : there e x i s t s a unique s e t {u,y,pl o f f u n c t i o n s such t h a t

OPTIMAL CONTROL OF NON-WELL-POSED DISTRIBUTED SYSTEMS

m(t)Ay = u

t

,-

3

t

m(t)Ap = y-zd

7

,

(1.8)

y = p = 0 on 1,

(1.9) (1.10)

Y(x,O) = 0, p(x,T) = 0 YyPE

(1.11)

L2(Q)

ptNu = 0

(1.12)

One can o f course e l i m i n a t e u by (1.8) (1.12). The system i n u,y,p o p t i m a l i t y system ; p i s a Lagrange m u l t i p l i e r . .

i s called the

The proof o f t h i s r e s u l t can be obtained as f o l l o w s ; one considers the penalized problem

(1.13) where where

2 2 L (Q), Z E L (Q), > 0 i s "small".

V E

E

$

t

2 m(t)AzE L ( Q ) and z(x,O)

= 0 ; z = 0 i n 1, and

Then (1.14)

= J,(u,,Y,).

i n f J,(v,z)

One defines p, by 1 P, = (Tt mAy,

-

-

u),

.

(1.15)

One v e r i f i e s t h a t

I-

t

p,(x,T)

m(t)Ap, = 0

, p,

y,-zd

i n Q,

(1.16) = 0 on 1

and t h a t pE

+ NU,

(1.17)

= 0 i n Q.

It follows from (1.13) and (1.17) t h a t

uEyyE,pE remain

, as E + O , i n a bounded s e t o f ( L2 ( Q ) )3 ,

(1.18)

and t h a t

T e r e f o r one can pass t o the l i m i t i n E -+ 0. One v e r i f i e s t h a t u,,yE L! (9)! x $L (9) and t h a t u,y,p s a t i s f y the o p t i m a l i t y system.

+

u,y i n

the

One can w r i t e a v a r i a t i o n a l formulation f o r the problem, as f o l l o w s : p i s solution of 1 (1.20) t m(t)AP, t m(t)Aq) t (psq) = -(zds t mAq) vq (2 2 (where (p,q) = p q dx d t ) where q E L ( Q ) , t m(t)Aq E L ( Q ) , q(x,T) = 0,

1,

-2

a

8

-#

8

J.-L. LIONS

1.3. We are now going t o show how t o "uncoup1e"the o p t i m a l i t y system.

The technique i s s i m i l a r t o the one i n J.L. LIONS C 11 b u t one deals now w i t h s o l u t i o n s , due t o the f a c t t h a t the " s t a t e equation" i s n o t w e l l posed.

weak

One considers a problem analogous t o (1.8) ...( 1.12) b u t i n t h e i n t e r v a l Is,lC O<s
,

$t

t

$(x,s)

1 J,

= h(x) i n

0

- 8t

,

R, $(x,T)

dJ, = $-zd

= 0 in

, i n Rxls,TC ,

R,

(1.21)

$ = J) = 0 on I ' x l s , T C .

I n (1.21) h i s given i n , say, the space o f C" smooth f u n c t i o n s w i t h compact support. Problem (1.21) i s the o p t i m a l i t y system f o r a problem e n t i r e l y analogous t o (1.6) (1.7) b u t w i t h Rxl0,TC replaced by Rxls,TC and w i t h z(x,O) = 0 replaced by $ ( x , s ) = h ( x ) . Therefore (1.21) admits a unique s o l u t i o n and J,(x,s)

i s uniquely defined.

(1.22)

The space where $ ( a ,s) belonqs depends whether s < to o r s > to. I f s < to, one deals w i t h a w e l l set system, pnd J , ( * , s ) c HJ(R) ; i f s < to, one has t o work w i t h weak s o l u t i o m ( * ,s) E H- (a). We have :

(1.23)

J , ( * , s ) = P(s)h t r ( s )

where P(s) i s a l i n e a r 0perator;one has

(1.24) I f we take i n (1.21) h = y(*,s) y ( s ) then (p,$ so t h a t (1.23) becomes (changing s i n t o t) :

P(t) = P(t)Y(t) t r ( t )

= r e s t r i c t i o n o f y,p t o Rxls,TC,

(1.25)

*

Using the L. Schwartz kernel theorem C 1 1 , one sees t h a t P(t)h =

I

R

P(x,S,t)

(1.26)

h(S) dS

where the kernel P(x,S,t) i s a d i s t r i b u t i o n on RxXRg. (1.8),..(1.12) one obtains f i n a l l y t h a t P(x,S,t) Using ( 1 . 2 5 ) ' o

-

ap

t

P(x,S,T)

m(t)(Ax+AS)P t

4

~ ~ ( x , ~ y t ) P ( ~ y ~= y6(x t ) d-C), C

satisfies

(1.27) (1.28)

0

and P(x,S,t)

= P(S,x,t)

V X Y S 6 Rxi-2

.

(1.29)

The Boundary conditions are o f D i r i c h l e t type, i n the usual sense f o r t o < t < T

OPTIMAL CONTROL OF NON-WELL-POSEDDISTRIBUTED SYSTEMS

and in a very weak sense f o r 0 < t
9

0 on

1.4. Formal Remarks.

Remark 1.1 : Equation (1.27) i s a non linear equation of Riccati's type. The operator i s well s e t f o r to< t < T ; when t < to, the linear part of the operator becomes non well s e t ( i n the backward time d'rection) b u t the whole operator re ains well s e t in a larger space; L(H-l(S2) ; H o ( R ) ) i s replaced by L ( H - l ( R ) ; 'H (R))..

1

T

Remark 1.2 : Of course one can consider functions m ( t ) which change sign in C0,TI an arbitrary number of times. Assume f o r instance t h a t m ( t ) i s given onlR, with eriod 1, and changing sign in C0,ll a t l e a s t once, and such t h a t m > 0 n e a r r h i ne (1.30)

m,( t ) = m( t / E ) .

All what has been said i s valid w i t h m replaced by mE as defined by (1.30). Therefore there exists P,(x,S,t) solution of (1.27) (1.28) (1.29) where m i s replaced by mE' Finally when 1.1 =

J:

E

+

0 , mE should be replaced by (1.31)

m(t)dt

and one obtains (formally) for the limit of P, : (1.32)

i t shows t h a t the of 1.1 ; one has P E 1.I > 0).

operties of regularity o f the l i m i t of PE depends on the sign H - l ( R ) ; Hk(R)) ( resp. P ~ l e ( H - l ( n ); H - l ( n ) ) ) i f v < O (resp.

T-t )6(~-6). If u=O, P(X,S,t) = fl ( t h -

Remark 1.3 : One can also consider the equation

z t A z = v

(1.33)

at

where Az = - a ( x az ) on R = R , with z(x,O) = 0 ( t h i s i s again a non well posed problem). The Zorresbonding Riccati's type equation will be : tm

P(x,S,t) 2. 2.1.

-

=

P(S4,t)

.

.

-m

Control of unstable systems. Distributed control w i t h o u t c o n s t r a m .

We are given v,z such t h a t

v,z

L ~ ( Q x)

L?Q) ,

(1.34)

10

J:L. LIONS

(z(x,o)

o

i n R, z =

o

on C . 2 Remark 2.1 : Given v i n L ( Q ) , problem ( 2 . 2 ) does n o t necessarily adm t a globa s o l u t i o n i n time. The s o l u t i o n , defined l o c a l l y , can blow up. =

I

Remark 2.2 : I f Z E L 6 ( Q ) then i t f o l l o w s from (2.2) t h a t az

- AZ

3

2

= V t Z E L (9) d

which imp1ies , w i t h the boundary conditions i n (2.2)

, that

n

The c o s t f u n c t i o n J(v,z)

i s given by

6 where zd i s given i n L ( Q ) and where N>O. I n the case w i t h o u t c o n s t r a i n t we want t o f i n d i n f J(v,z),

v,z subject t o ( 2 . 1 ) ( 2 . 2 ) .

One has f i r s t problem (2 . 6 ) admits a s o l u t i o n {u,y}. (There i s no reason why t h i s s o l u t i o n should be unique). For the proof, we consider a minimizin sequence vn,z5. By v i r t u e o f (2.5) a VnsZn r e a i n i n a bounded set o f L 2 ( Q ) x L ( Q ) . Then Vntzn remains i n a bounded s e t o f L ( Q ) SO t h a t Zn remains i n a bounded s e t of H ~ Y ~ ( Q Ther ). f o r e one can e x t r a c t a subsequence, ) ) s t i l l denoted by VnaZny such t h t v -P u i n L ( Q ) weakly, zq -t y i n H ~ Y ~ (nQL6(Q weakly. Since, i n p a r t i c u l a r , H$.1(1) + L ( 9 ) i s compact, Zn * y 3 i n L2(Q ) weakly and therefore

i

?

5

8-

AY

-

y3 = u, y(x,o)

One can then show t h a t p 6 L 2(Q ) and

-

Ay

-

y3 = u,

~ ( ~ 3 0 0, ) p(x,T)

=

if u,y

-3-

Ap

o , y=o on c.

rn

i s a s o l u t i o n o f (2.6) then t h e r e e x i s t s p such t h a t

-

2 5 3y p = (y-zd)

,

= 0,

(2.9)

y = p = 0 on X,

ptNu = 0 i n Q For the proof, one considers, as i n Section 1.2,

(2.10) the penalized problem

OPTIMAL CONTROL OF NON-WELL-POSEDDISTRIBUTED SYSTEMS

for

L2 ( Q ) ,

Ve

Zc

L6 ( Q ) , Z c H’J(Q),

11

z ( o ) = 0, z l C = 0.

We c ons ider i n f J€(V,Z) and we denote b y v,,yE t h i s p e n a l i z e d problem. We s e t

a s o l u t i o n (which e x i s t s ! ) o f (2.12)

We have 2 ~Y,P,

- at -

AP,

-

p,(x,T)

= 0

, p,

= (Y,-zd)5 = 0

on

i n Q, (2.13)

C,

and p,t

NU, = 0.

(2.14)

Therefore pE =-NU, it follows t h a t 3y,p,2

(y,-zd)5

t

s o t h a t p,

i s bounded ( a s E + O )

i n L2 (Q) ; s i n c e y,

i s bounded i n L6 (Q),

i s bounded i n L 6 l 5 ( Q ) hence t h e r e s u l t w i l l f o l l o w

i s bounded i n W 291;6/5(Q),

.b

2.2. D i s t r i b u t e d c o n t r o l w i t h c o n s t r a i n t s . L e t us t a k e a gain t h e s i t u a t i o n ( 2 . 1 ) ( 2 . 2 ) traints VE

Uad

, Uad

= c l o s e d convex subset o f L

o f S e c t i o n 2.1 ; we add now t h e cons2 (Q),

(2.15)

and we make t h e ( s t r o n g ) assumption : uad has a non empty i n t e r i o r .

(2.16)

We a l s o assume t h a t t h e s e t o f { v , z l n o t empty.

such t h a t (2. 1)(2. 2)(2. 15)

take place

,-&

By a s u i t a b l e a d a p t a t i o n of an i d e a o f

P. RIVERA C11 one c a n a g a i n show t h e e x i s tence o f a Lagrange m u l t i p l i e r p f o r a s o l u t i o n u,y o f J(u,y)

= i n f J(v,z),

s a t i s f y (2.1)(2.2)(2.15). 2 More p r e c i s e l y , t h e r e e x i s t s P E L (9) s a t i s f y i n g (2. 8) (2.9) and (ptNu,v-u)

’0

v,z

v v 6 uads

E

uad

.

(2.17)

(2.18)

2.3. Boundary c o n t r o l . L e t us assume now t h a t v and z s a t i s f y v E ~ ~ ( , 2z E) L ~ ( Q )

(2.19)

12

J.-L. LIONS

_ a~ - az at az

--i-

av

z3 =

o

i n Q,

v on 1,

(2.20)

z(x,O) = 0 i n R . If z

E

L6(Q) then

_ a' - A Z at

= z

3 E

2 L (Q)

so t h a t (we use :fractional"

(2.21) Sobolev spaces, as, f o r instance, i n J.L. LIONS,

E. MAGENES C 11) 'ZE

L2 ( o , T ; H ~ / ~ ( R ) );

(2.22)

then (2.21) gives az 2 T

L~ ( 0 ,T;H-~/* (0)

(2.23)

so t h a t z i s (a.e. equal t o a function) continuous from C O Y T I therefore

+

H112(.Q)

, and (2.24)

z 6 L~(o,T;H~/~(R)). The cost f u n c t i o n i s given by

(2.25) We want now t o f i n d

I

V E

, for

v,z subject t o (2.19)(2.20) 2 uad = closed convex subset o f L ( C ) ;

i n f J(v,z)

and (2.26)

we assume t h a t the s e t o f v,z s a t i s f y i n g these c o n s t r a i n t s i s n o t empty. This problem admits a s o l u t i o n Cu,y) , n o t necessarily unique.

I n order t o o b t a i n an o p t i m a l i t y system (and a Lagrange m u l t i p l i e r ) we s h a l l assume ( f o r technical reasons explained below) t h a t 2 RcR ,

(2.27)

and t h a t Zd E L10(o~T;t.6(fi))(1)

(2.28)

We have then t h e existence o f p such t h a t p c L ~ ( o , T ; L ~ ( R )n) L ~ ( o , T ; H ~ ( R ) )

(2.29)

( * ) I t would be s u f f i c i e n t t o assume t h a t z d c L6(Q) and such t h a t t h e r e e x i s t s B > o Such t h a t Zd6 L10(OJ;L~'~(R)).

OPTIMAL CONTROL OF NON-WELL-POSED DISTRIBUTED SYSTEMS

- AY - y 3 = o , - 8 - AP - 3y2p

13

(y-zd)5 i n Q, (2.30)

\y(x,o)

c

0, ~ ( x , T ) =

=

(p+NU)(V-U) dc

o 0

i n R, Vve uad

U E

uad.

(2.31)

L e t us sketch t h e proof. We s t a r t from t h e p e n a l i z e d problem : (2.32) VE

L2 ( I ) , Z

z(x,O)

= 0 and

where

Then

E

L6

az

(a), $ - Azc L2 (Q), = v on 1.

z s a t i s f i e s (2.22)(2.23)(2.24).

The problem

, VE

i n f J,(v,z)

uad and v,z as above

admits ( a t l e a s t ) a s o l u t i o n uEYyE. As

2 remain i n a bounded s e t o f L ( C )

u, ,y 1

E

(K 5% -

- y,)3

Ay,

(2.33) +

x

0 we have

6 L (9) ,

remains i n a bounded s e t o f L2

(2.34)

(a).

(2.35)

We i n t r o d u c e (2.36) We v e r i f y t h a t

12

=

0 on C,

P,(x,T)

=

o

in

(2.37)

n

and (2.38) I t f o l l o w s from (2.35)(2.34)

-

Ayf = f f

that

, fE bounded

2

i n L (Q),

(2.39)

and we have

av = u,

on C, y,(x,O)

= 0 i n R.

Therefore ( c f . (2.22) (2.23) (2.24) )

(2.40)

14

J.-L. LIONS

y,

remains i n a bounded s e t o f L2 (0,T;H3/2(R)

The re f o re , f o r e v e r y 0 such t h a t 0 s 9 y,

nLm(0,T;H1/2(R)).

(2.41)

1, one has 1/2+0 remains i n a bounded s e t o f L2le(O,T;H (0)). 5

B u t - s i n c e t h e space dimension e q u a l s 2 H1'2+9(Q)cL 6 ( Q ) i f 0 = 1

-

(2.42)

one has

.

(2.43)

Then (2 . 4 2) shows t h a t y,

remains

i6

a bounded s e t o f L

We s h a l l v e r i f y below t h a t p, p,

12

6

(0,T;L

(2.44)

(R)).

satisfies

remains i n a bounded s e t of Lm(O,T;L

2

(a)) n L2 (O,T,H 1(0)).

(2.45) 2

(F)

xL6(Q)x Th5n one fan pass t o t h e l i m i t . One s h o w s ' t h a t uE,YE,PE * u,y,p i n L X L (0,T;H ( Q ) ) weakly ( a n d a l s o i n weak t o p o l o g i e s o r weak s t a r t o p o l o g i e s c o r r e s ponding t o (2 . 4 1)(2 . 4 4 ) ( 2 . 4 5 ) ) , where u,y i s a s o l u t i o n o f (2.26) and where (2.29) (2.30) (2.31) h o l d t r u e .

I n o r d e r t o v e r i f y (2.45) l e t us change t i n t o T - t and l e t us s e t $ ( x y t ) = p,(x,T-t) (y,(X,T-t)-Zd(X,T-t))

42- y,(xyT-t) 5 = h,

= o,(xyt)

.

,

We have

I

(2.46)

Y! av

=

o

We s e t :

on C, ~ ( X , O )=

o

1$12

,ll~111~ = J

$2 dx

R

IV$12dx +

R

R

1$12

dx.

Tak ing t h e s c a l a r p r o d u c t o f (2.46) w i t h 9 , we o b t a i n I d Ta l W 2 + I l 9 ( t ) 1 I 2 - l W l 2 - (n$N (where ( f ,I)) =

1

R

f 9 dx) -

.

= (h&

(2. 47)

But

so t h a t (and t h i s i s v a l i d i n dimension n s 3 )

(2.48)

(where t h e C's denote v a r i o u s c o n s t a n t s ) . We have a l s o

OPTIMAL CONTROL OF NON-WELL-POSED DISTRIEUTED SYSTEMS

15

so t h a t

(2.49)

(2.50)

1 remains i n a bounded s e t o f L (0,T)

, i t follows L6W from (2.51) and Gronwall’s i n e q u a l i t y t h a t J, remains i n a bounded s e t o f Lm(O,T; i.e. (2.45). m L2(R)) n L2 (O,T;Hl(R)) and since i n particularllQ,ll

2.4. Various remarks. Remark 2.3 : By estimates analogous t o those o f S e c t i o n 2.3, one sees t h a t t h e Section 2.2 i s v a l i d w i t h uad n o t n e c e s s a r i l y s a t i s f y i n g (2.16) i f Remark 2.4 : L e t us consider the equation

at -

AyE

-

y,

3

+

EY,

ayE

av = 0 on I, y,(x,O) For

E

5

= v in

Q, (2.52)

0 i n R.

> 0 t h i s problem admits a unique s o l u t i o n y,(v).

One can then d e f i n e (2.53)

Then one can v e r i f y t h a t :

l i m ( i n f J E ( v ) ) = i n f J(v,z), E+O

v,z s u b j e c t t o (2.1)(2.2)(2.15).

(2.54)

VEUad

Remark 2.5 : One can o b t a i n s i m i l a r r e s u l t s f o r o p t i m a l i t y systems connected w i t h s t a t i o n a r y problems such as -AZ

- z’

= v in

R, (2.55)

and

18

J.-L. LIONS

BIBLIOGRAPHY

A.V.

FOURSIKOV C11 On some problems o f c o n t r o l . Doklady Akad. Nauk, 1980, pp. 1066-1070.

J.P. KERNEVEZ, J.L. LIONS and D. THOMAS C 11 To appear. J.L. LIONS 111 Sur l e c o n t r a l e optimal des systemes gouvernes par des equations aux derivees p a r t i e l l e s . Paris, Dunod-Gauthier-Villars, 1968 ( tn g l i.s h t r a n s l a t i o n by S.K. MITTER, Springer, 1971). C21 3ome aspects of the optimal c o n t r o l o f d i s t r i b u t e d parameter sysReg. Conf. Ser. i n Applied Math., S I A M 1 6 , 1972.

tems.

C31 Remarks on the theory o f optimal c o n t r o l o f d i s t r i b u t e d systems, i n Control Theory of Systems Governed by P a r t i a l D i f f e r e n t i a l tquations, ed. by A.K. A Z I Z , J.W. WINGATE, M.J. BALAS, Acad. Press, pp. 1-103. C41 Function spaces and optimal, c o n t r o l o f d i s t r i b u t e d systems. U.F.R. J . Lecture Notes, Rio de Janeiro, 1980. C51 Some methods i n t h e Mathematical Analysis o f systems and t h e i r conScience Pub. Company, BEIJING, 1981.

trol.

J.L. LIONS and E . MAGENES C11 Problemes aux l i m i t e s non homogenes e t a p p l i c a t i o n s , Vol 1,2. Dunod, Paris, 1968 English t r a n s l a t i o n by P. K t N " H , Springer Verlag , 1972.

.

.

P. RIVERA C11 To appear

L. SCHWARTZ C11 Theorie des noyaux. Proc. I n t . Congress o f Math. 1 (1950) 230.

J . SIMON C11 To appear.

, pp.

220-

Nonlinear Problems: Present and Future

17

A.R. Bishop, D.K. Campbell, 8. Nicolaenko Ids.) 0 North-Holland Publishing Company, 1982

COMPACTNtSS AND TOPOLOGICAL METHODS FOR SOME NONLINEAR

VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS P i e r r e L.

LIONS

Laboratoi r e d ' Analyse Numerique U n i v e r s i t e P. e t M. Curie 4, place Jussieu, 75230 Paris Cedex 0.5 FRANCE A general method t o solve some nonlinear v a r i a t i o n a l problems o f Mathematical Physics i s i l l u s t r a t e d on two examples : the so-called Hartree and Choquard equations. This method i s based upon t h e use o f , f i r s t , c r i t i c a l p o i n t theory and, second, symmetries i n order t o gain compactness. INTRODUCTION

We want t o present here some general methods t o solve some nonlinear v a r i a t i o n a l problems o f Mathematical Physics. I n general terms, we emphasize the use o f c r i t i c a l p o i n t theory i n order t o prove existence and m u l t i p l i c i t y r e s u l t s . But, since c r i t i c a l p o i n t theory needs some form o f compactness ( P a l a i s - h a l e c o n d i t i o n f o r example) and since i n many problems o f Mathematical Physics the N problem i s s e t i n an unbounded domain ( l i k e IR f o r example) compactness may be lacking : t o avoid t h i s d i f f i c u l t y we w i l l show below t h a t using t h e symmetries o f the problem, we may o b t a i n some form o f compactness. These general (and vague) coments w i l l be i l l u s t r a t e d here on two problems : i n section I below, we consider the general Hartree equation (and r e l a t e d questions)

u of :

t h a t i s we look f o r s o l u t i o n s

- Au -

-,$,.

u

t Xu

+

(I

2

dy)u = 0

u (x-y) IR3

u

E

H1( R3) ( l ) , u f 0

( o f course we want n o n - t r i v i a l s o l u t i o n s : the equation). Here X

E

IR, z

>

u

f 0, since u i 0 i s a s o l u t i o n o f

0 ( z i s the so-called t o t a l charge). As i t w i l l be

indicated below i n s e c t i o n I,we c a n t r e a t a s w e l l general Coulomb p o t e n t i a l s 3 zi w i t h xi E R , zi > 0 instead o f ), systems, Hartree-Fock equa-

'f

Ix-xilr

-&

t i o n s and r e l a t e d questions, I n section I below, we g i v e general r e s u l t s ( t h a t we

18

P. L. LIONS

b e l i e v e a r e o p t i m a l ) concerning e x i s t e n c e of m u l t i p l e s o l u t i o n s : those r e s u l t s a r e obtained by a simple use o f c r i t i c a l p o i n t theory. Section I 1 i s devoted t o the study o f t h e s o - c a l l e d Choquard e q u a t i o n , t h a t i s we look f o r solutions u o f

where

X E

IR. I n s e c t i o n 11, we g i v e optimal e x i s t e n c e and m u l t i p l i c i t y r e s u l t s

which a r e o b t a i n e d u s i n g , i n p a r t i c u l a r , c r i t i c a l p o i n t theory. A t t h i s p o i n t , l e t us make a remark on ( 2 ) : i f u i s a s o l u t i o n of ( 2 ) c l e a r l y un(x) = u ( x t nE)

1, 151 = l ) l i s a l s o a s o l u t i o n o f ( 2 ) , b u t un converges weakly ( f o r 2 o r i n H ) t o 0 and t h i s convergence cannot be s t r o n g ( s i n c e f o r 2 2 e x a m p l e t h e L n o r i n o f u n i s e q u a l t o t h e L normofI;). I n o t h e r w o r d s we have a se(with n

2

example i n L

quence of s o l u t i o n s ( w i t h t h e same norms i n every,say,

Sobolev space)

that i s not

compact ( i n any Sobolev space) : t h i s prevents t h e d i r e c t a p p l i c a t i o n o f c r i t i c a l p o i n t t h e o r y ( l a c k o f compactness). As i t w i l l be seen i n s e c t i o n 11, i n o r d e r t o be a b l e t o use c r i t i c a l p o i n t t h e o r y we w i l l use t h e symmetries o f problem ( 2 ) - t h a t i s o f course t h e s p h e r i c a l symmetry

-

t o o b t a i n some form of compactness.

F i n a l l y i n s e c t i o n 111, we g i v e b r i e f l y a l i s t o f o t h e r problems t h a t can be t r e a t e d by s i m i l a r methods ( i n c l u d i n g N o n l i n e a r S c a l a r F i e l d equations, R o t a t i n g Stars, Thomas-Fermi Problems, Vortex Rings, ...). We a l s o p r e s e n t some nonexistence r e s u l t s showing t h a t t h e symmetries of ( i n p a r t i c u l a r ) problem ( 2 ) a r e n o t o n l y e s s e n t i a l f o r t h e proof of e x i s t e n c e r e s u l t s b u t d i r e c t l y f o r e x i s t e n c e purposes. L e t us f i n a l l y mention t h a t t h e r e s u l t s presented i n s e c t i o n I a r e taken from P. L. L i o n s [26],

w h i l e those o f s e c t i o n I11 a r e from P. L. L i o n s C271. F i n a l l y

t h e s o l u t i o n o f a l l the problems l i s t e d i n s e c t i o n I 1 1 can be found i n H. B e r e s t y c k i and P. L. L i o n s [8],[91,[10], [20],[21]

P.L.

L i o n s [281, M.J.

Esteban and P. L. L i o n s

; and a systematic account o f a l l these problems i s g i v e n i n P. L. L i o n s

C291.

I. THE HARTREE EQUATION AND RELATED QUESTIONS L e t us r e c a l l t h e problem : f i n d s o l u t i o n s u o f

SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS

19

t h i s problem i s some k i n d o f model problem f o r general Hartree o r Hartree-Fock equations (see, f o r example, E. H. Lieb and B. Simon C251 f o r the relevance of such problems t o Physics) : these equations were introduced by D. Hartree C231 (see also J . C. S l a t e r C361) as an approximation method t o describe the Coulomb Hamiltonian o f electrons i n t e r a c t i n g w i t h s t a t i c

nucleii.

REMARK 1.1 : We could t r e a t as w e l l general Hartree o r Hartree-Fock systems (see

P. L. Lions [26],[29] f o r more d e t a i l s ) : i n p a r t i c u l a r we could replace the

Coulomb po t e n t i a 1

- -,$,-

by very general p o t e n t i a l s V(x) i n c l u d i n g f o r example Z

x.I (Xi E IR3 , zi E R , zi > 0 ) . i+ We w i l l i n d i c a t e below how our main r e s u l t s modify i n t h i s case (see P. L. Lions the general CbuTamb p o t e n t i a l : V(x) =

-

Z

C26IyC291 f o r more d e t a i l s ) . F i n a l l y we could consider t h e general Thomas-Fermil Von Weizacker equations (see R. Benguria, H. Brezis and E.H. L i e b [7 I f o r the i n t r o d u c t i o n o f these equations) :

{u

E

*i

-;I

Au

H1( R 3 ) ,

2 * "u1

u t B(u) t Au t ( u

= 0

in I R ~

u $ 0 ;

where x E R , zi > 0, xi E R 3 and B i s some increasing f u n c t i o n s a t i s f y i n g B ( 0 ) = B ' ( 0 ) = 0. The r e s u l t s are t o t a l l y s i m i l a r t o those concerning (1). ( 2 ) . REMARK I . 2 : An important r o l e w i l l be played by theeigenvalues o f the l i n e a -

r i z e d problem (around 0) namely by the r e a l (s) A such t h a t there e x i s t s u f 0 solution o f :

- Au -

z

,-$ t A u

= 0,

u

E

H1( R3).

(4)

I t i s well-known (see f o r example E.C.Titchmarsh C401) t h a t t h i s occurs i f a n d o n l y

2 i f A = x

k= ?

(k

2

1) and f o r such A = A k there e x i s t s o n l y one

m u l t i p l i c a t i v e constant) s o l u t i o n of (4). We may now s t a t e our main r e s u l t :

Vk

(up t o a

P.L. LIONS

20

THEOREM 1.1 :

i) ii)

If

X < 0

If, xktl

d

5

x

S(v) =

for a l l

2

l ) , there e x i s t a t l e a s t

< S(U2)

5

... 5 S(Uk)

IR3 - VI 1 1DvI2

1 3 H (R )

v

i i i ) j ~ -x = 0

k

.

distinct pairs

<

v2

< 0

(6)

$ v2 dx

t

t

1

7 \lR3xR3

v 2 ( x ) v 2 ( y ) I x - y l - l d x dy

.

, there

s_plutions o f (1) Such S(ul)

e x i s t s no s o l u t i o n o f (1)

o f s o l u t i o n s o f (1) such t h a t :

J kjsk

where

, there

?,>Al

< hk ( k

(+u.)

S(Ul)

f

of

e x i s t s a secquenceof d i s t i n c t u a i r s ( i Uk)kr

that (5)

S(u2) <

I n a d d i t i o n we have : Duk

and

... <

L2

W :

(6’)

k+t

S(uk)-+ 0 0

k

,

L6

uk

(6‘1

m

.

0

k

1 3 REMARK 1.3 : L e t us f i r s t e x p l a i n why S i s w e l l defined on H ( R ) : 1 3 indeed i f v E H ( R ) , v € L P ( R 3 ) ( 2 5 ~ 5 6 ) (Sobolevembeddings) therefore by c l a s s i c a l convolution i n e q u a l i t i e s : v 2

I t i s then very easy t o prove t h a t

S

E

C

1 1 (H )

1

E

LP( R3 )

.

REMARK 1.4 : I f we consider a general Coulomb p o t e n t i a l : V(x) =

-

1 zilx-xil-l

(xi

E

s t i l l holds w i t h

Xk

being t h e

o f the Schrodinger operator

>

0) ; then Theorem 1.1

k - t h eigenvalue

= ,lktr >lktrtl , one has i n f a c t f o r

REMARK 1.5 --

, zi

R3

i

- A t V

X

< Xk

Xk-l

> X k = Xktl

=

...

( k t r ) d i s t i n c t pairs o f solutions)

.

: Hartree equations have been studied

M. Reeken C341

(If

by many authors (see f o r example

, N. Bazley and R.Seyde1 C51 , N. Bazley and B. Zwahlen C61 ,

21

SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS

K. Gustafson and D. Sather [21],

C. A. S t u a r t [38],[39],

J. Wolkowisky [41]

b u t a l l these works e i t h e r g i v e l o c a l r e s u l t s (A near Al) o r use h e a v i l y t h e

-

spherical symmetry and reduce the problem t o some O.D.E. question in p a r t i c u l a r do n o t y i e l d any information i n t h e case o f a general Coulomb potential.

The most general known r e s u l t s

-

symmetry

-

i n absence o f spherical

were those o f E. H . Lieb and B..Simon [25] b u t were o n l y concerned

w i t h the existence o f ground s t a t e s o l u t i o n s ( e s s e n t i a l l y those s a t i s f y i n g (5)). We w i l l only sketch the proof o f Theorem 1-1, t h e d e t a i l e d p r o o f may be found i n [26].

REMARK

1-6:

that i f 0

5

Using a method due t o N. Bazley and R. Seyde [5], A < A1,

min

one can prove

t h e s o l u t i o n o f the minimization problem: S(v)

vE H i s unique (up t o a change o f sign) and i s thus u.l I n a d d i t i o n i t i s obvious t o deduce t h a t u,l

0 < A 5 A1 and t h a t as A

+

{ u E L6(J?),

and weakly i n

Du E L2($))

0, ul(A)

as a f u n c t i o n o f A, i s C 1 f o r

converges t o ul(0)

i n norm i n t h e space

H1(d).

We now i l l u s t r a t e Theorem 1-1 by a few " b i f u r c a t i o n diagrams": emphasize t h a t these diagrams are more o r l e s s formal. introduce some notations: t h a t Du E L 2 ( d ) , u E L6(& Du(x).Dv(x)dx.

0

....&-- A-J

let

X

l e t us

F i r s t we need t o

be t h e H i l b e r t space o f f u n c t i o n s

u

equipped w i t h t h e s c a l a r product ( ( u , ~ ) )

=

such

The b i f u r c a t i o n diagram i n X looks l i k e Fig. 1. below:

%gure

1.

But now; i f we look a t t h e b i f u r c a t i o n diagram i n H1 ( o r L2) we f i n d the f o l l o w i n g Fig. 2:

22

P.L. LIONS

Figure 2 The d i f f e r e n c e between these two b i f u r c a t i o n diagrams i s due t o the f a c t t h a t f o r A = 0, w h i l e uk

3 0 ,

Iu I

i s bounded away from 0.

L2 2z

We have proved t h a t :

'I2

(V k

2

1) and we conjecture t h a t f o r

A = 0 we have:

lu I

>

z1'2

(V k

1) and

L2 F i n a l l y l e t us mention t h a t i n [25], proved t h a t l u

lim lu I = z 1/2 k+@ L2 [7] (by two d i f f e r e n t methods) i t i s

> z 1/2.

I 12

L

REMARK 1-7: C l e a r l y t h e r e i s b i f u r c a t i o n from each eigenvalue Ak (which i s t o be expected i n view o f the general r e s u l t s o f M. G. Grandall and P. H. Rabino-

[la]).

A n i c e phenomenon i s what happens f o r t h e branches near A = 0: i n branches stop when they h i t the continuous spectrum o f - A ( ( A 5 0)). More p r e c i s e l y i t can be proved (P. L. Lions [29]) t h a t 1 u1 (=ul(A)) i s a C curve i n H1 and t h a t f o r a l l 0 < A < A1 the l i n e a r l i z e d oberator AA around ul(A) i.e. :

witz

some

sense t h e

H 1 ~ V + - A V - & V + A V + ( U 2~

1 * T;;T'v

+ 2

((up) *

i s coercive and thus i s an isomorphism from H1 onto H-'.

But f o r A = 0,

SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS

23

A. i s o n l y i n j e c t i v e and i s n o t o n t o and t h i s e x p l a i n s why t h e branch cannot be "extended i n t h e s e t { h < 0)".

We now t u r n t o t h e p r o o f o f Theorem 1-1: we w i l l o n l y g i v e some s k e t c h o f t h e p r o o f , f o r a d e t a i l e d p r o o f t h e r e a d e r i s r e f e r r e d t o P.L. L i o n s C261. Sketch o f t h e p r o o f o f Theorem 1-1 : We w i l l c o n s i d e r o n l y t h e case when 0 < h < h l ( t h e non e x i s t e n c e p a r t i s q u i t e easy and t h e case when 1 = 0 i s q u i t e t e c h n i c a l ) . As i t was s a i d b e f o r e , we w i l l a p p l y c r i t i c a l p o i n t t heory; and we w i l l j u s t i n d i c a t e what a r e t h e p r o p e r t i e s s a t i s f i e d b y S here which

enable us t o appl y

c r i t i c a l p o i n t theory.

S E C 1( H 1) s a t i s f i e s :

Therefore, we c l a i m t h a t

1) S(0)

, S i s even (and t h u s S '

0

=

we have on

3 ) S(u) 4) S

.+

t

my

BE : S(u)

<

IIu /IH1

as

0 +

for

t

E

-

E

(0) = 0).

( 0 , 1~ ( ~ f o r some

E

~

0)>

,

s a t i s f i e s the Palais- h a l e condition : If (

u

~

E

)H'

~i s ~such~ t h a t : S(un) isbounded,

S'(un)

-1' 0 H

t h en t h e r e e x i s t s a subsequence

of (u,)

which converges i n

H1

.

llull

is

small t h en S(u) behaves e s s e n t i a l l y as i t s q u a d r a t i c p a r t which i s j u s t "

the

Property

1) i s o b v i o u s ; t o understand 2 ) l e t us j u s t remark t h a t i f

q u a d r a t i c f o rm a s s o c i a t e d w i t h ( - A

+A)

and t h u s 2 ) i s c l e a r ( r e c a l l t h a t

m zT) .

i s the j - t h eigenfunction o f - A To e x p l a i n 3), l e t us j u s t i n d i c a v j t h e n t h e main d i f f i c u t y t o check 3) l i e s w i t h t h e t e t h a t i f 1 > hj+l j

w~

subspace Q IR v 21 o f H1 which i s f i n i t e d i mensional. B u t on t h i s subspace i=l a l l norms a r e e q u i v a l e n t and t h u s t h e t e r m R3 d x ' d y w i l l be

jR3x

predominant a t i n f i n i t y . F i n a l l y 4), which i s t h e fundamental s t r a i g h t f o r w a r d t o check u s i n g 3 ) .

c o n d i t i o n i n C r i t i c a l P o i n t Theory, i s

To conclude we j u s t need t o a p p l y a r e s u l t due t o L j u s t e r n i k and

24

P.L. LIONS

Schnirelman [ 3 l ] ( f o r the precise statement, see P.H.

Rabinowitz 1331, o r Krasnoleskii [24] ) : conditions 1 ) - 4 ) imply t h e existence o f a t l e a s t k

t i n c t p a i r s o f . c r i t i c a l p o i n t s o f S,that i s s o l u t i o n s of (1) f o r X

c

dis-

Xk ,

REMARK 1-7 : I f one c o n s i d e r s t h e time dependent H a r t r e e e q u a t i o n t h a t i s :

where 0 are complex valued functions. Then solutions o f (1) provide s o l i t a r y waves

f o r (7) : 0(x,t)

= e- i h t uk(x)

.

I n T. Cazenave and P.L. Lions C171,it i s proved t h a t the o r b i t (,-iht 0 5 X < X1) i s stable. Ul(X) ) t 2 0

(for

11. THE CHOQUARD EQUATION AND RELATED QUESTIONS We now look f o r s o l u t i o n s

-

Au t Xu

-

u of

1 (u2 *m)u

1 3 u c H (R )

,

= 0

in

R3

u 8 0 .

We w i l l also consider t h e "normalized" problem : f i n d (A,u) s o l u t i o n o f

r where M

-

Au t Xu

-

(u 2 * m1 ) u = 0

i s given > 0 ( f o r example M

Equations ( 2 )

-

R3

in

1)

.

( 2 ' ) were introduced by Choquard and t h e existence o f a

p o s i t i v e s o l u t i o n d f ( 2 ' ) was proved by

E.H.Lieb C241

. Problem (2')

a l s o i n the so-called mean f i e l d theory (see M. Donsker and S.R .S.

Some other references where such problems are considered are given i n P.L. Lions C271. Our main r e s u l t s concerning (2) o r ( 2 ' ) a r e :

occurs

Varadhan [19]).

SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS

26

THEOREM 11.1 :

i) ii)

If

x

2

x

> 0

there i s no s o l u t i o n o f (1)

0

of

there e x i s t s a sequence o f d i s t i n c t p a i r s ( ' u ~ ) ~ ~

solutions o f (2) such t h a t = ~ ~ ( 1 x >1 )0 ; u1

ul(x)

0 < S(ul)

S(v)

5

i s decreasing i n

for a l l

k(x) = u k ( l x l )

v

(8)

1x1

solution o f ( 2 )

(9)

( V k 2 1)

(10)

THEOREM 11.2 :

There e x i s t s a sequence

, (10)

such t h a t ( 8 )

J(ul)

(xk,

+uk)

( f o r any M > 0)

o f d i s t i n c t solutions o f ( 2 ' )

hold and : = min{J(v)/vE H

1

, IvI

(9'1

= M} < 0

L

<... < J ( u k )

J(ul) < J ( u ~ )

where

J(v) =

I,

and i n a d d i t i o n :

for

2 < p s 6 ,

ilDv12dx

xk

>

-1 4

0 (Yk

(11')

v2(y)lx-yl-' l),xk

2

2 L

fi

Duk-pO

k+O ++a dx dy

k z m O

,

(VV

E

H1)

uk-p 0

;

2

Lp

.

REMARK 11-1 : L e t us remark t h a t i n Theorem 11-1,s i s n o t bounded from below

t'qmm if v #

(and surely n o t from above) since

S(tv)

REMARK 11-2 : The existence of u1

i n Theorem 11.2.

ved by of :

E.H.

Lieb [24]

IvI 2 = My v L

E

H

E

H

.

( w i t h M = 1) has been pro-

. I n a d d i t i o n i t i s proved i n [24]

J ( v ) = minCJ(w)/w

0

t h a t any s o l u t i o n v

1 IWIL2 =

1

i s necessarily o f the form : v ( x ) =

some xo R 3 and where E = +1. can be found i n P.L. Lions C271.

, for

EU~(X-X~)

The d e t a i l e d proofs o f Theorems 11.1, 11.2

26

P . L LIONS

REMARK 11-3 : let

I f one c o n s i d e r s more g e n e r a l e q u a t i o n s t h a n ( 2 ) of t h e f o r m :

N> 2 -Au

where

V

t XU

-

(U

2

*V(lXl))u

satisfying :

f o r example i s some p o t e n t i a

v

Efl,(RN)

V

E

MP(RN)

t t

RN

in

V(x) = v ( l x 1 ) z 0

l s q < t my if N M q ( R N ) ( 3 ) f o r some T < p < q < t

t h e n Theorem 11-1 s t i l l h o l d s ( s e e P.L.

N

f o r some

Lq(RN)

5

m y

if

3 Nr4

(14)

L i o n s C271, i n C271 a r e a l s o g i v e n

c o r r e s p o n d i n g e x t e n s i o n s of ( 2 ' ) ) . I n p a r t i c u l a r , l e t us remark t h a t assumptions ( 1 3 ) , (14)

V =6,

in

R'

( t h e D i r a c mass a t 0) s a t i s f i e s

and we o b t a i n e x i s t e n c e o f m u l t i p l e s o l u t i o n s o f

-AutXu=u3

in

R3

,

1 3 u c H ( R ) ,

u

$

0 ;

t h i s had a l r e a d y been proved by s e v e r a l a u t h o r s (M.S. Berger C161, Z. N e h a r i C321, G.H.

Ryder C351

, W.

S t r a u s s C271). A l s o remark t h a t t h e above e q u a t i o n a r i s e s

when l o o k i n g f o r S o l i t a r y waves i n c u b i c n o n l i n e a r S c h r 8 d i n g e r e q u a t i o n s a r i s i n g i n o p t i c s ( s t u d y o f l a s e r beams). Before going i n t o

t h e p r o o f s o f Theorems 11-1,

11-2, l e t us g i v e t h e

b i f u r c a t i o n diagram c o r r e s p o n d i n g t o Theorem 11-1 ( w h i c h can be made r i g o r o u s ) :

a,

>

A

F i g u r e 3. I n p a r t i c u l a r we see t h a t t h e r e i s b i f u r c a t i o n f r o m (0,O) o f a c o u n t a b l e number o f branches

Gk

= {(X,uk(A))1 : t h i s phenomena occurs i n some n o n l i n e a r problems

when t h e r e i s some c o n t i n u o u s spectrum ( s e e a l s o H. B e r e s t y c k i and P . L . L i o n s Clll f o r some o t h e r example).

27

SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS

We now t u r n t o t h e p r o o f s ofTheorems 11.2, 11.3 : a g a i n we w i l l o n l y s k e t c h it.

Sketch o f t h e p r o o f o f Theorem 11.1 : We w i l l o n l y c o n s i d e r t h e case when X

>

0

( t h e non e x i s t e n c e p a r t i s q u i t e easy). Again we w i l l j u s t i n d i c a t e ( w i t h o u t p r o o f ) what a r e t h e p r o p e r t i e s s a t i s f i e d by S which e n a b l e u s t o use c r i t i c a l p o i n t t h e o ry.

We c l a i m t h a t S

E

1 1

C (H ) s a t i s f i e s :

1 ) S(0) = 0, S i s even ( a n d t h u s S ' ( 0 ) = 0 ) . 2)

]E

, 6 > 0 such

ifI I v l k l s E, v # 0

t h a t S(v) > 0

3 ) For any f i n i t e dimensional subspace X o f H1, we have : S(u)

+

-

m,

if u

E

X, I J u11 H

-+

+

m.

4) S s a t i s f i e s t h e Palais-Smale c o n d i t i o n i n H:

= (v

E

H1,

v ( x ) = v ( ( x 1)

I.

P r o p e r t i e s 1) and 2) a r e c l e a r ; and p r o p e r t y 3) i s e a s i l y deduced f r o m t h e f a c t t h a t on X a l l norms a r e e q u i v a l e n t . Now 4) i s s m e k i n d o f compactness p r o p e r t y t h a t we w i l l e x p l a i n below. I f one admits t h e s e p r o p e r t i e s , t h e n a p p l y i n g a c r i t i c a l p o i n t theorem i n t h e 1 space Hr due t o A. Ambrosetti and P.H. R a b i n o w i t z C11, m e o b t a i n s ( e s s e n t i a l l y )

t h e e x i s t e n c e o f s o l u t i o n s as i n Theorem 11.1.

To conclude t h i s s k e t c h o f p r o o f , we would l i k e t o emphasize t h e f a c t t h a t S does n o t s a t i s f y t h e Palais-Smale c o n d i t i o n i n H 1 ( i n p a r t i c u l a r because o f t h e d i f f i c u l t y due t o t r a n s l a t i o n s as mentioned i n t h e I n t r o d u c t i o n ) : b u t r e s t r i c t i n g o u r s e l v e s t o t h e f u n c t i o n s having a l l t h e symmetries o f t h e problem ( h e r e t h e s p e r i c a l s y m e t r y ) , we o b t a i n some compactness. A key lemma i n t h e p r o o f o f p r o p e r t y 4) i s t h e f o l l o w i n g L e m a which i s a consequence o f r e s u l t s due t o C371 (see a l s o [81, C291) :

1 LEMMA 11.1 : The r e s t r i c t i o n t o Hr o f t h e Sobolev embedding f r o m

i s compact f o r 2

< p
'fN

2

3 and f o r 2 < p <

+ if N m

w.

Strauss

H (IR ) i n t o Lp(R3) 1

3

= 2.

REMARK 11.4 : T h i s lemma c l e a r l y shows the r o l e p l a y e d by t h e symmetries i n o r d e r

t o g a i n some compacteness. E x t e n s i o n s o f t h i s Lemma t o more g e n e r a l Sobolev spaces

28

P.L. LIONS

than H

1 i s given i n P.L. Lions C291, 1301.

Sketch o f the p r o o f o f Theorem 11.2 : We j u s t need t o apply a general c r i t i c a l p o i n t theorem o f H. Berestycki and P.L. Lions C121 (see a l s o C91) : Theorem 11.2 i s then a consequence o f t h e f o l l o w i n g p r o p e r t i e s o f J : l e t Jb = {u E Hr,1 I u I = M I , we have : J E C 1(H1) and L

1) J ( 0 ) = 0 , J i s even, 2)

J i s bounded from below on

3) JI4

A.

s a t i s f i e s Palais-Smale condition.

( A c t u a l l y JI&

s a t i s f i e s (P.S.-)

condition, and thus something e l s e has t o be

checked, b u t we w i l l n o t e n t e r such t e c h n i c a l i t i e s ) . Property 1) i s obvious, and 3) comes e s s e n t i a l l y from Lemma Ir.1. L e t us prove 2) : indeed by w e l l - known i n e q u a l i t i e s :

Therefore, o n A

, we have

:

and t h i s proves t h a t J i s bounded from below on

4.

111. RELATED PROBLEMS 111.1

-

Symmetries.

I n t h i s section, we f i r s t mention t h a t i f , i n view o f Lemma 11.1, the spherical symmetry gives some compactness, some c y l i n d r i c a l symmetry a l s o gives compactness (see P.L. Lions C281, and a l s o C291, C301). I n p a r t i c u l a r , we can t r e a t equation (12) when V has some c y l i n d r i c a l symmetry (see C291). Now we would l i k e t o mention a few non existence r e s u l t s due t o M.J.

Esteban

and P.L. Lions C211 (see a l s o C201) t h a t seem t o i n d i c a t e t h a t some form o f symmetry i s needed i n order t o obtain non t r i v i a l s o l u t i o n s o f ( 2 ) o r (12).

SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS

29

For example consider the equations :

where

x

>

0, p

>

1, aij

E

3

Ci(lR j and

Remark -hat (15) has a l l symmetries i n (x, w i t h respect t o xl.

,...,xN)

: t h e o n l y broken symm-try i i

Something s i m i l a r happens when we assume (18) f o r t h e m a t r i x

a. . ( x ) . 1J

Then we have the f o l l o w i n g r e s u l t : THEOREM 111.1 (M.J. Esteban and P.L. Lions C211) :

i)The o n l y s o l u t i o n o f (15)

&

: u E 0.

i i ) The only s o l u t i o n o f (16) under assumptions (17)-(18)

&

: u :0.

In some sense, the symmetries i n equations l i k e (12) o r (2) keep t h e s o l u t i o n s from going t o i n f i n i t y and thus vanishing. I n the above resu1t"the s o l u t i o n s " s l i p

t o i n f i n i t y and disappear : and we end up w i t h the s o l u t i o n u :0.

111.2

-

Other probems :

We conclude by giving a l i s t o f other problems which can be t r e a t e d by s i m i l a r methods and where the symmetries are e s s e n t i a l t o g e t compactness :

1 ) Nonlinear scajar f i e l d equations : We look f o r solutions u o f

P.L. LIONS

30

-

N i n lF! , u ( x ) + 0, u 2 0

AU = g(u

1x1

(19)

-+03

where g i s some n o n l i n e a r i t y s a t i s f y i n g : g(0) = 0. O f course s o l u t i o n s o f (19) are c r i t i c a l p o i n t s o f the f o l l o w i n g a c t i o n : S(v)

=IRN 21DvI2 1

G(v)dx

and the equation (19) i s i n v a r i a n t by r o t a t i o n s and t r a n s l a t i o n s . I n H. Berestycki and P.L. Lions [8], 191 (see a l s o C131, 1141, 1151) t h i s problem i s solved under optimal conditions on g by developping new c r i t i c a l p o i n t theorems and using t h e spherical symmetry i n order t o obtain compactness. 2) Rotating s t a s :

all

L1(R3),

where M i s given

0,

>

1 3

I

3p(x)dx !? ), p 0, -p(x)dx =LM,(lthe function31 2 :

A model problem i s t o f i n d p

E

2

=

M minimizing,

over

0, j i s a nonnegative convex f u n c t i o n and V i s a given nonnegative

bounded f u n c t i o n depending only on r (where x = (r,e,z)

i n c y l i n d r i c a l coordinates).

This problem i s a c l a s s i c a l model o f e q u i l i b r i u m c o n f i g u r a t i o n s o f axisymmetric rotating fluids. I n P.L. Lions r2Pi (see a l s o 1301) t h i s problem i s solved under optimal cond i t i o n s on j , V (extending by a d i f f e r e n t method previous p a r t i a l r e s u l t s due t o

J.F.G. Auchmuty and R. Beals C31, see a l s o r21). The method r e l i e s on some new compactness

r e s u l t s using the c y l i n d r i c a l symmetry o f the problem.

3) General Thomas-Fermi Problems :

I

1 3 p(x)dx = M minimizing, A model problem i s t o f i n d p E L (lR ) , p 2 0, 1 3 : F(x)dx = M , t h e f u & t i o n a l over a l l E L (l!? ), t 0,

;

where M i s given

>

0 and f i s a given s p h e r i c a l l y symmetric f u n c t i o n . O f course

t h i s problem has a spherical symmetry. I n P.L. Lions C281, C291, t h i s problem i s solved under optimal conditions

SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS

31

u s i n g t h e s p h e r i c a l symmetry t o o b t a i n compactness. 4) Vortex r i n g s : A model problem i s t o f i n d u s a t i s f y i n g :

over a l l v s a t i s f y i n g J(u) = where

I,,.

F((u

{(r,z)

'~i

E

IT -

>

i I D u l 2dx =

and m i n i m i z i n g ,

1Du12 = q. t h e f u n c t i o n a l :

-

wr2

R2, r

IT

k)+)dx

01, w

(22)

> 0, k 2 0, q > 0 a r e g i v e n ; F i s a f u n c t i o n

s a t i s f y i n g : F ( 0 ) = 0. Using t h e symmetry ( w i t h r e s p e c t t o z ) o f t h e problem, i n H. B e r e s t y c k i and P.L. Lions 1101, t h i s problem i s s o l v e d under o p t i m a l c o n d i t i o n s .

5) General n o n l o c a l problems : Here, one wants t o f i n d t h e s o l u t i o n s o f : K

*

u = f ( u ) i n l RN

where K ( x ) = K ( l x l ) , f i s a g i v e n n o n l i n e a r i t y s a t i s f y i n g : f ( 0 ) = 0. T h i s problem i s s o l v e d i n P.L. L i o n s C291. Footnotes : (1) H

1

m3

= {U

E

L 2 P 3 ) , Du

E

L2(R3)}

( 2 ) The a u t h o r would l i k e t o thank E.H.

Lieb f o r having pointed o u t t o h i s a t t e n t i o n

equ at io n ( 3 ) . ( 3 ) MP(!RN) denotes t h e u s u a l M a r c i n k i e w i c z space.

References C11

Ambrosetti, A. and Rabinowitz, P.H'.

, Dual v a r i a t i o n a l methods i n c r i t i c a l

p o i n t t h e o r y and a p p l i c a t i o n s , J. Funct. Anal. 14 (1973) 349-381. C21

Auchmuty, J.F.G.,

E x i s t e n c e o f axisymmetric e q u i l i b r i u m f i q u r e s , Arch. Rat.

Mech. Anal. 65 (1977) 249-261. C31

Auchmuty, J.F.G.

and Beals, R.,

V a r i a t i o n a l s o l u t i o n s o f some n o n l i n e a r

32

P.L. LIONS

free boundary problems, Arch. Rat. Mech. Anal. 43 (1971) 255-271. C43

Bader, P., Variational method for the Hartree equation of the Helium atom, Proc. Roy. SOC. Edim. 82 A (1978) 27-39.

C51

Bazley, N. and Seydel, R . , Existence and bounds for critical energies of the Hartree operator, Chem. Phys. Letters 24 (1974) 128-132.

C61

Bazley, N and Zwahlen, B., A branch of positive solutions of nonlinear eigenvalue problems, Manuscripta Math. 2 (1970) 365-374.

C71

Benguria R., Brezis, H. and Lieb E.H., The Thomas-Fermi-von Weiz'dcker theory o f atoms and molecules, to appear in Comm. Maths. Phys.

C81

Berestycki, H. and Lions, P.L., Existence of solutions for nonlinear scalar field equations. I. The ground state, to appear in Arch. Rat. Mech. Anal.

C93

Berestycki, H. and Lions, P.L., Existence of solutions for nonlinear scalar field equations. 11. Existence of infinitely many bound states, to appear in Arch. Rat. Mech. Anal.

I101

Berestycki, H. and Lions, P.L., to appear.

C111

Berestycki, H. and Lions, P.L., Existence of stationary states of nonlinear scalar field equations. In Bifurcation phenomena in Mathematical physics (Reidel, New-York, 1979).

C123

Berestycki, H. and Lions, P.L., Existence d'ondes solitaires dans des problemes nonlineaires du type Klein-Gordon, Cmptes-Rendus Paris 288 (1979) 395-398.

C131

Berestycki, H. and Lions, P.L., Existence of a ground state in nonlinear equations o f the type Klein-Gordon. In Variational Inequalities (J. Wiley, New-York, 1979).

C141

Berestycki, H. and Lions, P.L.., Une methode locale pour l'existence de solutions positives de problemes semi-lineaires elliptiques dans RN, J. Anal. Math. 38 (1980) 144-187.

C151

Berestycki, H., Lions, P.L., and Peletier, L.A., An O.D.E. approach to the existence of positive solutions for semi-linear problems in lRN, Ind. Univ. Math. J. 30 (1981) 141-157.

33

SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS

C 161

Berger, M.S.,

On t h e existence and s t r u c t u r e o f s t a t i o n a r y states f o r a

nonlinear Klein-Gordon equation, J. Funct. Anal. 9 (1972) 249-261. C 171

c 181

Cazenave, T.,

and Lions, P.L.,

Crandall, M.G.,

t o appear.

and Rabinowitz, P.H.,

B i f u r c a t i o n from simple eigenvalues,

J . Funct. Anal. 8 (1971) 321-340. C 191

Donsker, M.,

and Varadhan, S.R.S.

c 201

Esteban, M.J.,

and Lions, P.L.,

, to

appear.

Non existence de s o l u t i o n s non t r i v i a l e s

pour des problemes semilineaires dans des ouverts non born&.

Comptes-

Rendus Paris 290 (1980) 1083-1085. c211

Esteban , M. J

.

and Lions, P.L.,

Existence and non existence r e s u l t s f o r

semi1 inear e l i p t i c problems i n unbounded danains, t o appear i n Proc-Boy. SOC. Edim.

c221

Gustafson, K.

and Sather, D.,

Branching analysis o f the Hartree equations,

Rend. d i Mat. 4 (1971) 723-734. C231

Hartree, D . , The wave mechanics o f an atom w i t h a non-coulomb c e n t r a l f i e l d . Part 1. Theory,and methods, Proc. Camb. P h i l . SOC. 24 (1928) 89-132.

C241

K r a s n o s e l ' s k i i , M.A.,

Topological methods i n t h e theory o f nonlinear i n -

t e g r a l equations (Mac M i l l a n , New-York, 1964). C251

Lieb, E.H.,

and Simon, B., The Hartree-Fock theory f o r Coulomb systems,

Comn. Math. Phys. 53 (1977) 185-194. C261

Lions, P.L.,

Sane remarks on Hartree equation, t o appear i n Nonlinear Anal.

T.M.A.

C271

Lions, P.L., The Choquard eouation and r e l a t e d questions, Nonlinear Anal. T.M.A. 4 (1980) 1063-1073.

C281

Lions, P.L., 236-275.

Minimization problems i n L'm'),

C291

Lions, P.L.,

Topological and canpactness methods f o r some nonlinear v a r i a -

J . Funct. Anal. 41 (1981)

t i o n a l problems o f Mathematical Physics, t o appear.

34

C30]

P.L. LIONS

1 3 Minimization problems i n L (IR ) and a p p l i c a t i o n s t o some f r e e

Lions, P.L.,

boundary problems. I n Free boundary problems, Pavia, 1979 (Roma, 1980).

1311

L j u s t e r n i k , L.A.,

and Schnirelman, L.G.,

Topological methods i n t h e c a l -

culus o f v a r i a t i o n s (Hermann, Paris, 1934).

C321

Nehari, Z.,

On a non l i n e a r d i f f e r e n t i a l equation a r i s i n g i n nuclear phy-

s i c s , Proc. Roy. I r i s h Acad. 62 (1963) 117-135.

C331

Rabinowitz, P.H.

, Variational

methods f o r nonlinear eigenvalue problems.

I n Eigenvalues o f nonlinear problems (Cremonese, Rome, 1974).

C341

Reeken, M.,

General theorem on b i f u r c a t i o n and i t s a p p l i c a t i o n t o t h e

Hartree equation o f the helium atom, J. Math. Phys. 11 (1970) 2505-2512.

1351

Rvder, G.H.,

Boundary value problems f o r a class o f nonlinear d i f f e r e n t i a l

equations, P a c i f i c J . Math. 22 (1967) 477-503.

C361

S l a t e r , J.C., A note on H a r t r e e ’ s method. Phys. Rev. 35 (1930) 210-211.

C37]

Strauss, W.A.,

Existence o f s o l i t a r y waves i n higher dimensions, Cmm.

Math. Phys. 55 (1977) 149-162.

[38]

Stuart, C.A.,

Existence theory f o r the Hartree equation, Arch. Rat. Mech.

Anal. 51 (1973) 60-69.

C391

Stuart, C.A.,

An example i n nonlinear f u n c t i o n a l a n a l y s i s : t h e Hartree

equation, J. Math. Anal. Appl. 49 (1975) 725-733.

C403

Titchmarsh,

E.C.,

Eigenfunctions expansions, P a r t I 1 (Oxford Univ. Press,

London , 1962).

C411

Wolkowisky, J . , Existence o f s o l u t i o n s o f the Hartree equations f o r N electrons. An a p p l i c a t i o n o f t h e Schauder-Tychonoff theorem. Ind. Univ. Math. J . 22 (1972) 551-558.

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North-HollandPublishing Company, 1982

35

ON YANG-MILLS FIELDS

I.M. Singer Department o f Mathematics U n i v e r s i t y o f C a l i f o r n i a ] Berkeley Berkeley, C a l i f o r n i a 94720 U. S. A.

INTRODUCTION For t h i s conference on nonlinear problems, I was asked t o survey the developments i n global nonlinear p a r t i a l d i f f e r e n t i a l equations a r i s i n g i n quantum f i e l d theory. I w i l l , i n f a c t , t r y t o g i v e an expository account o f the most s i g n i f i c a n t recent development: t h a t o f t h e c l a s s i f i c a t i o n o f t h e s e l f dual Yang-Mills equations o f motion. I propose n o t t o be technical because o f t h e diverse backgrounds o f the p a r t i c i p a n t s . I n f a c t , as a l i s t e n e r , I o f t e n g e t l o s t e a r l y i n a t a l k because the speaker assumes I know more than I do. While he i s exposing technical d e t a i l s , I am sadly f r u s t r a t e d s t r u g g l i n g w i t h some basic questions whose answers I ' m supposed t o know. So I p l a n t o be simple. 1owi ng questions:

For the t o p i c I ' v e chosen I ' d l i k e t o address t h e f o l -

(1) What s c i e n t i f i c problem l e d t o the nonlinear equation? (2) What methods have provided t h e s o l u t i o n ? ( 3 ) Are these methods applicable t o other f i e l d s ? Our general experience i n mathematics shows t h a t new i n s i g h t s i n one area o f t e n spread and have other applications. So i t i s q u i t e useful t o understand what the breakthroughs are i n a f i e l d using n o n l i n e a r i t y ; t h e methods occasionally have other uses.

(4)

How does n o n l i n e a r i t y show i t s e l f i n t h e special p r o p e r t i e s o f t h e solutions?

and f i n a l l y ,

(5)

Do the s o l u t i o n s r e a l l y shed l i g h t on t h e o r i g i n a l s c i e n t i f i c question?

Often a nonlinear problem, o r any mathematical problem ends up having a l i f e o f i t s own f a r removed from i t s place o f o r i g i n . And one r a r e l y asks i f i t s solut i o n s are r e l e v a n t t o the o r i g i n a l m o t i v a t i n g question. OUTLINE My t o p i c i s gauge t h e o r i e s i n quantum f i e l d theory.

Iw i l l

Review the basic setup f o r gauge f i e l d s , I n d i c a t e where t h e nonlinear problem comes from, Describe t h e problem t h a t has been solved, Point t o t h e new methods t h a t lead t o the s o l u t i o n s , E x h i b i t t h e s o l u t i o n s and some new features they r e f l e c t , L i s t some remaining problems, Comment on the usefulness o f the s o l u t i o n s i n quantum f i e l d theory. Rather, I should say, comment on t h e K u s e f u l n e s s i n quantum f i e l d theory t o date.

I.M. SINGER

36

The new features e x h i b i t e d are (a) corn a c t i f i c a t i o n ; t h e s o l u t i o n s , which appear t o have s i n g u l a r i t i e s (sometimes a-yl' l i v e on compact manifolds and e x h i b i t topological-geometrical p r o p e r t i e s . Thus a t o p o l o g i c a l discreteness o r quantization appears i n t h e form o f t h e Pontryagin charge. Another new feat u r e i s (b) complexification; t h e i n s i g h t t h a t l e d t o the s o l u t i o n s stems from the complex p r o j e c t i v e " t w i s t o r " program o r i g i n a t e d by Roger Penrose. F i n a l l y , t h e r e i s (c) e x c i t a t i o n , e s p e c i a l l y o f some o f us geometers and t o p o l o g i s t s n o t accustomed t o f i n d i n g our research a p p l i c a b l e t o h i g h energy physics. GAUGE FIELDS

L e t me f i r s t b r i e f l y review gauge f i e l d s over Euclidean 4-space R4. Euclidean gauge theory on R4 i s e a s i e r than t h e theory over Minkowski (3 + 1) space. There i s a theorem (Osterwalder-Schrader [18]) t h a t says c e r t a i n c o n s t r u c t i v e f i e l d t h e o r i e s i n the Euclidean domain w i t h t h e c o r r e c t p o s i t i v i t y assumptions can be a n a l y t i c a l l y continued back t o r e a l time. We are s t i l l f a r away from a c o n s t r u c t i v e gauge theory i n 4 dimensions. S t i l l considerable i n s i g h t i s gained studying quantum f i e l d theory i n t h e Euclidean domain. Let A be a vector p o t e n t i a l - - o r connection, as i t i s c a l l e d i n mathematics--with values i n a simple L i e algebra--say, su(N), t h e skew-adjoint complex matrices o f t r a c e 0. [When I r e f e r t o electromagnetism, t h e L i e algebra i s u ( l ) , t h e pure imaginary numbers.] So A has components A and t h e A have values i n t h e L i e algebra su(N): CI' c1

A = A dx

CIP

and

A

= AaT

IJ

P a

'

where Ta i s an orthonormal basis o f su(N). Associated w i t h A i s i t s f i e l d o r curvature F, a two form w i t h values i n su(N):

-

Note t h a t the bracket [A A 3 = A A A A makes F a nonlinear f u n c t i o n o f A. A l l the t r o u b l e and a l l p t h e u i n t e k t t st8mPfrom t h e nonlinear term. I t i s a feature o f t h e non-Abelian L i e algebra. The a c t i o n S i s t h e L2 norm o f t h e f i e l d

We are i n t e r e s t e d i n vector p o t e n t i a l s w i t h f i n i t e a c t i o n ; so l e t A be t h e s e t o f a l l vector p o t e n t i a l s w i t h f i n i t e action:

Before we can proceed we s h a l l need a few o t h e r concepts. d i f f e r e n t i a l , DA i s t h e operator

F i r s t the covariant

37

ON YANG-MILLS FIELDS

1 .

(5)

Next, a gauge transformation i s a f u n c t i o n $ ( X I w i t h values i n t h e group G--say, SU(N), the group o f N x N u n i t a r y matrices o f determinant 1. L e t G denote t h e group o f a l l gauge transformations. G acts on A: a gauge transformation $ ( X I acts on a vector p o t e n t i a l A t o g i v e t h e vector p o t e n t i a l $-A where

or, e q u i v a l e n t l y

It i s easy t o check t h a t

Since $ has values i n a compact group--here SU(N)--formula (8) i m p l i e s t h a t t h e a c t i o n i s i n v a r i a n t under G:

F i n a l l y , we need some geometry o f 4-space. L e t R4: 12 -+ 34, 13 + -24, 14 + 23, and * 2 = + l . and (*F)14 = F23. Also l e t

*

denote d u a l i t y o f two forms i n So (*F)12 = F34, (*F)13 = -Fe4

D u a l i t y decomposes t h e f i e l d F i n t o i t s s e l f dual F+ and a n t i - s e l f dual Fcomponents; these components are the (+1)and (-1) eigenvalues under *. Ordinary two forms--not j u s t two forms w i t h values i n SU(N)--split t h e same way. Two forms a r e denoted by A 2 , the space o f skew symmetric 2-tensors. The s p l i t t i n g gives

where A2 are the self-dual and a n t i - s e l f - d u a l o r d i n a r y forms respectively. since t h e Formula 411) gives the decomposition o f sO(4) i n t o two sD(3)'s--or, L i e algebras are t h e same--into two sU(2)'s.

I.M. SINGER

38

Note t h a t

For l a t e r use we define

Observe t h a t * i s conformally i n v a r i a n t i n 4 dimensions, so t h a t S(A) i s a conformal i n v a r i a n t . S h o r t l y we s h a l l see how t h e 4-sphere S 4 , the conformal comp a c t i f i c a t i o n o f R4, enters. THE PROBLEM We can now s t a t e the mathematical question: What a r e t h e s t a t i o n a r y p o i n t s o f S(A)? The m o t i v a t i o n f o r the problem stems from t h e saddle p o i n t method o f e v a l u a t i n g the Feynman-Kac path i n t e g r a l

where

F(A)

i s a gauge i n v a r i a n t q u a n t i t y : f(g.A) = f ( A ) .

I t i s a fundamental problem o f quantum f i e l d theory t o "make sense" o f t h e i n t e g r a l above--one r e a l l y doesn't know what DA, " i n t e g r a t i o n over a l l f i e l d s " means. I n the saddle p o i n t method one f i x e s on t h e s t a t i o n a r y p o i n t s . Then one w r i t e s t h e a c t i o n as a quadratic term i n A p l u s a remainder, and uses a perturbat i v e expansion f o r the remainder term. One emulates quantum electrodynamics (QED) where t h e method has been spectacularly successful. It i s essential, then, t o know the s t a t i o n a r y solutions. It i s easy t o compute the Euler-Lagrange equations f o r t h e s t a t i o n a r y points:

where + i n d i c a t e s the a d j o i n t operator; i n components,

We seek, then, t h e s o l u t i o n s A t o (16) w i t h f i n i t e a c t i o n on R4. EXTENSION TO S4 One o f t h e p o i n t s I wish t o emphasize i n t h i s l e c t u r e i s t h a t c o n s i d e r a t i o n o f

ON YANG-MILLS FIELDS

30

the problem above leads n a t u r a l l y t o t h e 4-sphere, t o topology and f i b r e bundles; these concepts are n o t introduced a r t i f i c a l l y . To t h i s end I c i t e a theorem o f Karen Uhlenbeck [20]. Theorem:

Suppose

(i)S(A) < m, (ii) A s a t i s f i e s the equation o f motion, i . e . , (iii)A has an i s o l a t e d (point) s i n g u l a r i t y .

i s a s t a t i o n a r y p o i n t , and

Then the s i n g u l a r i t y i s removable by a gauge transformation i n t h e f o l l o w i n g sense: i f t h e s i n g u l a r i t y i s a t p, t h e r e i s a b a l l (D ) around p and a gauge transformation I$ defined on D -p so t h a t @ * A i s e x t e n d h l e t o a l l o f D [See Fig. 1.1 P P'

0 * A EXTENDABLE

Figure 1: The domain D around a p o i n t p used i n extending t h e v e c t o r p o t e n t i a l A t o remove 8 p o i n t s i n g u l a r i t y . We have s p l i t R4 i n t o two parts: the B a l l D , and i t s complement; t h e i r common boundary i s t h e 3-sphere S3. On D , t h e v e c t b p o t e n t i a l @ * A i s nonsingular and there i s a mismatch between A [email protected] on the common boundary S3. We use i t t o b u i l d a new object--a f i b r e bundle. Before describing the f i b r e bundle i t i s worth n o t i n g t h a t we can a l l o w t h e s i n g u l a r i t y t o be a t i n f i n i t y . Because o f conformal invariance, i n f i n i t y i s no d i f f e r e n t than any other p o i n t i n R4; t h a t i s , a conformal transformation brings i n f i n i t y back t o the o r i g i n . So a f i n i t e a c t i o n , s t a t i o n a r y solution-w i t h no s i n g u l a r i t i e s on R 4 , b u t n o t defined a t i n f i n i t y - - c a n be considered t o have an i s o l a t e d s i n g u l a r p o i n t a t a. By a conformal transformation, b r i n g m back t o the o r i g i n and apply Uhlenbeck's theorem t o remove t h e s i n g u l a r i t y . Figure 1 i s now modified and becomes Fig. 2. The b a l l D i s t h e upper hemisi t s compliment i s the lower hemisphere D-, andPtheir common boundary phere D,, i s the equator S3. Now we are ready t o b u i l d a p r i n c i p a l bundle over t h e 4-sphere, p o t e n t i a l associated w i t h it.

and a vector

40

I.M. SINGER

D, x G

I

\

s3

D-xC

Figure 2:

The c o n s t r u c t i o n o f a f i b r e bundle over t h e 4-sphere.

The simplest p r i n c i p a l bundle over t h e 4-sphere i s S4 x G. The t r o u b l e i s t h e vector p o t e n t i a l A i s s i n g u l a r . To remove t h e s i n g u l a r i t y , we break A up i n t o $ * A i n t h e upper hemisphere and A i n t h e lower hemisphere producing a mismatch between A and 4 - A along the equator. We compensate f o r t h i s mismatch by b u i l d i n g a new bundle. On t h e upper hemisphere i t i s 0, x G. We patch o r paste along the equator by i d e n t i f y i n g the s e t x x G o f D, x G w i t h x x G o f D- x G by l e f t m u l t i p l i c a t i o n w i t h $(x). We g e t a f i b r e bundle, t h a t i s , a c o l l e c t i o n o f f i b r e s , groups i n t h i s case, one f o r each p o i n t o f S4. But t h e f i b r e s have been sheared along t h e equator by t h e gauge transformation $(x), x e 53. Since t h e gauge transformation $(x) on S3 w i t h values i n G can be t o p o l o g i c a l l y nont r i v i a l , i . e . , can wind around G, the newly constructed p r i n c i p a l bundle P can be complicated. The p o i n t o f i t s c o n s t r u c t i o n i s t h a t i t compensates f o r t h e mismatch o f A and $.A on t h e equator 9. We have produced a new vector potent i a l AuI$.A on P w i t h o u t s i n g u l a r i t i e s . s3

Uhlenbeck's theorem says, i n short, t h a t any s t a t i o n a r y p o i n t o f t h e a c t i o n on R4 w i t h i s o l a t e d s i n g u l a r i t i e s can be promoted t o a nonsingular v e c t o r potent i a l on a p r i n c i p a l bundle over S4. The new vector p o t e n t i a l i s a s t a t i o n a r y p o i n t f o r the a c t i o n on t h i s p r i n c i p a l bundle. One can recapture t h e o r i g i n a l p i c t u r e on R4 by stereographic p r o j e c t i o n . A DIVERSION F i b r e bundles have become so common i n h i g h energy physics (grand u n i f i c a t i o n schemes, symmetry breaking, and dimensional reduction), t h a t perhaps a s h o r t digression i n t o t h e i r geometry i s i n order. A few p i c t u r e s i n t h e simplest cases are worth many formulas, i n my view. Consider, then Table I and Fig. 3.

ON YANG-MILLS FIELDS

ICI

Figure 3: F i b r e bundles over the c i r c l e S1: (a) the torus; (b) the K l e i n b o t t l e ; (c) an a l t e r n a t i v e d e s c r i p t i o n o f the F i b r e bundle leading t o a K l e i n bottle. TABLE I FIBRE BUNDLES 1.

C i r c l e bundles over a c i r c l e S1:

torus, K l e i n b o t t l e

2.

C i r c l e bundles over a 2-sphere 52:

The Chern-Gauss-Bonnet formula and Dirac Magnetic Monopoles

3.

P r i n c i p a l bundles over a 4-sphere S4: .

Our previous examples; c l a s s i f i c a t i o n i n terms o f t h e Pontryagin charge o r second Chern class

41

1. M. SINGER

42

On the c i r c l e (Sl) t h e r e are two c i r c l e bundles, one o f which i s t r i v i a l and one t h a t i s n ' t . You have seen them both. The f i r s t i s a torus. Take a c i r c l e as shown i n Fig. 3a, and b r i n g i t around t h e base c i r c l e . The second i s a K l e i n b o t t l e (Fig. 3b): when t h e f i b r e c i r c l e i s brought around t h e base c i r c l e t o the s t a r t i n g p o i n t , the i d e n t i f i c a t i o n i s made w i t h t h e i n i t i a l f i b r e by 0 -0 instead o f 0 0 . A s l i g h t l y d i f f e r e n t way o f l o o k i n g a t t h i s c o n s t r u c t i o n generalizes t o higher dimensions. Take t h e base S1 and break i t up i n t o two pieces, so t h e r e i s an upper hermisphere (arc) and lower hemisphere. Each hemisphere i s an i n t e r v a l and the "equator" c o n s i s t s o f two points. The o n l y issue i s how t o patch the f i b r e c i r c l e s a t t h e equator. A t one end patch by the i d e n t i t y and a t t h e other end patch by 0 -0 g i v i n g t h e K l e i n b o t t l e (see Fig. 3c).

-

-

-

Next consider t h e two sphere ( S 2 ) (Fig. 4), decomposed i n t o t h e upper hemisphere (D,) and the bottom hemisphere (D-). We want t o c o n s t r u c t a bundle o f c i r c l e s by patching o r p a s t i n g D, x S1 and D- x S1 along the equator. To do so r e q u i r e s a map Q f r o m t h e equator (a c i r c l e ) i n t o t h e c i r c l e S1. We know these maps Q are determined t o p o l o g i c a l l y by t h e i r winding number. Patching by I$ a t x on the equator means i d e n t i f y i n g the c i r c l e from t h e northern hemisphere w i t h t h e c i r c l e from t h e southern hemisphere by a r o t a t i o n through angle Q(x). The topology o f the bundle i s completely determined by t h e winding number o f Q which has the formula

1J-f 2n . 1

As an example, consider the c i r c l e bundle o f u n i t tangent vectors o f S2 (Fig. 4a). The f i b r e a t each p o i n t i s the c i r c l e o f u n i t tangent vectors emanating from t h a t p o i n t . What i s t h e winding number f o r t h i s bundle? We can e a s i l y i d e n t i f y a l l the f i b r e s i n the upper hemisphere by adopting a r e l a t i v i s t i c p o i n t o f view. An observer a t the n o r t h p o l e c a r r i e s a c l o c k (a u n i t tangent vector) along w i t h him as he moves along a g r e a t c i r c l e p a t h (geodesic) t o any o t h e r p o i n t i n the northern hemisphere. Since the u n i t tangent v e c t o r o f a geodesic are a u t o - p a r a l l e l , the d i r e c t i o n i s carried i n t o the direction , determining the i d e n t i f i c a t i o n o f t h e c i r c l e l a b e l e d C ' w i t h the one l a b e l l e d C (Fig. 4b). Do the same f o r an observer a t the south pole; a l i t t l e surface theory, t e l l s you the shear i s t w i c e the s o l i d angle R. Thus, t h e winding number i s two.

$'

Figure 4: Construction o f a f i b r e bundle on t h e 2-sphere, S2.

43

ON YANG-MILLS FIELDS

Here i s another geometric example t o help b u i l d your i n t u i t i o n f o r t h e f o u r dimensional case w e ' l l need l a t e r . Think o f t h e 3-spherel S3, as the s e t o f p a i r s o f complex numbers (Zl,Z2) subject t o

(18) Map the 3-sphere t o the two sphere by sending a p a i r (Zl,Z2)

s3 .I

n

= [(Z,,Z2)

SI

S2

:

.IQ

lZllL

i n t o Z1/Z2.

n

I Z 2 f = 11

+

(eiezl,eiez2)

z1/z2

because Z1/Z, c'f8 For our present purposes, S2 equals t h e complex plane p l u s be = i f Z, = 0 and lZll = 1. Since 1Z112 + 1Z2I2 m y b t i p l y i n g by e Z2. Moreovph i f r o t a t e s vectors i n S3 b u t preserves t h e r a t i o : Z1/Z2 = e' Zl/e Z1/Z2 = W1/W, and (Z1/Z2) i s o f the same l e n g t h as (Wl/W2)l then Z . = e W., j = 1,2. So we have e x h i b i t e d S3 as a f a m i l y o f c i r c l e s over S2:J a c i r c l e bundle where each c i r c l e p r o j e c t s i n t o the same r a t i o o r p o i n t o f S 2 . This bundle has winding number one, though t h a t ' s n o t easy t o see. I t ' s computable by the Gauss-Bonnet formula which we now discuss.

=.d,

L e t ' s again t h i n k o f the two sphere as u n i t vectors i n t h r e e space. Dirac found a l l c i r c l e bundles over S2, motivated by magnetic monopoles [lo]. He s t a r t e d o u t w i t h Maxwell's equations f o r an electromagnetic f i e l d F. I n concise mathematical notation, F s a t i s f i e s dF = 0 d*F = j,

the current

.

Dirac wanted t o reverse t h e two equations above t o o b t a i n dF = pM(0), d*F = 0

a magnetic pole a t t h e o r i g i n

.

He was i n t e r e s t e d i n t h e s t a t i c case, so t h e equations reduce t o t h r e e space. Away from t h e o r i g i n , say on a sphere o f f i x e d p o s i t i v e radius, dF = 0. On a region t h a t i s c o n t r a c t i b l e , one could conclude t h a t F f dA. But the sphere i s I n fact, we know t h a t t h e not c o n t r a c t i b l e so F = dA on the e n t i r e sphere. i n t e g r a l o f F over the two sphere has t o be the f l u x o f the magnetic f i e l d and i t i s not zero. But the upper hemisphere, D+ i s c o n t r a c t i b l e and we can w r i t e

44

1. M. SINGER

S i m i l a r l y , on the lower hemisphere, F I D = dA-. along t h e equator; t h e r e i s a mismatcl. dA+ = F = dA- so d(A+

-

However, A+ and A - w i l l n o t agree

I n a l i t t l e band about t h e equator

.

A_) = 0

One can show t h a t

A+

-

A-

=

=I-

9 1$

on t h e equator

-

f o r some f u n c t i o n t$ t h a t doesn't vanish. [ I n general J(A A 1 around t h e c i r c l e could be any number. But i f matter f i e l d s a r e introauced-and A and Aare t o be vector p o t e n t i a l s , J A+ A- i s an i n t e g e r and Eq. (24) holas, thus q u a n t i z i n g the magnetic pole a t the orig'in.] The f l u x w i l l be (up t o a u n i versal f a c t o r ) equal t o the winding number o f t$. That i s

-

-- c

A+

-

A- =

C

J

* (0

= c winding no. o f $

.

The f u n c t i o n $ can be used t o b u i l d a c i r c l e bundle as I have already described. That i s , c i r c l e s from the t o p hemisphere a r e patched w i t h c i r c l e s from t h e bottom hemisphere by $ / 1 $ 1 . Now A and A- f i t together t o g i v e a v e c t o r potent i a l on t h i s c i r c l e bundle over 9. The c i r c l e bundle i s determined by t h e winding number which we can compute i n two d i f f e r e n t ways. F i r s t , we can c a l c u l a t e i t as t h e f l u x ; t h i s i s the generalized Chern, Gauss-Bonnet theorem, and gives t h e winding numbers as an i n t e g r a l formula, t h e averaged curvature f i e l d . Second, given t h e vector p o t e n t i a l , by p a r a l l e l t r a n s p o r t along geodesics from the n o r t h and south poles (as we d i d w i t h t h e tangent c i r c l e bundle), we can compute the mismatch a t t h e equator and thus compute t h e winding number. Now l e t ' s r e t u r n t o t h e 4-sphere, S4, and reconsider Fig. 2. We had two hemisspheres, D+ and 0, and the equator 5 3 . The SU(N) bundles are determined by maps o f S3 i n t o SU(N). When N = 2, SU(2) i s the same as t h e u n i t quaternions, i . e . , S3. The maps o f S3 -t S3 are determined up t o homotopy by an i n t e g e r , t h e winding number k. (This remains t r u e f o r a r b i t r a r y N.) We have already seen t h a t t h e v e c t o r p o t e n t i a l s on R4 w i t h i s o l a t e d s i n g u l a r i t i e s s a t i s f y i n g t h e e uations of motion can be promoted t o vector p o t e n t i a l s on these bundles over S What i s t h e corresponding formula f o r k i n terms o f an i n t e g r a l formula fqr the f i e l d ? The ansEer i s again t h e Chern-Gauss-Bonnet formula: i n terms o f F , the self-dual, and F , t h e a n t i - s e l f - d u a l p a r t o f t h e f i e l d

9.

45

ON YANG-MILLS FIELDS

F i n a l l y , l e t me end the geometric discussion w i t h an example o f an SU(2) bundle over S4 t h a t generalizes (19). We use quaternions i n s t e a d o f complex numbers.

.L s3

Consider S7 as p a i r s o f quaternions with t h e sum o f absolute values squared = 1. Since q1 and q, are f o u r dimensional, t h e r e s t r i c t i o n 1q1l2 + 1q,I2 = 1 defines a sphepq i n eight-dimensional space, i . e . , S7. Map S7 t o S4 by mapping (ql,q2) i n t o q, ql, a l s o a quaternion. i s t o be i n t e r p r e t e d as quaternions p l u s i n f i n i t y , because when q, = 0, q ql-equals i_nfinity. As before, i f I q l = 1, then (qq,,qq,) i s also on S7 and (qq,) lqql = q,'ql. Furthermore, i f two p o i n t s on S7 g i v e the same r a t i o , they d i f f e r by a u n i t quaternion r a t i o . Hence we g e t the 3-sphere bundle, S7 over S4.

-s4

One i n t e r e s t i n g f a c t t h a t emerges from our discussion i s t h a t t h e t o p o l o g i c a l quantization i s inherent i n the problem. We have shown t h a t c e r t a i n s t a t i o n a r y solutions could be promoted t o S4 as vector p o t e n t i a l s on bundles over S4. A vector p o t e n t i a l w i t h f i n i t e a c t i o n may o s c i l l a t e near i n f i n i t y , and may n o t be promotable t o S4. Nevertheless, the right-hand s i d e o f Eq. (26) makes sense. I t ' s a r e a l number and n o t i n f i n i t y because i t i s l e s s than t h e action. One can ask whether i t ' s always an i n t e g e r as i n t h e s t a t i o n a r y case. The answer i s yes, i f m i l d decay conditions are assumed f o r A as x + m. [Then AWI- $ la $ and $, as a f u n c t i o n o f rays, has a winding number.] So t h e s e t o f most" Y i n i t e a c t i o n vector p o t e n t i a l s s p l i t s i n t o a d i s c r e t e s e t (quantized by the i n t e g e r k) c a l l e d Pontryagin sectors. THE SOLUTION I have taken considerable time describing our problem and i t s promotion t o a geometric-topological context. Now I want t o t a l k about s o l u t i o n s . We s t i l l do It i s n o t know a l l the s t a t i o n a r y s o l u t i o n s (up t o a gauge transformation). known [7] t h a t a l o c a l minimum, i n a given Pontryagin sector, i s a minimum. One can now say a great deal about the minima. By s u b t r a c t i n g Eq. (26) from Eq. (12), we get

We

can assume k > 0, since $ change o f o r i e n t a t i o n changes the s i g n o f k. Thus equal t o 0, i . e . , F = F , assures us a minimum. Such F are c a l l e d " s e l f dual" f i e l d s . What do these self-dual . f i e l d s l o o k l i k e ? Note t h a t , whereas being s t a t i o n a r y leads t o a second-order equation, t h e c o n d i t i o n f o r a s e l f dual minimum gives a f i r s t order equation; i t i s much easier t o deal with.

F

Theorem.[6]

Self-dual s o l u t i o n s e x i s t f o r N = 2 and f o r a l l k

There's an Ansatz f o r them:

I f f i s a harmonic function,

2

0.

1. M. SINGER

46

i s self-dual. The m a t r i x q - - I ' m using physics n o t a t i o n here--is j u s t t h e e onto A: = su(2) so t h a t AO! i s a L i e algebrap r o j e c t i o n o f t h e two-form valued form. The s p e c i f i c h a h o n i k f u n c t i o n s used a r e

cup-

and g i v e the " m u l t i - i n s t a n t i o n " s o l u t i o n s [13,14]. They have i s o l a t e d singul a r i t i e s a t p.. By Uhlenbeck's theorem, t h e s i n g u l a r i t i e s can be removed; these vector p d t e n t i a l s have Pontryagin charge k. One can count t h e number o f degrees o f freedom f o r t h e s o l u t i o n s above. finds:

One

TABLE 2 k

Number o f parameters

0 1 2 k > 2

unique s o l u t i o n 5 degrees o f freedom 13 degrees o f freedom 5k + 4

(N = 2)

One can a l s o count the number o f degrees o f frqedom by l i n e a r i z i n g the d i f f e r e n t i a l equation f o r s e l f d u a l i t y . The l i n e a r equation i s a standard e l l i p t i c one and the index theorem counts the number o f degrees o f freedom. For SU(2), i t gives 8k 3. For SU(N) we have t h e

-

Theorem:[3]

The s e l f - d u a l s o l u t i o n s a r e a maniform o f dim 4Nk

-

-

N2

+ 1.

For N = 2 we f i n d 8k 3 > 5k + 4 when k > 3, so t h e r e must be a d d i t i o n a l solut i o n s f o r k 2 3. What do they look like?- S u r p r i s i n g l y , t h e answer comes from algebraic geometry. And i t i s the i n s i g h t o f the Penrose t w i s t o r program t h a t t r a n s l a t e s the s e l f - d u a l problem t o a holomorphic one [4,19,21]. B r i e f l y B what happens i s t h i s . The sphere S7 i n (28) i s acted upon by t h e c i r c l e e (see Eq. (31). CP3 denotes complex p r o j e c t i v e space, t h e complex l i n e s i n complex 4-space; t h a t i s [(z1,z2,z3,z4) (Az1,Az2,Az ,Az4); A & f, and A # 01. T h i s space i s a f i b r e bundle over S4 w i t h f i b r e S2. ?he map i s given by (z1,z2,z3,z4) + z1 + ~ 2 j / z 3 + z 4 j .

-

s4

(zl

+

z2j/z3 + z 4 j )

47

ON YANG-MILLS FIELDS

The SU(N) bundle P over S4 p u l l s up t o a bundle over CP3, and t h e f i b r e can be complexified t o SL(N,c) [see Eq. (3211.

That i s . the SWN) f i b r e over a D o i n t D o f S4 i s l i f t e d t o a l l t h e S2 p o i n t s i n CP3 over p. The-group SU(N) i s enlakged t o t h e complex special 1inear group SL(N,C), constructing a new p r i n c i p a l buedle Pc over CP3. The v e c t o r p o t e i t i a i A can be promoted t o a vector p o t e n t i a l A on P . S e l f d u a l i t y f o r A t r y s l a t e s immediately d n t o the Nirenberg-Newlander conditions [17] f o r making P , with connection A , i n t o a holomorphic bundle. That i s t h e i n s i g h t provided by t h e Penrose Twister Program. Put more concretely, the r a t i o s z1/z2, z2/z4, 23/24 are coordinates f o r CP3 (when 24 # 0). Two r a t i o s would s u f f i c e t o c o o r d i n a t i z e S4 l o c a l l y ; b u t a l l three complex coordinates are needed t o t r a n s l a t e s e l f dual i n t o holomorphic (even though the t h i r d coordinate i s redundant and a c t s somewhat l i k e gauge freedom). There i s a s i m i l a r e f f e c t i n two-dimensional a b e l i a n Yang-Mills/Higgs theory [151. There the a c t i o n S(A,$) depends on an a b e l i a n ( G = U(1)) v e c t o r p o t e n t i a l A and a complex scalar f i e l d 0:

C r i t i c a l p o i n t s are complicated.

DI;F = Imag.

$D~+,

The equations o f motion are:

(D~DI; + DI;D~)$ = 1-1l2 4 .

(34)

.+

Note t h a t t o have f i n i t e action, 141 + 1 as r + 01 and 110 $ 1 1 + 0 as r + 00. So A, = $ %,d + O ( l / r 2 ) . Again the f i n i t e a c t i o n c o n f i g u r a t i o n s f a l l i n t o classes 1 determined by the winding number k = 1 F e d2x. ;,I$'/$ = 6 A, = ,,v ,,v

rn

For A = 1, t h e minima are much easier t o determine.

DA$ = i*DAd,

rn h2

The equations are:

*F =

Equation (35) says t h a t

(35)

;i[(Al+iAZ)/2]

($)

= 0.

That i s e-W$ i s holomorphic

where aw = (Al+iA2)/2. The s o l u t i o n s @ are o f t h e form $ = h(z) ll w i t h the winding number k = ha. a

(z

- z a ) nQ

The reduction t o a holomorphic problem i s again p r e d i c t a b l e from t h e t w i s t e r program, which also leads t o new non-Abel i a n magnetic monopoles [151.

I.M. SINGER

48

Once the self-dual problem i s t r a n s l a t e d t o a holomorphic one on CP3, i t becomes a problem i n algebraic geometry. The a l g e b r a i c geometers have been studying t h i s matter extensively; t h a t i s , studying t h e c l a s s i f i c a t i o n o f holomorphic bundles over CP3 [5,121. Those t h a t come from t h e s e l f - d u a l v e c t o r p o t e n t i a l s on t h e f o u r spheres can be c l a s s i f i e d . We describe them i n two ways [2]. F i r s t , t h e a b s t r a c t d e s c r i p t i o n f o r the c f m p l e t % k q t o f s o l u t i o n s i s t h i s : consider l i n e a r maps, T(z), z c C4, from C t o C ; t h e s + $ r e 2k + 2 by k has a symplecmatrices, depending l i n e a r l y on z. Assume the second space C t r i c s t r u c t u r e given by a skew-adjoint m a t r i x , J. Also assumf t h a t T(z) s a t i s f&$2these conditions: (1) I t ' s o f maximad rank so i t maps C i n j e c t i v e l y i n t o C ; t h a t i s , f o r any vector v i n T(z)(C ), (Jv, v> = 0. Then

k i s two dimensional. For T(z)(C ) i s i s o t r o p i c and l i e s i n s i d e the J- orthogonal complement which i s o f dimension k + 2. The two-dimensional space E(z) i s independent o f the scale and t h i s c o n s t r u c t i o n assigns t o every l i n e i n C4 o r every p o i n t o f CP3 a two-dimensional space. The SU(2) bundle i s t h e s e t o f orthonormal frames i n each E(z). Theorem. -this

Every SU(21-bundle w i t h s e l f - d u a l v e c t o r p o t e n t i a l can be repreway.

For a thorough discussion o f t h i s theorem and i t s g e n e r a l i z a t i o n t o SU(N) see [l]. Despite the a b s t r a c t construction, t h e s e l f - d u a l v e c t o r p o t e n t i a l s can be r e a l i z e d q u i t e concretely, as i l l u s t r a t e d by t h e Ansatz I describe next. F i r s t l e t me remark t h a t any vector p o t e n t i a l can be w r i t t e n as

A

CI

= T

+ aT

K ,

(37)

P

where T i s an L by N m a t r i x f u n c t i o n w i t h T+T = IN (see Narasiman and Varadan [16]). These A w i l l be self-dual i f T s a t i s f i e s c e r t a i n conditions. F i r s t , f $ r the k t h P o k r y a g i n sector, L i s N + 2k. Next, note th$g2khe c o n d i t i o n T T = IN mean3+@e columns o f T are N orthogonal vectors i n C . Choose 2k o r t j o g o n a l t o the columns o f T. Denote t h i s N + 2k x 2k vectors i n C m a t r i x by A. Hence, T A = 0. Self duality f o r A

P

A(x) = a + bx AA '

i s given by t h e f o l l o w i n g equations i n A:

,

and

(38a)

commutes w i t h m u l t i p l i c a t i o n by quaternions.

I n Eq. (38a), t h e p o i n t x i n i s , x = xo + xli + x 2 j + x3k matrices. Also, a = (al,a2) matrices. Equation (38b) says

(38b)

R4 stands f o r quaternionic m u l t i p l i c a t i o n . That w i t h i, j, k represented by the usual 2 x 2 and b 7 (blrb2) w i t h a., b. constant N + 2k x k t h a t A A i s a k x k m a t i i x ti I2 [a].

49

ON YANG-MILLS FIELDS

I t i s easy t o check t h a t vector p o t e n t i a l s s a t i s f y i n g t h i s Ansatz are s e l f dual. One can also check t h a t t h e number o f degrees o f freedom i s 8k 3 f o r N = 2. The algebraic geometry assures us t h a t a l l s o l u t i o n s are gauge equivalent t o t h e exhibited solutions. I t i s amusing t o o t e t h a t i f one could prove d i r e c t l y t h a t the space o f a l l solutions i s connected, then the detour through geometry would be unnecessary. For the e x h i b i t e d solutions form a complete manifold o f the proper dimension.

-

OPEN PROBLEMS We close w i t h a l i s t o f open problems. (a)

Are there i r r e d u c i b l e s t a t i o n a r y s o l u t i o n s which are n o t s e l f ( a n t i ) dual? See [11,23] f o r a t w i s t o r d e s c r i p t i o n o f the equations o f mot ion.

(b)

What are the complex s o l u t i o n s o f the Yang-Mills equations o f motion, i . e . , s o l u t i o n s f o r C4 w i t h G = SJ?(N,C)?

(c)

I s the space o f self-dual s o l u t i o n s connected f o r k

(d)

I f no decay assumptions are mage, does 3 vector p o t e n t i a l A with I I F I t 2 ) an i n t e g e r ? f i n i t e a c t i o n have k = 1/8n2 ( I I F 1 1 2

(e)

What are the a p p l i c a t i o n s t o quantum gauge theories?

2 3?

-

So f a r , the existence o f solutions o f d i f f e r e n t t o p o l o g i c a l type, i.e., t h e existence o f d i s t i n c t Pontryagin sectors, describes t h e vacuum s t r u c t u r e o f t h e The s p e c i f i c forms f o r t h e s o l u t i o n as displayed by theory ( a l l a tunnelling). Eqs. (37) and (38) imply a n i c e closed form [9] f o r t h e Green's f u n c t i o n of t h e Dirac operator $A coupled t o A:

and t h i s form has been useful i n computing determinants i n t h e weak coupling limit. But attempts t o i n t e r p r e t o r compute Feynman namics using the s e l f - d u a l s o l u t i o n s have n o t " d i l u t e gas approximation'' doesn't work and, as r a t i o n s o f s e l f - and anti-dual s o l u t i o n s t h a t been found.

i n t e g r a l s i n Quantum Chromodybeen successful t o date. The f a r as I can t e l l , t h e configudominate the i n t e g r a l have n o t

REFERENCES

[l]Atiyah, M. F., Geometry o f Yang-Mills Fields, Fermi Lectures 1979 (Pisa). [2]

Atiyah, M. F., D r i n f e l d , V. G., H i t c h i n , N. J., and Manin, Yu. I . , Con(1978). pp. 185-187. s t r u c t i o n o f instantons, Phys. L e t t .

[3]

Atiyah, M. F. , H i t c h i n , N. J . , and Singer, dimensional Riemannian geometry, Proc. Roy. pp. 425-461.

[4]

e

I. M., S e l f - d u a l i t y i n f o u r SOC.

London,

A362 (19781,

Atiyah, M. R. and Ward, R. S., Instantons and algebraic geometry, Commun. Math. Phys. 55 (1977), pp. 117-124.

I.M. SINGER

Earth, W., Some p r o p e r t i e s o f s t a b l e rank-2 vector bundles on Pn, Math. Ann. 226 (19771, pp. 125-150. Belavin, A., Polyakov, A., Schwartz, A . , and Tyupkin, Y., Pseudoparticle s o l u t i o n s o f the Yang-Mills equations, Phys. L e t t . 598 (1975), pp. 85-87. Bourguignon, J.-P., Lawson, H. B., and Simons, J., S t a b i l i t y and gap phenomena f o r Yang-Mills f i e l d s , Proc. Nat. Acad. Sci. U.S.A. 76 (1979), pp. 1550-1553. C h r i s t , N., Stanton, N., and Weinberg, E. J., General s e l f - d u a l Yang-Mills s o l u t i o n s , Columbia U n i v e r s i t y p r e p r i n t , 1978. Corrigan, E., F a i r l i e , 0. B., Goddard, P., and Templeton, S., A Green's Paris). f u n c t i o n f o r the general s e l f - d u a l gauge f i e l d , p r e p r i n t (E.N.S., Dirac, P.A.M.,

Proc. Roy. SOC.

s, 60 (1931).

Green, P. S., Isenberg, J., and Yasskin, P., Phys. L e t t . 788 (19781, pp. 462-464.

Non-self-dual

gauge

Horrocks, G., Vector bundles on the punctured spectrum o f a l o c a Proc. London Math. SOC. (19641, pp. 684-713.

14

ields, ring,

Jackiw, R., Nohl, C . , and Rebbi, C., C l a s s i c a l and semi-classica s o l ut i o n s t o Yang-Mills theory, Proceedings 1977 B a n f f School, Plenum Press. Jackiw, R., Nohl, C., and Rebbi, C., Conformal p r o p e r t i e s o f pseudop a r t i c l e configurations, Phys. Rev. 150 (19771, pp. 1642-1646. J a f f e , A. and Taubes, C.,

Vortices and Monopoles, Birkhauser (1980) Boston.

Narasimhan, M.S. and Ramanan, S., Existence o f universal connections, Amer. J. Math. 83 (19611, pp. 563-572. Newlander, A. and Nirenberg, L., Complex a n a l y s t i c coordinates i n almost complex manifolds, Ann. o f Math. 65 (1957), pp. 391-404. Osterwalder, K. and Schrader, R., Axioms f o r Euclidean Green's f u n c t i o n s I, 11, Comm. Math. Phys. 31 (1973); 42 (1975). Penrose, R., pp. 65-76.

The Twistor programme, Rep. Mathematical Phys.

2

(1977),

Uhlenbeck, K., Removable s i n g u l a r i t i e s i n Yang-Mills f i e l d s , B u l l . Amer. Math. SOC. 1 (19791, pp. 579-581. Ward, R.S., Ward, R.S.,

79 (1981),

On s e l f - d u a l gauge f i e l d s , Phys. L e t t .

61A (19771, pp.

81-82.

A Yang-Mills-Higgs monopole o f charge 2, Commun. Math. Phys. pp. 317-325.

Witten, E., An i n t e r p r e t a t i o n o f c l a s s i c a l Yang-Mills theory, Phys. L e t t . 778 (1978), pp. 394-398.

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 9. Nicolaenko (eds.) 0 North-Holland Publishing Company, 1982

51

GAUGE THEORIES FOR SOLITON PROBLEMS

D.H. S a t t i n g e r

*

Department o f Mathematics University of Minnesota Minneapolis, Minn. U.S.A.

Lax

1.

equations

and i s o s p e c t r a l deformations.

Gardner, Greene, Kruskal, and Miura [41 showed t h a t t h e eigenvalues of t h e Schrodinger operator

L=b

evolves according t o t h e

2

x

+ -61

KdV

u

are invariant i f the potential

equation

.

u +u +uu = O t xxx x

u=u(x,t)

P. Lax [ 5 ] gave

i s a linear

t h i s observation t h e following g e n e r a l formulation: Suppose L(u) operator depending on t h e f u n c t i o n u i s u n i t a r i l y equivalent when

evolves according t o a nonlinear e v o l u t i o n equa-

u

.

t i o n &(u,ux,uxx,ut,uxt, . . ) = O ily of u n i t a r y operators

U(t)

such t h a t t h e family of o p e r a t o r s [ L ( u ( t ) ) ]

This means t h a t t h e r e i s a one parameter fam-

such t h a t

U-'(t)L(u(t))U(t)

.

U-kU

Differentiating

with r e s p e c t t o

t

"he u n i t

U = B U , where

a r y operators must s a t i s f y a d i f f e r e n t i a l equation of t h e type B*=-B

.

=L(u(O))

t

we o b t a i n t h e equation

atL = [B,Ll

(1)

I n t h e context of s o l i t o n t h e o r i e s t h i s equation has sometimes been c a l l e d t h e

Lax equation, e=O

.

and t h e operators B

I n t h e case of t h e

1 B=-b(b 3 + x 4

x

+ 81 "XI

check t h a t t h e commutator Thus

L =[B,L]

t

s o l u t i o n of t h e

The KdV

*

and

KdV

.

a Lax p a i r for t h e e v o l u t i o n equation

equation Lax gave f o r

Then

[B,L]

L

b L= t

g1 ut

B

the operator

( m u l t i p l i c a t i o n ) and one can

i s t h e m u l t i p l i c a t i o n operator

1 - g(uxxx+

i s an a p e r a t o r equation which i s s a t i s f i e d whenever KdV

u

uux)

is a

equation.

equation can be solved e x p l i c i t l y by t h e method of i n v e r s e s c a t t e r i n g

This research was supported i n p a r t by NSF Grant #MCS 78-00415.

52

D. H. SA TTINGER

applied to the scattering problem (L- A ) v = O

.

Some of the other striking

properties of the K d V equation are its Hamiltonian structure,an infinite number of conservation laws, an associated invariant (isospectral) scattering problem, and multi-soliton solutions.

Since the first investigations of the K d V

equation it has become apparent

that the K d V equation is not alone in possessing these properties. Other evolution equations with many of the same properties are the nonlinear Schrodinger equation i Yt + Yxx + I Y I

2

Y =0

, which was shown to share similar properties by

Zakharov and Shabat [ 101; and the Sine-Gordon equation, which was investigated by Ablowitz, Kaup, Newell, and Segur [l].

These three equations, as well as others, can all be obtained in the context of a gauge theory with gauge group SL(2.R) will be discussed in $4 ; and in

.

The gauge theory of these equations

$5 we discuss isospectral problems from the

general point of view of gauge theories on fiber bundles. In [81 we derived a generalization of the nonlinear Schrodinger equations for the gauge group

sC(3,C)

.

In $6 we derive multi-component analogues of the sine/sinh-Gordon

equations for a general semi-simple Lie algebra. We illustrate these equations explicitly for the case of algebras of rank two, namely A2,B2

and Gp

.

Similar equations have been obtained by Fordy and Gibbons [ 3 ] and Mikhailov

c.&1.

[61.

In the gauge theory point of view the Lax equation (1) arises as the condition of zero curvature of connection [D D ) x' t the form [Dx,Dt]= O

.

on a principal bundle: thus (1) takes

The role of the scattering operator L is then played

by Dx and that of the infinitesimal generator of the unitary transformations by Dt

.

We begin in $52,3 with a discussion of Wahlquist and Estabrook's notion of

prolongation albegras and show how the isospectral problems and the connection may be introduced from their point of view.

GAUGE THEORIES FOR SOLITON PROBLEMS

53

I wish t o thank Leon Green and P e t e r Olver f o r many u s e f u l d i s c u s s i o n s concerning t h e work i n t h i s paper.

f i o l o n g a t i o n Albegras [ 9 1

2.

The f i r s t s t e p i n t h e Wahlquist-Estabroak program i s t o r e p l a c e t h e evolution This

equation by an i d e a l of 2-forms closed under e x t e r i o r d i f f e r e n t i a t i o n .

procedure i s b e s t i l l u s t r a t e d by an example, say t h e sine-Gordon equation

.

u =sin u xt then, i n

We w r i t e t h i s as a f i r s t order system

x,t,u,p

, Cr2=dp A

d x + s i n u dx A d t

The f o u r dimensional space with coordinate

IR = [u,p]

over

(x,t,u(x,t)

,

X

space we introduce t h e two forms

a l = d u A d t - p dx A d t

2

u = p , p t = s i n u : and

R 2 = [ x , t]

.

p(x,t))

.

x,t,u,p

.

i s regarded as a bundle:

A s e c t i o n of t h i s bundle i s a graph of t h e form

When Crl

and

a r e r e s t r i c t e d t o such a s e c t i o n

Q12

they t a k e t h e form E1=(ux-p)dx A d t Thus,

Crl

B2=(sin u-pt)dx A d t

.

and a2 vanish on s o l u t i o n manifolds of t h e sine-Gordon equations.

fi1=E2 = O a r e equivalent t o t h e

According t o C a r t a n ' s theory, t h e equations p a r t i a l d i f f e r e n t i a l equation

u =sin u xt

i s closed under e x t e r i o r d i f f e r e n t i a t i o n . d Z 2 = c o s u dx A C r

clC2 = d t A Cr2

1

i f f t h e i d e a l generated by

(al,Cr2]

A simple computation shows t h a t

1 '

s o indeed t h i s i d e a l i s c l o s e d .

The Cartan i d e a l ( t h e i d e a l generated by t h e two forms; i n t h e c a s e above, t h a t generated by variables

Cr

1

and

) i s now prolonged by introducing a d d i t i o n a l

Q2

; one-forms

yl,...y,

(uk=dyk+Fk d x + G k d t 1 n and r e q u i r i n g t h a t [(u , . .(u ,

.

iation. variables

The functions u,p

Fk

(21

; Cx1,Cr2,.

and Gk

..]

be c l o s e d under e x t e r i o r d i f f e r e n t -

a r e assumed t o depend only on t h e dependent

( a s t h e case may b e ) and t h e

k k h =dF

A

k dx+dG A d t

y

1

. ..y n

.

I n t h e c a s e a t hand,

54

D. H. SA TTINGER

=

#

5 k

du R d x + #

U

P

dp A d x +

= G k du A d t +Gk dp A d t

P

du A d t = ’ Y l + p dx A d t

.

and

CX1

k

+

bY and

From (2) d y j A d x = o j A d x - G j d t A dx

Also, from t h e d e f i n i t i o n of

d y j A dx

aY

d y j A d t = d wj A d t - $ d x

,

(r2

.

dyj A d t

j

dp A d x = ’ Y 2

- sin

A

.

dt

u dx A d t

and

The prolongation equations a r e obtained by s u b s t i t u t -

i n g t h e s e i d e n t i t i e s i n t h e expression above f o r modulo t h e i d e a l generated by

(r1,a2 , and ok

and then s e t t i n g

duk

.

duk

I

0

I n t h e c a s e of t h e Sine-Gordon

equation we o b t a i n G =O

G =O

P sin u F = O P Prolongation equations f o r o t h e r w e l l known e v o l u t i o n equations have a l s o been U

[G,F] +flu-

derived. the

Wahlquist and Estabrook have derived t h e p r o l o n g a t i o n equations for and n o n l i n e a r Schrodinger equations. The problem now i s t o solve t h e

KdV

prolongation equations f o r v e c t o r f i e l d s

F

by Wahlquist and Estabrook f o r t h e

equation

KdV

.

and G

This was o r i g i n a l l y done

[ g l . The c a s e of t h e Sine-

Gordon equation i s much simpler ( s e e [ 8 1 ) .

3. I s o s p e c t r a l Problems Let us t u r n now t o t h e problem of f i n d i n g a s s o c i a t e d i s o s p e c t r a l problems f o r a given e v o l u t i o n equation.

We suppose t h a t v e c t o r f i e l d s

F

and

G

have been

found which s a t i s f y t h e prolongation equations and a r e l i n e a r i n t h e y - v a r i a b l e s : F=yjFjk

and

G =yjGjk

hY

3

.

This g i v e s us an isomorphism between t h e L i e

a l g e b r a oF-matrices and of v e c t o r f i e l d s . matrices

F=Fjk

and

G=G

k

3

by

L=bx-G

A Lax p a i r i s given i n terms of t h e

and

The a s s o c i a t e d

j

i s o s p e c t r a l s c a t t e r i n g problem i s given by

Ly = 0

Let us v e r i f y t h i s f o r t h e sine-Gordon equations. equations we have L =Fu +Fp =Fp t u t p t p t and

[B,L] = -[G,L] = -[G,ax

.

B=-G

- F]

or

bx

-F2yk = 0

.

From t h e p r o l o n g a t i o n

GAUGE THEORIES FOR SOLITONPROBLEMS

=

axG - [F,Gl

=

flu(flusin u F 1 =sin u F P P

55

The sine-Gordon equation thus comes from the Lax equation Lt = [ B , L ]

.

We can

get explicit realizations of the isospectral problems by taking specific representations of the Lie algebra so ( 3 ) .

For example, by choosing the representation of

so ( 3 ) given by the Pauli spin matrices

one is led to the scattering problem introduced by Ablowitz et. al. [11. (See [81).

4. Connections on Fiber Bundles; s t ( 2 , E ) The Lax equations can be interpreted from the point of view of gauge theories, as follows. Consider the case of the sine-Gordon equation, and let bx

, bt

be

the total derivatives on the jet bundle x,t,u,p

+... + p b bu + p x h p +... b = A+ u - b t bt t bu +Pthp Putting Dx = b, - F , Dt = at G , we have, from equations (3) b =- b

x

bx

-

[Dx,Dtl=[bx-F, bt-Gl =btF-bxG+[F,GI

-

=F

- sin u

p Gup +flu P t = (p, sin u)F

-

.

P

F

P

Thus the curvature [Dx,Dt] of the connection vanishes on an integral manifold

of the sine-Gordon equation.

It is useful to compare this formalism with the gauge theories of force fields. For example, in electromagnetic theory one introduces the connection D =b +eA +

P

P

where ~ = 0 , 1 , 2 , 3 ; the b

P

are the derivatives

, and the bxp

electromagnetic &-potentials. In electromagnetic theory the A

P

A

P

are the

are scalars,

for the gauge group of electromagnetic theory is the abelian group U(1) non-abelian gauge theories the A

LL

; but in

take their values in a Lie algebra. The

D.H. SATTINGER

56

commutator [D , D v ] = F I-1

F

=e(b A

CIV

P V

-b

.

A ) V P

gives the field strengths. In electromagnetic theory

I.Lv

Thus the condition of zero curvature [D , D V ] = O

means

CI

zero external field. The Lax equations may be interpreted as the vanishing of the curvature of a connection, o r , as we shall see below, as an integrability condition.

To see how to proceed let us take sL(2,lR)

as gauge group; sL(2,R) gauge

theories lead to the sine-Gordon, K d V and nonlinear Schrodinger equations. As basis for sC(2,IR) we may take

These matrices have the commutation relations [o1,a21 =2u2 [u~,,J~] =al

D,

.

, [u,,~~]=-20 3 '

Form the connection

= bx + i@,

- qu2 - ra3

Dt = b t - k 1 - b 2 - C b

3 -

Setting the curvature [Dx,D 1 = O we are led to the system of equations t -hxA + qC - rB = 0 qt - bxB - ?(qA

+ i@)

=0

r - b C+Z'(rA+iGC) = O t x which were obtained by Ablowitz &.

(5)

g.

[2].

The coefficients A,B,C depend

only on q,r and their derivatives. Ablowitz

e.&. construct solutions of

( 5 ) by expanding N

N N A=C CnAn , B = C , C = Z GnCn ; n=O n=O n=O these series may be substituted into ( 5 ) and the coefficients An'Bn.Cn for recursively.

6'

=

me

evolution equations are obtained as the coefficents of the

1 terms.

Examples: (1.1 Nonlinear Schrodinger equation r = 1 , q=V 2

2

A=-2i6 -ilYI B=26Y

+iYx

c=2
solved

GAUGE THEORIES FOR SOLITON PROBLEMS

57

r=-1,q=u A = -4163 +2i6u - u

X

2

2

B = 4 6 u+2i6u - 2 -~Uxx 2 C = - 4 6 +2u

.

[DX ,Dtl=(~t+6~~x+~x,)u2

The sine-Gordon equation is obtained by putting A = ?i Dx = bx + i p l + uX

(

‘2-‘3

6

2

,

b

c= c

:

6

B = - Q ’

)

D = b - 1 cos u u sin u (02+u t t 16 1 3 [D~,D,I= (uxt - sin u) [ u q ] . In this formalism

Lax equations

[DX ,Dtl = O

isospectral problem c-) D ( 6 ) q = 0 X Unitary group

Dtcp=O

.

That is, as q and r evolve according to the nonlinear evolution equation the eigenfunctions (solutions of D ( G ) ( p = O )

must evolve according to the

X

differential equation D t ~ = O .

5. Connections on Fiber Bundles; general. We consider a principal bundle of a linear (matrix) Lie group G over IR2 ; and let [Dx,Dt) be a connection on this principal bundle. Let @ fundamental solution matrix of D ( g ) @ -0 X

parameter 6

.

(We assume Dx

be the

contains the

as one of the coefficients, as in the case of sC(2,R) above).

Proposition. If [Dx,Dt]= O the equations DY(6)ip = O the Lie group G

.

ip

there exists a single-valued solution @(x,t,c)

of

, D+,+= O . For each point x,t CP takes its values in

is thus a section of the principal bundle of G over

Proof. The equation DxCP = O

is of the form

D.H. SATTINGER

50

bxQ = Iw where

U

l i e s i n the Lie algebra

.

g

This can be regarded as a d i f f e r e n t i a l

c-

equation on the Lie group

.

G

The mapping A - A @

I

matrix

.

p

to

on

Since A @

.

d U=UP

multiplication, the Jacobian of t h i s mapping i s

takes the i d e n t i t y

G

ip

i s matrix

Hence t h e

d i f f e r e n t i a l equation can be i n t e r p r e t e d as d@ = ax where

dQU

i s a tangent vector t o

$U

D ip = O

t

.

0

Since

Dx

these operators commute; and ip(x,t) (xo,to)

,

.

Q

at

[Q(x,t )]

group, t h e e n t i r e t r a j e c t o r y t o the equation

G

Hence, i f

lies in and

Dt

; and t h a t a t

g

.

Then

u+x

@we

depends only on i t s value a t a base point

as

bx-U+

U+

where

i s an element of t h e Lie

implies

x - + m

Choose Q

so t h a t

x+-00 S

such t h a t

If [U+,V+] = O

a-e

u+x

S(t,C)

as

xd+

m

.

We suppose

the time evolution of the " s c a t t e r i n g data''

i s given by St-V+S=O

Proof.

-

component tend t o zero

Dx

-

as

Then t h e r e i s a matrix

S(t,C)

Dx

are constant matrices.

A*

Proposition 2 .

x=*

DxQ = O

U+" @ N e A+where t h e

The same argument applies

not on the path.

1x1 -m

algebra

l i e s i n the

commute, the flows generated by

Now suppose t h a t the c o e f f i c i e n t functions i n the as

.

G

$(xo,to)

.

We may assume the asymptotic behavior

u+x

@-e

S(t,c)

i s uniform i n

and t h a t t h i s r e l a t i o n may be d i f f e r e n t i a t e d i n % h e . Since D,@ = O we have Dte u+x S ( t , g ) = 2 + x S t - V + 2 + x S = eu+x (St-V+S)-O a s 'x-.~ .

t

GAUGE THEORIES FOR SOLITON PROBLEMS

Since

i s independent of

S -V S

t

+

x

and

e

59

u+x i s nonsingular,

St-V+S=O

It i s p r o p o s i t i o n 2 which accounts i n part f o r t h e success of t h e i n v e r s e s c a t t e r i n g method.

One can s o l v e f o r t h e e v o l u t i o n o f t h e s c a t t e r i n g d a t a

no knowledge of t h e s o l u t i o n s of t h e nonlinear e v o l u t i o n equations. [U+,V+] = 0

may check t h a t t h e condition st(2,R)

S

with

The reader

i s s a t i s f i e d i n each of t h e cases f o r

described above.

6. Multi-component sine-Gordon equations.

G . S . [21

a r r i v e d a t t h e sine-Gordon and sinh-Gordon equations by a(q,r) , e t c . i n ( 5 ) . These equations a r e of t h e form A(C,q,r) = 6

Ablowitz assuming

u = f ( u ) , where xt

i s a sum of two exponentials.

f

t h e i r method t o a g e n e r a l semi-simple L i e a l g e b r a

a system of equations ials.

ui

xt

= f i ( u I...uk)

I n t h i s s e c t i o n we extend g

where each

t o o b t a i n , as an analogue,

i s a sum of exponent-

Pi

Some such systems have been discussed by Fordy and Gibbons [31, and A.V.

Mikhailov

g . a.[ 6 ] ,

o f v a r i a b l e s i n our system i s equal to t h e rank of t h e semi-simple

The number k algebra

g

b u t by o t h e r methods.

( i . e . t h e dimension of t h e Cartan subalgebra).

The number of

independent exponentials i s determined i n a more complicated way from t h e r o o t s of

g

.

I n t h e case

st(3,C)

,

f o r example, we w i l l have two independent v a r i a b l e s and

t h e nonlinear terms will be sums of t h r e e exponentials.

is

.

A2

algebra

We c o n s t r u c t t h e equations f o r

G2

A2

, B2 ,

The a l g e b r a

and t h e e x c e p t i o n a l L i e

.

We begin our c o n s t r u c t i o n with t h e a n s a t z Dx = ax

where

i

- 60, - uiui

runs f r o a 2 t o n

.

Setting

[Dx,Dt] - 0

st(3,C)

we g e t

D.H. SATTINGER

60

u + c u i a j [o,,ojl = O i=2

j

btu 1ui + a3[al,a 1 = 0 I n both equations

i s summed over

j

1 to

n

and

i

over t o

to

2

n

.

The f i r s t can be w r i t t e n i bxA= C [uoi,A1 if1 . We t r y t o i n t e g r a t e t h i s system by s e t t i n g

i d

u = hx

j

. . . ,cp n ) .

2

A = A ( ,~

gi =

and

We g e t

[ui,Al

bcp

The necessary c o n d i t i o n s f o r i n t e g r a b i l i t y o f t h i s system a r e

[aj’[oi,Al1 = [ u i , [ o j . A ~ ~ for

i,j=2,

.

...,n

B y t h e Jacobi i d e n t i t y we may r e p l a c e t h i s by

The e a s i e s t way t o s a t i s f y t h i s c o n d i t i o n i s t o choose t h e

x

where

(h.)

i’ J t o l i e i n an abel-

i

We a r e t h u s l e d t o try

i a n subalgebra. D =b

u

[[u u ] , A 1 4

x

- u’hi - La

D =b

t

t

-

bj -

c‘j

are elements i n t h e Cartan subalgebra and u,u

are root vectors.

j

(There i s no l o s s i n g e n e r a l i t y (proof omitted) by o m i t t i n g components along t h e Cartan subalgebra i n The condition

[ Dx,Dt] = 0

).

Dt

then leads t o

bxbj = uiaj(hi)bj

(no sum over

b ui h i + b j [a,ajl = O t Putting

i

u =Vxi

j

.

we g e t

j % = aj(hi)bJ

,

b’p whose s o l u t i o n i s

a

b 3 -- B3 e j The

(h i

Bj being c o n s t a n t s .

(7)

9

(We sum over

i

i n t h e exponential.)

We now t u r n t o t h e i n t e g r a t i o n of t h e second equation i n ( 6 ) . algebra

g

t o be semi-simple and t h a t t h e v e c t o r s

hi

i n t h e Weyl-Chevalley normal form ( s e e [ 71, pp. 46-52). roots i n the vector figure, with

a-i

=-‘pi

.

We expand

and Let

We assume t h e Lie a r e chosen t o be

Ui

0

i

run over a l l

GAUGE THEORIES FOR SOLITON PROBLEMS

.

k u=c u

bJck=O

I n t h e case choose

(6) i s

Then t h e second equation i n

i h. +bJck[u u 3 = O 'Pxt 1 k' j Consequently if

sh(3,C)

b1,b3

,

-2

vXt = c 3

vxt = c (Recall t hat

0

J

+Q

and

b-2

.

i s a non-zero r o o t ,

k

M=[al.a3,a-2)

take

61

t o be non-zero.

.

c 1= c 3 = c -2 = O and

We s e t

This l e a d s t o t h e e a u a t i o n s

-2('P1 +2v2 + 'P3) 2 -2 b =c2 e

(9)

-3 3

2(-'P1+'P2 + 2 q b = c - ~e

h-i =-hi

and

'p-i=-pi

so, f o r example,

a . ( h . ) c p i = 2 ( a ( h )(pl+a ( h )(p2 +aj(h3)cp3 1.1 J I 3 1 J 2 The a s s o c i a t e d i s o s p e c t r a l problem i s

Equations

(9) may be r e w r i t t e n , a s follows. Put u = Q (h

)(pi

v=ag(hi)cp i

1 i

Then

u + v = (a ( h ) 1 i u xt

+a3 ( h i ))cp i = a 2 ( h i ) 4

, and

. equations ( 9 ) can b e reduced t o

e U - 2 c2 e -(uw)- 2c-3eV

(10)

v = - 2 c - l e U - 2 c2 e -(u+v) +hc-3,V xt

Now consider t h e algebra B

.

I t s v e c t o r fig1 *e and t a b l e of Cartan i n t e g e r s

hl a

are given below:

1

h4

h2

h3

2

2

0

1

2

1

0

0

2

2

2

0

1

2

a

3

-

2

a4 -

1

62

D.H. SATTINGER

M={a ,a ,a ] , then we must s e t

We take

aicM

.

1 2 3 Consequently, we must have

c =O j

whenever

c - =~ c 4 = c - ~= c

-3

Cr

1

+Cr

E

j

.

= c =O -1

P f o r some Our equations

then a r e 1 -2(2% + 2%) yxt = -c b-' = 2 1 -cl e2(,,+2,, +$ = -c2b-' = -c2 -2 (2y2 + 2a3) 'p;lt = -c b-3 = -c

vxt

3

3 e

Again, defining

u = Q (h )cpJ=2(2y1+2v2) 1 j v = a (h )cpj=2(2y2+2y3)

3

j

-u

1 we have

cpxt=-c 1e

U+V -( 7)

2

, 'P,t=-C2

, Tit

e

= -c3e-'

and we a r r i v e a t the equations

where

c 1,c2,c3

Now we consider

are a r b i t r a r y constants. G2

,

the exceptional Lie algebra of rank 2.

and t a b l e of Cartan integers f o r

G2

is

a2

hl a

t

a

1 2

a2 3

a

-5

"-4

a-l

-

h3 1

h4

5

3

0

h6 -3

1

3

3

0

2

0

3

3

a4

1

2

1

1

0

2

1

0

1

1

1

2

1

"6 -1

0

1

-1

1

2

5

-3

h2

1

a 3 - 1 1

a

a

The vector figure

-

1

GAUGE THEORIES FOR SOLITON PROBLEMS

=c

as usual, we reduce these by introducing u=Q( (h. ) yi =2(2y1+'pg + 3 q + ) 1 1

v = Q (( h ) yi = 2 ( c + + c p 4 + 2 9 ) 5 i Then since Q( +Q( = m 4 and 0 +a =a2 @e g e t 1 5 4 5 i u+v u+3v a2(hi);= ( 2 aq(hi)y = ( 2 so o w equations become

u+3v

u =-4c xt

vxt=-2c

1 e-u

-2c

2

4 7 ) e

u+3v

2

e

-(

2 1-2c 4

4 - 6 ~ U+V

e

-3

=c-4=c

63

-5

=c6=c6=o

-

.

D.H. SATTINGEA

64

References

1. Ablowitz, M . J . , Kaup, D . J . , Newell, A . C . , Segur, H . , "Method f o r solving the the sine-Gordon equation,'' Phys. Rev. L e t t . 30, l.262-1264 (1973). 2.

3.

, "The inverse s c a t t e r i n g transform-Fourier analysis Math. 2 (1974) 249-313. for nonlinear problems," Studies A&. Fordy, A.P., Gibbons, J., "A Class of Integrable Nonlinear Klein-Gordon Equations i n Many Dependent Variables,'' Commun. Math. Phys. 77 (19801,

21-30.

-

14.

Gardner, C.S., Greene, J.M., Kruskal. M.D., and Miura, R . M . , "Korteweg de Vries Equation and Generalizations, VI. Methods for Exact Solution." Communications Pure and Applied Mathematics. 7 ,97-133.

5.

Lax, P.D., "Integrals of Non-linear Equations of Evolution and S o l i t a r y Waves," Corn. Pure A p p l . Math. el (19681, 467-490.

6. Mikhailov, A.V. Olshametsky, M.A., Perelomov, A.V. "Two Dimensional Generalized Toda Latice," P r e p r i n t , Landau I n s t i t u t e f o r Theoretical Physics, Moscow, 1980.

-

7.

Sarnuelson, H . , Notes on Lie Algebras, Van Nostrand Studies #23, New York, 1969.

8.

Sattinger, D.H. "Prolongation Algebras and Gauge Theories f o r I s o s p e c t r a l Problems, " Proceedings, 2nd. I n t e r n a t i o n a l Symposium on Dynamical Systems, Gainesville, Fla, 1981.

Rheinhold Mathematical

9. Wahlquist, H . D . , and Estabrook, F.B., "Prolongation s t r u c t u r e s of nonlinear evolution equations," J o u r . Math. 16 (19751, 1-7.

x.

10. Zakharov, V.E. and Shabat, A.B., "Exact Theory of two dimensional s e l f focusing and one-dimensional s e l f modulation of waves i n nonlinear 34 (19.9721, 62-69. media," Soviet Phys. JEW -

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 6.Nicolaenko (eds.) 0 North-Holland Publishing Company, 1982

65

THE INVERSE MONODROMY TRANSFORM I S A CANON I CAL TRANSFORMAT I O N H. Flaschkaf'and A.C.

Newell*

*Clarkson CoI
'I. INTRODUCTION AND GENERAL DISCUSSION

I n e a r l i e r papers C11, C21, we used deformation theory t o study t h e PainlevC equatons which govern the s e l f - s i m i l a r s o l u t i o n s o f the modified Korteweg deVries (MKdV) and sine-Gordon (SG) equations. I n doing so we introduced t h e Inverse Monodromy Transform (IMT) which p a r a l l e l s the Inverse S c a t t e r i n g Transform (IST); whereas the l a t t e r i s used t o l i n e a r i z e the i n i t i a l value problem f o r c e r t a i n classes of nonl i n e a r e v o l u t i o n equations, the former allows us t o f i n d , by l i n e a r methods, an important class of solutions , the multiphase s i m i l a r i t s o l u t i o n s t o these equations. They are t o the standard (one-phase] simi l a r l t y solutlo% wnat m u f t i s o l i t o n o r f i n i t e gap p e r i o d i c s o l u t i o n s are t o s o l i t o n s and cnoidal waves. We begin i n s e c t i o n 2 by i n t r o d u c i n g the family o f i n t e g r a b l e e v o l u t i o n equations

Pt

= P.(P,P,,---P.

j

J

JX

),

(Pjx means the jth d e r i v a t i v e o f P w i t h respect t o x) which are i n t e g r a b i l i t y conditions f o r a c e r t a i n eigenvalue problem

Vx = (SR+P(x,tj)lV,

-DXX<~,

(1.2)

and associated f a m i l y

Applying a f u r t h e r c o n s t r a i n t

v5

= s(c;P,---P(n-2)x

)V

(1.4)

w i t h S r a t i o n a l i n 5 , defines a f i n i t e dimensional manifold of s o l u t i o n s common t o a subset j=O,l,--n-1 of the equation s e t (1.1). The manifold i s defined by

66

H. FLASCHKA, A.C. NEWELL

a nonautonomous, nonlinear o r d i n a r y d i f f e r e n t i a l equation i n x, the c o e f f i c i e n t s depending on x and t , which i s the analogue o f t h e Lax-Novikov (LN) C3,41 equation j t h a t defines the f i n i t e - g a p and m u l t i s o l i t o n s o l u t i o n s of the members o f (1.1). There a r e many p a r a l l e l s between the LN equation and i t s nonautonomous (NLN) counterpart. I n t h i s paper, we (1) Introduce t h e NLN equation and t h e h i e r a r c h y o f time flows w i t h which i t i s associated. (2) Show that, w i t h respect t o a given symplectic form and Poisson bracket, the NLN equation ( t h e x-flow) and the companion time flows a r e generated by a sequence o f commuting Hamil tonians, c l o s e l y r e l a t e d t o those which generate the mu1t i p e r i o d i c f 1ows ; (3) Define and i l l u s t r a t e how t h e IMT provides a mapping from the o l d coordinates P,Px,--t o new coordinates, the monodromy data M associated w i t h the s o l u t i o n o f (1.4), which admit t r i v i a l i n t e g r a t i o n . I n t h e case considered, S i s polynomial i n 5 of degree (n-1) and so M c o n s i s t s of t h e Stokes m u l t i p l i e r s 2n {sjIjzl defined i n connection w i t h the rank n i r r e g u l a r s i n g u l a r p o i n t a t c=". (4) Prove t h a t t h e map i s canonical and o b t a i n expressions f o r the Poisson brackets o f t h e Stokes m u l t i p l i e r s . Along the way, we

( 5 ) Develop expressions f o r t h e i n f i n i t e s i m a l changes i n t h e new coordinates i n terms o f a n a t u r a l i n n e r product between t h e o l d coordinates and a s e t o f (2n-2) vectors Vk which a c t as a basis i n the space o f dependent v a r i a b l e s and which are formed from the "squared eigenfunctions". These expressions may be used t o c a l c u l a t e t h e e f f e c t s o f p e r t u r b a t i o n s on the NLN equations ( f o r example, add a p e r t u r b a t i o n t o t h e Painleve equation; does i t s t i l l r e t a i n i t s special prop e r t i e s ? (see C11)) and provides a s t a r t i n g p o i n t f o r an attempt t o prove a (Kolmogoroff, Arnold, Maser) theorem - t h e f l o w does not, i n general, take place on a compact manifold. (6) Develop expressions f o r t h e new coordinates themselves as i n n e r products between t h e o l d coordinates and Vk. (7) W r i t e expansions f o r t h e i n f i n i t e s i m a l changes i n t h e o l d coordinates and f o r the o l d coordinates themselves i n terms of the basis vectors Vk. The l a t t e r provide expressions f o r s o l u t i o n s o f the Painleve and r e l a t e d equations i n terms of contour i n t e g r a l representations. (8) The r e l a t i o n w i t h analogous expressions which a r i s e i n connection w i t h t h e i n verse s c a t t e r i n g transform are explored. A more general discussion of t h e ideas presented here, together w i t h a d d i t i o n a l m a t e r i a l on mu1 t i p e r i o d i c systems, p a r t i c l e systems and perturbed systems w i 11 be given i n a s e r i e s of forthcoming papers C51. Our work on these questions was s t i mulated by several works by Sato, Miwa, Jimbo, Mori and Ueno [61 r e l a t i n g t o t h e c o r r e l a t i o n f u n c t i o n s of e x a c t l y solvable models i n s t a t i s t i c a l mechanics and quantum f i e l d theory. I n C61, they discuss the Hamiltonian s t r u c t u r e o f deformation equations.

2.

DEFINING THE EQUATIONS

The inverse s c a t t e r i n g transform focuses p r ' n c i p a l a t t e n t i o n on (1.2) where 5 i s the eigenvalue, R a constant m a t r i x and P(x,)) a m a t r i x o f p o t e n t i a l s which evolve i n time ~=(to,tl,---tn-l) according t o ( l . l ) , j=O,--n-1. From (1.2) and (1.3), we have

67

THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION

(2. l a )

P t j - Q ( x~ ) + c ~ R + P , Q ( ~ )=I 0, j = 0,1,2,--

(j)t k ‘tj (k)+[Q(J!Q(k)] = 0, j,k = 0,1,2 where [ , 3 i s the comnutator. i n 5 o f degree j

,---

We seek s o l u t i o n s o f (2.la) f o r Q ( j ) polynomial

5 k , ( t h e Qj+l-k ( j 1 are independent o f j)

& j )= R5j +j -c1 Q ( j+l-k j) k=O

(2.lb)

(2.2)

and t h e Qj+l-k ( j1

a r e solved from the c o e f f i c i e n t s o f 5 k ( t h e p a r t o f Q!’) ~ + -k1 which does n o t comnute w i t h R) and o f gk-’ ( t h e p a r t t h a t does) i n ( 2 . l a ) (see (71). T h e ( 3 y t equation i n the sequence gives the e v o l u t i o n equation (1.1). The p a r t of Qj+l-k which commutes w i t h R i s determined up t o a constant i n x ( i t can depend on tO,tl,--tj-l)

and we take t h i s constant t o be zero w i t h o u t l o s s o f gen-

erality.

I f nonzero, we can make i t zero by a transformation i n t h e time coordinates

Example:

Define

Finite-gap s o l u t i o n s o f (1.1) are obtained by s u b j e c t i n g the vector V i n (1.2) and (1.3) t o a f u r t h e r (algebraic) c o n s t r a i n t n- 1 1 u.v .+uovx = hV 1 J ~ J

(2.5a)

or (Q-AIIV

= 0, Q = “lu.

Q(~)+U,(CR+P).

(2.5b)

1 J

To motivate t h i s choice, consider s o l u t i o n s o f (2.6) q = q(x-ct3),

x

=

x-ct3,

(2.6)

whence q(X) s a t i s f i e s 1 2 -@qxxx+3/2q qx = cqx. I n t h i s case Q(3) i s a f u n c t i o n o f 5 and X. Introduce the change o f variables X=x-ct3, T=t3 and (1.2), (1.3) w i t h j = 3 become V X = (
(2.8)

68

H. FLASCHKA. A.C. NEWELL

AT respectively. Now s e t V+e V and o b t a i n 2.5) w i t h u =1, u =c, u.=O j # 3 . Indeed the i n t e g r a b i l i t y c o n d i t i o n of (1.2) and 2.5b) f o r t i e valaes chdsen i n t h e examI n general, the i n t e g r a b i l i t y c o n d i t i o n i s p l e i s (2.7). n

From ( l . l ) , we see (2.9) i s a nonlinear o r d i n a r y d i f f e r e n t i a l equation i n x f o r the m a t r i x P. The f i n i t e dimensional s o l u t i o n manifold defined by (2.12) i s l e f t i n v a r i a n t by t h e flows (1.1). By t h i s we mean t h a t f i n i t e - g a p s o l u t i o n s considered as f u n c t i o n o f x s a t i s f y (2.9) f o r a l l values the time parameters. I t i s w e l l known (and d e t a i l s have been worked o u t i n several cases [a]) t h a t as f u n c t i o n s o f x, i t i s a completely i n t e g r a b l e autonomous Hamiltonian system. Solutions o f (2.9) (as w e l l as t h e i r time e v o l u t i o n under (1.1)) can be constructed e x p l i c i t l y ; they a r e a b e l i a n functions. They are c l o s e l y connected with t h e Riemann surface R defined by t h e polynomial r e l a t i o n I’(5,X) = 0 between 5 and X which comes about by t h e vanishing o f the determinant o f Q-11. The p o i n t s o f t h e Riemann surface parametrize V(x,t) i n (1.2), (1.3) and (2.5b). By proper normalization o f V as eigenvector o f (2.5b), one can arrange t h a t i t s a t i s f i e s (1.2) and (1.3) and a t t h e p o i n t ( s ) a t i n f i n i t y on R, V - Cl+O(i)lexp ( kx+j$Rj(k)t J.)

(2.10)

k being a l o c a l parameter. The R . ( k ) a r e polynomials, u s u a l l y t h e d i s p e r s i o n r e l a t i o n s o f t h e l i n e a r i z e d e q u a t i o i s (1.1). I n a d d i t i o n , V as f u n c t i o n o f ( < , A ) on R has poles pS which a r e independent o f x and t.. S p e c i f i c a t i o n o f t h e surface R, the poles p , the asymptotics (2.10) and of a crkrtain normalization c o n d i t i o n determines V h q u e l y . Knowing V one can, by d i f f e r e n t i a t i o n w i t h respect t o x, r e cover P from (1.2). Now l e t us t u r n t o a new c l a s s o f s o l u t i o n s defined by adding another k i n d o f cons t r a i n t on t h e f u n c t i o n V(x,t,C). T h i s time we w i l l ask i t t o s a t i s f y an o r d i n a r y d i f f e r e n t i a l equation i n 5, 5V5 = Z j t . V +xVx, J tj =

(2.1 l a )

( f jtjQ(j)+x(
(2.11 b)

Equation (2.11) has c o e f f i c i e n t s which a r e r a t i o n a l functions o f 5 ; a l l dependence on x,t. i s o n l y parametric as f a r as 5 i s concerned. The i n t e g r a b i l i t y c o n d i t i o n 3 on (2.11) and (1.2) gives us t h a t (we omit t h e to f l o w ) n (2.12) jzljtjPtj+(xP)x = O which shows t h a t

P i s a f u n c t i o n o f t h e phases

’”’ A,.,, I

-x( n hI

j = 1 ,--n-1.

(nt,,

Equation (2.12), t h e analogue o f (2.9),is an o r d i n a r y d i f f e r e n t i a l equa o f order n w i t h c o e f f i c i e n t s which depend on t If we begin a t time j. s o l u t i o n o f (2.12) and then l e t i t evolve f o r a time %-?(o) i n the flows ( l . l ) , then a t time 1, i t w i l l again s a t i s f y (2.12) w i t h t h e c o e f f i c i e n t s tly---tnevaluated a t the new time The choice o f (2.11) can be motivated w i t h t h e a i d of the example ( 2 . 4 ) which has the s e l f s i m i l a r s o l u t i o n

THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION

69

(2.13a) and f ( X ) , X = X/(3t3)1/3, 1 -(f 4

xx- 2 f

3

)

=

s a t i s f i e s t h e Painlev6 e q u a t i o n o f t h e second k i n d

Xf-v.

(2.13b)

I f we choose q t o have t h e f o r m (2.13.a) i n (1.2) and (1.3) w i t h j = 3 , then V i s a

f u n c t i o n o f x,t3 and 5 o n l y through t h e combinations X = X’(3t3)1/3,

5 = < ( 3 t 3 ) 1/3

.

Then (1.3), w i t h j = 3 , becomes (2.11b) w i t h t.=O j f 3 . I n t h e c o n t e x t o f t h e example, the i n t e g r a b i l i t y c o n d i t i o n of (2.11b) and (11.2) i s (2.13b). Another c l a s s o f s o l u t i o n s t o (1.1) can be found by adding t h e c o n s t r a i n t V

= (?

1

5

B Q(J-’)txR)V j

(2.14)

which leads t o t h e i n t e g r a b i l i t y c o n d i t i o n n (2.15) I n order t h a t = Px i s t r a n s l a t i o n . = CR,Pl i s t h e s c a l i n g f l o w and P to t1 (2.14) and (1.3) are compatible (examine t h e asymptotic behavior o f V as 5-), Equation (2.15) then i m p l i e s t h a t P(tn-l,tn-2,--t2yxyt ) i s a function 8. = j t 0 J J’ o n l y o f t h e n phases

Recall P

P = P (Tn-2, ---T

where

{T.}~-’

(2.10)

1 Jo)

a r e t h e i n t e g r a t i o n constants d e f i n e d by s o l v i n g t h e equations

J O

(2.17)

.

It i s this where w i t h o u t l o s s o f g e n e r a l i t y we can t a k e n a t system 5R+P = which we discuss when (1.2) i s t h e Zakharov-Sha“Sl’

By t a k i n g t h e c o e f f i c i e n t m a t r i x i n t h e c o n s t r a i n t e q u a t i o n t o be a more complicated combination o f r a t i o n a l f u n c t i o n s o f 5, we can o f course b u i l d more e l a b o r a t e classes o f s o l u t i o n s . I n C21, we b r i e f l y i n d i c a t e d how t o i n c l u d e a combination of s o l i t o n s and s e l f - s i m i l a r s o l u t i o n s . I n C51, we d e s c r i b e how t o i n c l u d e t h e x dependence o f t h e NLN system i n d i f f e r e n t ways. 3.

3a.

THE ZAKHAROV-SHABAT SYSTEM The equations:

Here ( 1 . 2 ) , (1.3) and t h e c o n s t r a i n t equation (2.14) a r e

I

(3.la)

(3.lb)

70

H. FLASCHKA, A.C. NEWELL

1 aj v, -a b

5vt =

(3.1~)

the i n t e g r a b i l i t y conditions f o r which are ax = rb-qc+ic, bx+2isb

3h.

2qa, cx-2icc = -2ra,

The Solutions To (3.2):

We seek s o l u t i o n s t o (3.2) polynomial i n 5.

Set

(3.5) We f i n d al=al

constant, a2 can be omitted.

The o r d i n a r y d i f f e r e n t i a l equations 1 and 6 r e s p e c t i v e l y .

(2.12) o r (2.15) are formed by terminating t h e s e r i e s a t '5 We f o l l o w the l a t t e r course and then t h e equations a r e

where bjy cj, j=Z,--n+l bj+l

= $jx-ia

a r e defined

J.q+xq6j ,ny~j+l= -$jx-iajr+r&j

The c o e f f i c i e n t s a

jy

j=3,--n,

(3.7)

are found by n o t i n g t h a t the f u n c t i o n

2

0 = a +bc-2i61Xadx

i s independent o f x.

, n , j = ~ ,---n<

(3.8) We s e t t h i s equal t o (3.9)

and equate the powers o f gntain (3.8). For L=n-2 t o L = l , t h i s equation defines i n terms o f t h e b.,c. already determined t o t h i s l e v e l from (3.7) and an+l-R J J an+l -I, i s e f f e c t i v e l y the a d d i t i v e constant associated w i t h i n t e g r a t i n g a i n (3.2). For b=O t o b=-n+2, t h i s equation determines a,+l-p. ( c ~ ~ + ~ , - - a i~n~t - h~ )s o f t h e a l ready known a t\ ,c , j s n and (x, a3,a4,--an). For 11<-n+2, we simply choose c ~ , , + ~ - ~ j'j 1 I,<-n+2 a r e zero. These choices p l a y no r o l e i n such t h a t t h e c o e f f i c i e n t s o f cnt', the l a t e r analysis. We now l i s t the f i r s t s i x c o e f f i c i e n t s .

71

THE INVERSE MONODROMY TRANSFORM ISA CANONICAL TRANSFORMATION

a

b

C

U

0

-ialq

-ial

al

-a1 --r 2 x ial 2) - i a 3 r +rxx-2qr

P X

-ia4r

--

-ia4q

i 16 1(qxxx-6qrqx)x

3c.

The Solutions o f (3.3):

i

.-a 16 1 (rxxx-6qrrx)x

We seek polynomial s o l u t i o n s t o (3.3) i n the form

I t i s simple t o v e r i f y t h a t a$]-,,

when n=j, a l =i,a k=O, k=3--j.

(3.11

bn+l-ky (j)

C(

j)

For example, a?;jI?c3?-

are simply a n+l-k, bn+l-ky $(rqx-rxq). The

cn+l-k bal-

c0

ance i n (3.3) gives the e u t i o n equations f o r q and r as functions o f t.. Because of the normalization o f ausl, the c o m p a t i b i l i t y o f (3.2 and (3.3) require3 t h a t

an+,-j

= i j t . j = 2,--n.

J

(3.12)

flow. We w i l l show very s h o r t l y how t h i s dependenn-1 ce can be reincorporated. I n essence, t h e t flow i s simp1 a restatement pf t h e NLN equations (3.6) once the p a r t i c u l a r choi& o f s c a l i n g i n ~od3t2,...tn-l 1s made. We can take n t n = l . Also we w i l l take tl=O as t h i s f l o w simply mimics the x-flow. We w i l l often f o r convenience c a l l x by tl. We are l e f t w i t h the x-flow given by We l i s t and discuss the cases n=3,4. (3.6) and the time flows qtj, rtj, j=0,2,--n-2. We take tn-l=O

n=3:

and omit t h e t

The x-flow i s

b3x = 2(a3+ix)q, c3x = -2(a3+ix)r or z(qxx-2q2r) i = zixq, Ti( r x x - a r 2 ) = 2 i x r .

(3.13)

72

H. FLASCHKA, A.C. NEWELL

The o n l y time f l o w i s qto = -2iq, rt0 = 2 i r .

(3.14)

Now l e t us comment on what t h i s s o l u t i o n has t o do w i t h s o l u t i o n s o f

-- 17(qxX-2q2r ) ,

qt2

rt2 =

1 2 -T( rxx-2qr 1.

(3.15)

Ifwe had l e f t a2#0 and indeed chosen i t t o be 2 i t 2 , then (3.13) would.read w i t h al'l, qt + 2 t q +xq = 0, rt +2t r + x r = 0 2 tl to 2 2 tl to

(3.16)

which means t h a t q and r are f u n c t i o n s o f T1 =

2

X-t2y

t + -t 2 3- X t 2 0 3 2

To

Now impose (3.17),

(3.17)

and since (3.14) holds, we have

(3.18) which when subs i t u t e d i n t o (2.15) has f and g s a t i s f y i n g (3.13) w i t h a,=i and x replaced by x - t 2 .

i

Thus i n what follows we simply ignore t h e equation f o r qt and f i n d q , r rt as functions of toyx,t2,--tn-2. Then t o incorporate tn-lnZ?mplyndplace 9

to,x,t2,---tn-2

by

T~,T~---T,,-~

which are found by i n t e g r a t i n g (3.19)

n=4

The x-flows a r e

b4x = 2(a4+ix)q, c4x = -2(a4+ix)r or

-~(qxxx-6qrqx)-ia3qx-2ixq= 0

-&rxxx-6qrqx)-ia3rx+2ixr

= 0.

(3.20)

The associated time flows a r e qto = -2iq, rt 0

q

=

$qxx-2q2r),

= 2ir

rt2 = -S(rxx-2qr 2 1.

Again we may a l s o i n c l u d e s o l u t i o n s t o

(3.21) (3.22)

THE INVERSE MONODROMY TRANSFORM ISA CANONICAL TRANSFORMATION

73

(3.23)

by 3 4 t2t; 3 q ( t o - 4+ 2 Xt3’ x+t3-t2t3,

-

3t; t2-

and 3tf: t2t; 3 r(to - -t 2 xt3, x+t3-t2t3,

-

4

3t; -+.

t2-

I n t h e corresponding p e r i o d i c problem, gV i n ( 3 . 1 ~ ) i s replaced by XV. T h i s r e moves t h e i g x term i n a, t h e f u n c t i o n R i z a2tbc and t h e i d e n t i f i c a t i o n (3.12) i s no l o n g e r necessary; t h e a . a r e constants. The Lax-Novikov equations a r e J (3.24) bnx = 2anq, cnx = -2a,r. 3d. The Hamiltonian S t r u c t u r e : of t h e flows. I n p a r t i c u l a r H . = 4antltj,

The f u n c t i o n R generates t h e H a m i l t o n i a n f o r each

j = 0,---n-2.

(3.25)

The conjugate v a r i a b l e s a r e

15~. Cnt2-j where

6

j’

j=Z,--n

C,

(3.26)

a r e t h e c o e f f i c i e n t s o f 5-jt2,

j=2,--n

i n t h e asymptotic expansions f o r

as 5-. We can i d e n t i f y t h e c o r r e c t c h o i c e o f conjugate v a r i a b l e s from t h r e e sources : (i)

they a r e t h e same as those o f t h e p e r i o d i c problem &the d e t a i l e d a n a l y s i s l e a d i n g t o these choices i s g i v e n i n C 61; ( i i ) t h e Hamiltonian H which generates t h e s c a l i n g f l o w must decompose i n t o a sum of products o f coRjugate v a r i a b l e s ; (iii) t h e choices a r e necessary i n o r d e r t o define c e r t a i n i n n e r p r o d u c t s c o r r e c t l y . For now, we w i l l simply take (3.27) as given. The Hamiltonians c o n t a i n t h e indeWe make t h e system autonomous by i n c l u d i n g these pendent v a r i a b l e s x,t2y---tn-2. as dependent v a r i a b l e s and adding conjugates T1=X,T2,--Tn-2 t h e term I X a . d x i n R j=3,--n J H . = T.+A.(S ,C ,x,t2--tn-2). J J J k k From t h i s choice we see t h a t

which a r e d e f i n e d f r o m

i n such a way t h a t

(3.28)

H. FLASCHKA, A.C. NEWELL

74

and so t . i s the "time" v a r i a b l e f o r t h e f l o w generated by J

Hj '

L e t us look i n d e t a i l a t some examples. a1 = i, a3 = 0.

n=3:

Ho = -2i(b2c3+b3c2)

b$c: H1 = - 2 i ( b 3 c 3 - T

-

xb2c2+Ixb2c2dx).

The conjugate v a r i a b l e s a r e b2,b3,

X = -2iIXb2c2dx.

x and c3,c2,

It i s easy t o v e r i f y t h a t H ,H generate the flows (3.14) and (3.13),respectively. Also,both Ho and H1 a r e con8taAts w i t h respect t o to and x.

n=4:

'

a1 = i, a3

6,

where and t h e

H1

= b2,

6,

2 i t 2 , a4 = 0.

b3, 64 = b 4

=

-

a r e d e f l ed analogously.

CIS

8

- 2 i { 6 3 ( C 4 62c2 +8-

2 b2C2

e2 b4- 8 t2b2 a3+"3

=

6:E2

+ t,E2)+?3(64~t?62)

+$2i t2+$j2E2)

( 62C,+63E2)-x62E2+IX62C2dxl

+1~(L~C~+6~E~)-2t~(6~t~t6~t~+6~E~)1. Note t h a t the l a s t term i n H2 comes from -4ia3a5 appearing i n t h e c o e f f i c i e n t of 6 2 i n (3.8). The conjugate p a i r s a r e (62,E4), (63,C3)y (ti4,C2), (~,X=-Zi/~6~E~dx), (t2,T2 = -2iIX(6,C3+6,C2)dx). I n p a r t i c u l a r t h e form o f T2 may be obtained by n o t i n g t h a t , since H1 generates the x flow, dT2 dx

-

aH

1 - - 2 i ( 6,C3+6,C2).

at2

I n t h i s way, t h e form o f Tky k=2,--n-2

can always be found from t h e corresponding H1.

THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION

Before we continue, a convention o r notation.

75

df means d i f f e r e n t i a t e a l l variables dtj af

i n f which depend on t.,including t . itself,keeping t f t ( t l = x ) constant. J J k j means we d i f f e r e n t i a t e f only w i t h respect t o e x p l i c i t t . dependence.

atj

J

by leaving o u t the contributFor the multiperiodic case, we define Ho, H --H ions from 2i<Jxadx. We a l s o leave the a4 c o n s b i t s . Then the conjugate v a r i a b l "j es are simply (3.26), the equations are the same with i x replaced by constant an. 3e.

Poisson Brackets And The Commutability of {Hj}Y:Z.

I t i s natural t o define the Poisson bracket ( t l = x y T1=X) s

(3.29) The f i r s t term i s the Poisson bracket f o r the periodic problem and we write t h i s C F y G l p . Note that i f

v

=

J =

a

(-,cy-,y-

a

a

a aC2 a62 a;,

ati3

a a$

a

~

(3.30)

Y

31-

36,

(3.31)

-TOO--------

then

[F,J

=

where i s

F

(3.32)

,

n

jg2

u j v j y the usual inner product.

Also, i f

= F(b.y?.yt.yT.)y

J

J

J

J

then -dF =

CF,Hkl.

(3.33)

dtk

In L51, we show that for each j , k = 0,--n-2, CHj,Hkl

= 0.

(3.34)

T h u s each H i s a-constant of the motion f o r each of the flows generated by the j

H. FLASCHKA, A.C. NEWELL

76

.

other H We t h e r e f o r e have ( n - 1 ) independent c o n s t a n t s o f t h e m o t i o n i n i n v o l u t i o n . t h e p e r i o d i c system, w h i r h has (2n-2) dependent v a r i a b l e s , i s t h e n completel y i n t e g r a b l e . The m u l t i p h a s e s e l f - s i m i l a r s o l u t i o n s on t h e o t h e r hand a r e 4n-4 dimensional and t h u s we need a n o t h e r ( n - 1 ) independent c o n s t a n t s of t h e m o t i o n . These new c o n s t a n t s a r e i n t r o d u c e d i n t h e n e x t s e c t i o n . The I n v e r s e Monodromy T r a n s f o r m (IMT) :

4.

Consider ( 3 . l c ) ,

g i v e n by (3.5) w i t h antl=bntl=cn+,

w i t h a,b,c,

0.

Then

5=m

i s an i r r e g u l a r s i n -

g u l a r p o i n t o f rank n and t h e fundamental s o l u t i o n m a t r i x o(xytj;c)

=

($,$I

(4.2)

admits t h e f o r m a l a s y m p t o t i c e x p a n s i o n (see Appendix I ) as 5-,

~ -= i(It'? ?)e-+ eS (4.3)

s=l5 where

e

illo n n-2 anr;+o( 1 = L i tn-2': + - - + i t .cJ+i <x+i t o + T n J

9

= O ( l ) , Ho = +4C(,l

, (4.4)

I

h w been n o r m a l i z e d s o as t o s a t i s f y ( 3 . l a ) and C 3 . l b ) . Now, i t i s known t h a t i n g e n e r a l @ w i l l n o t have t h e a s y m p t o t i c e x p a n s i o n 3 i n e v e r y neighborhood o f I;=-. T h i s neighborhood n a t u r a l l y d i v i d e s i n t o 2n equal s e c t o r s s e p a r a t e d b y rays, c a l l e d a n t i - s t o k e s l i n e s , on w h i c h Ree=Reicn = 0. D e f i n e Sk as 1c;1sI>p, E ( k - l ) i A r g g < + l 0

and Rk as Argy= :(k-l). asymptotic behavior c a y i n g whereas q.-(;)e-'

$

L e t Q=(J,,$) b e a fundamental s o l u t i o n m a t r i x o f ( 4 . 1 ) w i t h 0 i n S1, The s o l u t i o n $-(l)e i s r e c e s s i v e ( a s y m p t o t i c a l l y dei s dominant ( i t i s u n i q u e as i t i s d e f i n e d on t h e i n i t i a l

r a y c=O,where 8 i s i m a g i n a r y ) .

C o n t i n u i n g t o s e c t o r S2, t h e r e c e s s i v e s o l u t i o n $

becomes t h e dominant s o l u t i o n T2 i n t h a t s e c t o r ; however, i n s o l u t i o n J, i n S1 become t h e r e c e s s i v e s o l u t i o n $ i n S, we f a c t o r ( t h e stokes m u l t i p l i e r ) t i m e s t h e r e c e s s i 6 e s o l u t i o n mental s o l u t i o n m a t r i x Q. v h i c + has a s y m p t o t i c b e h a v i o r T Jtl preceding neighbor b y Qjtl

=

tJjMj,

o r d e r t h a t t h e dominant must add a c o n s t a n t $. I n g e n e r a l , t h e fundai n Sjtl i s r e l a t e d t o i t s

(4.5)

$. dominant, t h e n t h e nonzero o f f - d i a g o n a l element o f M . occupies t h e (1,Z) p o s i t i o n . J

J

77

THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION

We c a l l t h e s e t o f Stokes m u l t i p l i e r s { s j } &

t h e monodromy d a t a M f o r (4.1).

Since

t h e t o t a l monodromy around c=m i s t h e i d e n t i t y (as i t equals t h a t about c=O, an ord i n a r y p o i n t ) , o n l y (211-2) o f t h e s e t V a r e independent, F o r reasons o f symmetry we w i l l choose t o work w i t h t h e s e t i s k } f o r ksZ={l ,Z,---n-l , n + l Y - - 2 n - 1 ~ . One o f t h e c e n t r a l r e s u l t s o f o u r p r e v i o u s papers i s : g i v e n 0 j = l ,--2n w i t h asympjy t o t i c behav iors 3 which s a t i s f y ( 3 . l a ) , ( 3 . l b ) , ( 3 . 1 ~ )and ~ V . ( S.) d e f i n e d by (4.5),

J

J

then t h e stokes m u l t i p l i e r s a r e c o n s t a n t s , independent o f to,x,---t,-2. i s again pro v ed i n S e c t i o n 5.

T his r e s u l t

We t h e r e f o r e have a map f r o m o l d v a r i a b l e s

t o new v a r i a b l e s

new v a r i a b l e s t h e e q u a t i o n s a r e

k = 0,1,--n-2,tl

sj, tk = 0,

je2,

H j, t k =

j = 0,--n-2,

tjytk

0, = 6

jk

= x.

k = 0,1,--n-2.

j , k = 0,---n-2.

Our n e x t goal i s t o show t h a t t h i s map i s c a n o nical. I n o r d e r t o do t h i s , i t i s necessary t o express t h e i n f i n i t e s i m a l v a r i a t i o n s 6Q, 6P i n terms o f 6q, 6p. 5.

5a.

IYFINITESIMAL VARIATIONS AND THE t j DEPENDENCE o f Infinitesimal variations:

Sk.

Take t h e i n f i n i t e s i m a l v a r i a t i o n o f

and s o l v e b y v a r i a t i o n o f parameters. I n t e g r a t e t h e r e s u l t between cl, which l i e s n e a r <=- on Rl,and G2,defined s i m i l a r l y . Since @ i s e n t i r e , any p a t h w i l l do. We find

Now use (4.3) and t h e f a c t s t h a t on Rly@-i

, on

R 2 C i M;l,and

compare t h e (2, l)

ele-

ments o f (5.2) t o o b t a i n

Where we have

, the point a t

m

on R1.

We cannot s e t r,2=-2,for

neither side

70

H. FLASCHKA, A.C. NEWELL

o f (5.3) converges. (3.1):

However, we make use o f t h e f o l l c w i n g i d e n t i t i e s d e r i v e d from

(5.4a)

Also from (A.2 ) , we may show t h a t t h e arguments o f t h e r i g h t hand s i d e s o f (5.6a) and (5.6b) have asymptotic behaviors -2i52s1+O(1) and - 2 i $ s l t o ( l ) respectively.

as c2’Ko0n R2

Therefore we may w r i t e (5.5)

where da = 6A-n?2 a t . 6 t (tl=x) j =o J j ’

and db, dc a r e d e f i n e d s i m i l a r l y .

The i n t e g r a l on t h e r i g h t hand s i d e o f (5.5) has an asymptotic expansion which cons i s t s o f ( a ) terms

l i k e e-+2e5 p when pCn-1

, which

R2 and (b) a term p r o p o r t i o n a l t o l / y on R2. c a n c e l l e d by 2Sl

.&ant, Iln 2.

i s w e l l defined.

a r e i n t e g r a b l e a l o n g t h e r a y s R1 and

When i n t e g r a t e d , t h i s term i s e x a c t l y

-

Thus t h e l i m i t 5,-

along R2 may be taken and -6sl

The c a l c u l a t i o n may be repeated f o r any s e c t o r . I n t h e odd numbered s e c t o r s , one compares t h e (2,l) elements o f (5.2); f o r t h e even onescompare t h e (1,2) elements. T h i s r e s u l t s i n a change of sign. I f cR(mR) i s a p o i n t which tends t o c=m along Rk, then,

where J, =

(:;)

i s always the dominant s o l u t i o n i n Sk.

Our n e x t t a s k i s t o r e w r i t e (5.6) t a k i n g account o f t h e dependence o f aj on t h e sequence o f b ’ s and c ’ s .

L e t us b e g i n w i t h t h e case n=3.

THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION

79

(5.7) Define

'k

(C$>i

b2$1b 2 ) d c - i s kb3Rn5k+l

Def in ing J as i n (3.31) and dR = (dS2,d62,---dt D e f i n i n g J as i n (3.31) and dR = (dc2,d62,--.-dt g a t e v a r i a b l e s o f S e c t i o n 3, we have

(-1)

k

,

dSk =

where =

n

5 uivi.

n

,d6 ) T ,where n

( E . ,,66..)) a r e t h e conjuJ J (5.9)

keZ,

By r e w r i t i n g t h e i d e n t i t y

(5.10)

(Lk,tkIy we f i n d

i n terms o f

-sk = ,

k
(5.11) I n Appendix 2, we f i n d t h e o r t h o g o n a l i t y r e l a t -

where F i s t h e m a t r i x {(-l)i+l&..}.

1J

ions f o r t h e sequence of v e c t o r s V = Nkl, k c Z , Mk

=

<JV ,V > = T(Gk,&-ltSkSR) k

R

,k
(5.12)

The s e t V forms a b a s i s i n 42n-2 and t h e r e l a t i o n s (5.9), (5.11) and (5.12) a l l o w us then t o expand dR and FR i n t h e b a s i s V, L e t N=M";

and (5.14)

~

H. FLASCHKA, A.C. NEWELL

80

where i n h o t h cases, t h e sumnation i s o v e r

j,REZ.

Several remarks a r e now i n o r d e r . We can, by t h e Gram-Schmidt p r o c e s s , choose a new s e t o f v e c t o r s U=Uky ktZ,such

1.

The t r a n s f n r m a t i n n h e t w w p I 1 and V w i l l denend np t h a t <JU ,U > = 6k,L-,,k
2.

9 changes i n such a way such t h a t dR=O t h e n

n- 2 6R = -JVHo6to-JVH1 Gx-i=2JVHk6tk

t h a t i s , R changes as a l i n e a r c o m b i n a t i o n o f t h e f l o w s g e n e r a t e d by R. i f E;Sk=O, t h e n dR=O. 3.

Conversely

Notice that i f dR = -aFR

k t h e n 6sk = ( - 1 ) ask. Howeverythe r i g h t hand s i d e s t i l l i s j s u t t h e s c a l i n g flo! and i s p r o p o r t i o n a l t o JVHo. The t i m e t dependence (6= d/dt say) of sk ( = s k ( t = O ) e (-1) at can always be removed by c o r r e c t l y n o r m a l i z i n g t h e a s y m p t o t i c expansion ( 4 . 4 ) . 4.

E q u a t i o n (5.9) i s t h e s t a r t i n g p o i n t f o r a p e r t u r b a t i o n t h e o r y . may c a l c u l a t e how a p e r t u r b a t i o n

Using i t we

dR = EF(R), o < ~ < < l ,

(5.15)

a f f e c t s t h e Stokes m u l t i p l i e r s . I n p a r t i c u l a r , (5.15) may n o l o n g e r have t h e s p e c i a l p r o p e r t y t h a t as f u n c t i o n s o f t h e phases, t h e o n l y moving s i n g u l a r i t i e s o f q , r a r e p o l e s . How i s t h i s r e f l e c t e d i n t h e change o f S.? More g e n e r a l l y , i t i s v e r y naJ t u r a l t o ask i f an analogue t o t h e KAM ( K o l m o g o r o f f , A r n o l d , Moser) theorem o b t a i n s . The f i n i t e d i m m s i o n a l s o l u t i o n m a n i f o l d f o r t h e s e f l o w s i s n o t n e c e s s a r i l y compact, i s n o t a t o r u s and so t h e KAM theorem does n o t d i r e c t l y a p p l y . The p o t e n t i a l connect i o n between a p o s s i b l e p r e s e r v a t i o n o f t h e s o l u t i o n m a n i f o l d and t h e p r e s e r v a t i o n o f t h e P a i n l e v C p r o p e r t y i s an i n t r i g u i n g one. Also,what w o u l d be t h e analogue o f t h e t o r i w h i c h a r e n o t preserved? A r e t h e r e resonances? The o n l y f r e q u e n c i e s w h i c h appear i n t h e u n p e r t u r b e d problem a r e c o n s t a n t s , e i t h e r z e r o o r one. We now r e t u r n b r i e f l y t o some f u r t h e r d e t a i l s i n t h e d e r i v a t i o n o f (5.91. g e n e r a l , does t h e e x p r e s s i o n

Why, i n

E = 1/ (2da$l$2-db$2+dc$1) 2 2

5 d6 7 The answer i s t h a t t h e b a s i s j y j' v e c t o r s Vk must be c a r e f u l l y c o n s t r u c t e d i n o r d e r t h a t t h e i r i n n e r p r o d u c t s can be c a l c u l a t e d . F o r example, when n=4, i g n o r i n g t h e terms a r i s i n g f r o m 6 t j' 2 2 2 2 E = d c 2 ( ~$2+ib2~$1$2+ih3$1$2). db2(< $ 1 - i c 2 ~ $ 1 $ 2 - i ~ 3 $ 1 + 2 )

n a t u r a l l y i n v o l v e t h e c o n j u g a t e v a r i a b l e s dc

-

+

d c 3 ( 5 ~ : t i b 2 J 1 1 $ 2 ) - d b ~ ( C $ ~ ~ i c 2 ~ l $ 2 1 + d c 4 $ ~ d b q $ ~I .f One were t o d e f i n e Vk by

THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION

81

T and JVky t h e f orw r i t i n g E as an i n n e r p r o d u c t between (dc2,db2,dc3,db3,dc4,db4) mula (A.14 of Appendix 2 would n o t a p p l y , I t t u r n s o u t u e m l r s t add 1 2 1 2 2 1 1 2 2 -la3$; - $2$2 and -?a3$, jjc2q1 t o t h e c o e f f i c i e n t s of dc2 and -db2. S u b t r a c t i n g

-

these terms from t h e r e s t o f t h e e x p r e s s i o n E leads t o a r e d e f i n i t i o n o f b 4 and c 4 A l s o t h e r e are terms p r o p o r t i o n a l t o 6 t l e f t over which combine i n t o 6, and j j u s t t h e c o r r e c t way t o make (5.9) h o l d . I t now a l s o becomes c l e a r why anti must f a c t o r i n t o a sum o f p r o d u c t s o f t h e c o n j u g a t e v a r i a b l e s . These terms cancel t h e p o t e n t i a l l o g a r i t h m i c divergences. For n=4, t h e b a s i s v e c t o r s a r e

c4.

(5.16)

Vk=

S

k+l d,gJ2-i 2 Skc2!2nCk+

mk

R e c a l l t h a t t h e v e c t o r $ = (q1&$

T

i s t h e dominant s o l u t i o n i n s e c t o r Sk,

5b. Poisson b r a c k e t s o f t h e Stokes m u l t i p l i e r s . We have a l r e a d y d e f i n e d t h e Poisson b r a c k e t i n (3.29). depend on 6X, 6T2,---6Tn-2,

Since 6Sk does n o t

we have

(5.17)

= -, = - < ( - l ) k J V k , ( - l ] nJ. 2V>,

frm (5.91,

82

H. FLASCHKA, A.C. NEWELL

Then t h e ( k , R ) element o f P = {CskysRl1, t h e m a t r i x o f Poisson b r a c k e t s , T k+n. i s (-l)k t %k e. D e f i n e N=M-"; t h e n t h e (k,!t) element o f (P-') i s (-1) .Mkk 5c. P r e s e r v a t i o n o f t h e s y m p l e c t i c form. We now c a l c u l a t e t h e s y m p l e c t i c form

(5.18)

i n terms o f t h e i n f i n i t e s i m a l w r i a t i o n s o f t h e new c o o r d i n a t e s

Sk,6x,6t

First, recall that

6R

n- 2 dR-JVH 6 t -JVH16X- C JVHk6tk. 0 0 2

6Hl,6H.

jy

J'

(5.19)

Second, observe t h a t s i n c e

(5.20)

6T. = 6t' - t (
(5.21)

j'

I n (5.20), (5.21) Hlx e x p l i c i t x.

means

aH1 ax, the

A l s o i n b o t h 6X,6Tj,

<JVHo,VHk> = [Hk,Hol appears anywhere).

partial

a r e o f t h e form t h e terms p r o p o r t i o n a l t o &to

= 0 ( s i n c e Ho i s independent o f tk and to never e x p l i c i t l y

I n these e q u a t i o n s we have a l s o r e p l a c e d VA,,

Hkx s i n c e t h es e a r e r e s p e c t i v e l y e q u a l .

= -. w =<- dlR,Jd2R>

d e r i v a t i v e o f H1 w i t h r e s p e c t t o

Akx

by VHk and

T A l s o we r e c a l l t h a t s i n c e J =J,<Ju,v>=

A f t e r a l i t t l e c a l c u l a t i o n we now f i n d , n-2 + 6x"6+H1 + 6t.A~H j* J

5

The terms p r o p o r t i o n a l t o 8 ( A 6 t . and 6 t . A 6 t k have as c o e f f i c i e n t s CHj,H1l J J r e s p e c t i v e l y , w hic h a r e zero.

(5.22) and CH H k'

3

83

THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION

But from (5.13), -

=

-

F,(,

= j c,ReZ (

-1 ) j b J L j 6 p j V e >

Nj,,G1SL62~j from

(5.91,

= ($S)T =

where P is t h e m a t r i x o f Poisson brackets.

By a j u d i c i o u s choice o f b a s i s U guch

t h a t <JUkyU,,> = 6 k , R - 1 , k
=

such t h a t

n-2

)A6gj(Sk) j-1 6f.(S J k

-+ 6x~6H, t

6 t A Hj .

(5.23)

t h e only v a r i a b l e s which depend on x,ti

I n (5.23), themselves. = tk = - aHk aHk

a r e t h e v a r i a b l e s x,ti J

For t h e tk f l o w (tlZx)

fj,gj,H1

j=2,--n-2 J

,Hk,k=2,--n-2

are c o n s t a n t and

1.

5d. F o u r i e r expansions and contour i n t e g r a l r e p r e s e n t a t i o n s . We focus on t h e case n=3 and take to=O. qx,-2q 2 r = xq ,rxx-2qr2=rxr.

The equations (3.13) a r e 5.24)

From (5.141, FR =-Zj

5.25)

,LNkjS jVn. Y

w i t h V g i v e n b y (5.8). I n t h i s case, t h e r e a r e s i x s e c t o r s a t 5=m. We w i l l choose V as being formed from s e c t o r s 1,2, 4 and 5 and t h e r e l e v a n t Stokes m u l t i p l i e r s are S 1 , S 2 , S 4 , s 5 . In 1 =

nllts,s2) ( lts4s5$--

l+SlS2

and (ccl=i)

0

84

F'

H. FLASCHKA, A.C. NEWEL1

& -1

-b3

Case 1.

S

S

l t S

5 I t s 1 s 2 v2t1+s 4 s5 v 4

s

1 2

S

4 I t s s v5. 4 5

(5.26)

The connection w i t h t h e squared e i g e n f u n c t i o n expansions o f i n v e r s e scattering,.

O r i g i n a l l y we were l o o k i n g f o r a f o u r parameter (complex) f a m i l y o f s o l u t i o n s f o r L e t us l o o k a t t h e two parameter f a m i l y formed by s e t t i n g S2=s5=0. Then, (5.24).

(5.27)

where $ r e f e r s t o t h e u n i q u e l y d e f i n e d dominant s o l u t i o n i n each o f t h e Sectors 2 and 5. B u t the J, i n S2 i s s i m p l y t h e r e c e s s i v e o l u t i o n i n S1 and, s i n c e s2=0,

37

Hence $e

a l s o t h e r e c e s s i v e s o l u t i o n i n S3.

i s a s o l u t i o n o f b o t h ( 3 . l a ) and

as c- i n ImC>O. ( 3 . 1 ~ ) which i s a n a l y t i c f o r I m p 0 and asymptotes t o i s p r e c i s e l y t h e s o l u t i o n JI we d e f i n e d i n papers r 9 , l O l i n connection w i t h t h e 3/ S i m i l a r l y t h e s o l u t i o n +e " t e r i n g problem ( 3 . l a ) . o f S5 i s what we c a l l e d C91 and C101. Now s i n c e t h e $ o f S2(S5) i s r e c e s s i v e i n S1,S3 (S4,S6), we may

(;p

t e n d t h e end p o i n t s o f the i n t e g r a t i o n paths i n (5.32) t o a x i s . Thus (5.27) becomes

(-:I

I

m

=

-m

%3 ele3 2

f:( Id< +

which i s p r e c i s e l y equation

6.55

1 -m

--

and

03

This scat-

$ in ex-

on t h e r e a l

s433

5

(5.28)

-i-

o f reference C l O l with (5.29)

But i f we do d e f i n e Wa, 6 / ; i n t h e usual way f o r (3.la),

and so t h e Stokes m u l t i p l i e r s s1 and s4 a r e s i m p l y b/,(c=O) ively.

We showed i n [ll,t h a t i f r=q, s4=s1.

6(5)=-b(-c)

and a(c)=a(-s) which i s c o n s i s t e n t .

then ( 3 . 1 ~ ) shows t h a t

and -b-/a(r=O)

respect-

We a l s o know i n t h i s case t h a t

THE INVERSE MONODROMY TRANSFORM I S A CANONICAL TRANSFORMATION

Case 2.

Contour I n t e g r a l r e p r e s e n t a t i o n s o f P a i n l e v 6 f u n c t i o n s .

Again we l e t r = q .

1; A)$(-<)

S4(S5) i s

(2).

a5

From p r e v i o u s work [ I 1 we know s 4 = s , , s5=s2 and t h e Q of where t h e l a t t e r r e f e r s t o t h e r e c e s s i v e s o l u t i o n o f s e c t o r 1

Changing t h e i n t e g r a t i o n v a r i a b l e i n V 4 and V 5 of (5.26), we o b t a i n

(5.31)

where +,($) a r e t h e dominant ( r e c e s s i v e ) s o l u t i o n s o f S1. Now t h e c o n t o u r o f t h e f i r s t i n t e g r a l can be extended back t o m6 as 6 i s r e c e s s i v e i n S 6 . S i m i l a r l y , we can s e t t h e

m2

o f t h e second i n t e g r a l t o

wl.

Now t h e c o n t o u r s a r e t h e same as

those used i n t h e i n t e g r a l d e f i n i t i o n s o f Airy f u n c t i o n s . I n p a r t i c u l a r , i f s2=0 which we have shown i s e q u i v a l e n t t o p i c k i n g t h a t c l a s s o f s o l u t i o n s of (5.24) which decay a t x=+m and admit c o n s t r u c t i o n by i n v e r s e s c a t t e r i n g , (5.32) I n t h e l i m i t of s m a l l amplitudes,

and (5.

is m

e 2 i 5x+2i c3/3 d5

q = -s,[

‘I-m

which i s

Ai(”4)

-S,T

These r e p r e s e n t a t i o n s w i l l be v e r y i m p o r t a n t when we t r e a t t h e e q u i v a l e n c e between P a i n l e v 6 e q u a t i o n s and p a r t i c l e systems w h i c h p a r a l l e l s t h e analogy between t h e f i n i t e gap s o l u t i o n s and p a r t i c l e systems C51. Appendix I :

A s y m p t o t i c Expansions.

We w r i t e e q u a t i o n ( 3 . 1 ~ ) i n t h e form V = -(-aYl+bY2+cY3)V 1 5 5 i n t h e n o t a t i o n (2.3). Let

v

(I +

F +)e-eY1-+

m c

with

e5

=

u15n-1tt3~n-’3t.. .+(tn+ix)

+2 5 ntl+---

H. FLASCHKA. A.C. NEWELL

86

'n+3

'53

$3 =

----.

(A.4)

I n (A.3), t h e i n t e g r a t i o n constant i s it,, show t h a t f o r k n ck = where

i n (A.4) i t i s zero.

By induction, we

- ~(tjk+.lY2-tk+lY3) 1

fik+l, tk+-,are

(A.5)

defined r e c u r s i v e l y as

S2=b2 ,63=b3 ,c2=t2 ,t3=c3. A1 so

hntl-J

0.

= antl-j

j=Z,---n.

(A.7)

A f t e r t h i s stage, t h e terms from d i f f e r e n t i a t i n g t h e bracket term on t h e r i g h t hand side o f (A.2) enters and d i s r u p t s t h e pattern.

I n p a r t i c u l a r we o f t e n have occasion t o need t o c a l c u l a t e $1$2 on R1 and R2 as 5-.

On R,,

$-(o)e 1

, and

thus $1$2-0.

On Rp,

however, $ - ( ~ ) e e - s 1 ( ~ ) e - ' and thus

I n order t o c a l c u l a t e t h e behavior o f t h e components o f Vk, we need t o $l$2--s1. keep more terms i n t h e expanslons. Appendlx 11: The i n n e r product (5.12). We w i l l i l l u s t r a t e t h e means o f p r o o f i n t h e case n=3.

L e t us examine

$, the recessive s o l u t i o n o f S1, i s the domiWe take 5; t o be c l o s e r t o I;=- than c2 on R2. The product i n i n (A.9) has three sets o f terms. F i r s t , the terms i n products o f logorithms cancel.

$ i s the dominant s o l u t i o n o f S1 and

nant s o l u t i o n o f S2.

+ c3 are

Next, the t e n s proportional t o +iS2110g

which i n t h e l i m i t o f dix I ) .

c1,c2 tending t o

~ = m on

R1,R2

r e s p e c t i v e l y i s -sl

S i m i l a r l y t h e terms proportional t o - i s l f i n 5 2 are -s2.

+

(use Appen-

Hence these two

Next we examine t h e i n t e g r a l terms which

terms together c o n t r i b u t e -iS, S2fin53/ 52' may be w r i t t e n

(A.ll)

-ic2($$j l$iJi i -2$

$2) I

where 5 l i e s i n the path between C1,C2

+ +

and '5 on the p a t h between C2,C3.

Now the

integral i s

( A . 12)

H. FLASCHKA, A.C. NEWELL

00

I f H,B,C

We have an i d e n t i t y :

satisfy (A.13)

A 5 = KtyB, B6t2& = 2 6 , C 5 - 2 d = 2YA s a t i s f y t h e same equations w i t h 5 replaced by f , then

and AC,BC,C’

(@-a,) ( BC’-B’C)

t

d (d t 7)(2AA

=

d5

( B-B’)(A’c-AC’) A

t

(Y-Y’) (A-B-AB‘)

.

.

(A.14)

-B C-BC ) .

d5

NOW, i n our case A=$l$2y

2 2 B=Q1, C=$2y a&

5’

f3zb’<

y=c!

5

and thus ( A . l l ) i s

(A.15)

where W(535.)

=

-

($1(6NJ2(6’) - l J 2 ( 6 ) $ l ( 5 v 2 .

I n t e g r a t i n g we o b t a i n (A.11) i s

-;

t

J62 w(csc3 51

+

5-c3

t

w(c,ch

dt,

+

5- c2

Now, we take the path between

-

jz:

2

c1

and

W(52,C’)--

W(S1

35.1

dc’

.

(A.16)

q-6’

52-5’

c2 t o be i n along

R1 t o the 5=0 and o u t along

R2. To i n t e g r a t e the t hti r d i n t e g r a l , we recognize t h a t 5, i s recessive i n S1 and t o Arg 5=0, then i n along R1 and o u t R3 b r i n g t h e contour from c2 from Arg 5 t

to 6

.

This means we must take account o f t h e p o l e a t f = S 2

Res,

w(62.60

5”G2

C2-5‘

= -W(S2y62) = -1,

and since

since $1$2-$1$2 i s t h e Wronskian.

Hence from

the t h i r d i n t e q r a l we o b t a i n t h e e x t r a term - 1 / 2 ( - 2 n i ) ( - l l = 71. Now on the v e u t r a l rays, t h e o n l y terms which w i l l c o n t r i b u t e as 6 1 , 6 2 . 6 ~ , 5 ~ t e n d t o i n f i n i t y a r e those along rays where W(5,C’) tends t o a constant. This o n l y happens when 5 l i e s on R2,

C’on

R3 whence W(5.6++2S1S2,

and thus the remaining terms i n (A.16) tend asympto-

I t i s easy t o see t h a t i f S and S (k
contiguous sectors, the c o n t r i -

THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION

09

(A. 17) This formula a l s o holds f o r a l l n>3. Acknowledgements: The authors g r a t e f u l l y acknowledge support f o r t h i s work Grant #DAAG29-81-K-0025, N00014-76-C-0867 and MCS-7903498-01.

References [11 Flaschka, H. and Newell, A.C. , Monodromy and Spectrum Preserving Deformation I , Comm. Math. Phys. 76 (1980) G5-116. [21 Flaschka, H. and Newell, A.C. , Multiphase S i m i l a r i t y Solutions o f I n t e r a b l e Evolution Equations, Lectures i n Pure and Applied Mathematics 54 (19807 373395. Also, Proc. Kiev Conference on Solitons, September 1979, t o appear Physica D July, 1981. C31 Lax, P.D., Periodic Solutions o f t h e Korteweg deVries equation, Lectures i n Applied Math. 15 (1974) 85-96. C41 Novikov, S.P., The Periodic Problem f o r t h e Korteweg deVries Equation, Funkts. Anal. Prilozen 8 (1974) 54-66. 153 Flaschka, H. and Newell , A.C. , Monodromy and Spectrum Preserving Deformations 11, 111. I n preparation. C61 Newell , A.C. , The General Structure o f I n t e g r a b l e Evolution Equations, Proc. ROY. SCO. A 365, (1979) 283-311. x ,T., Mori, Y. and Sato, J . , Density Matrix o f an Inpenetrable [71 Jimbo, M., M Bose Gas and t h e F i f t h Painlevf? Transcendent, Physica g (1980) 80-158. C81 Gel'fand, I.M. and D i k i i , L.A., I n t e g r a b l e Nonlinear Equations and t h e (1979) 8-20. . L i o u v i l l e Theorem, Funkt. Analiz P r i l o z . C91 Ablowitz, M.J., Kaup, D.J., Newell, A.C. and Segur, H., The Inverse Scattering Transform, Stud. Appl. Math. 2, (1974) 249. C l O l Newell, A.C., The Inverse Scattering Transform, Topics i n Current Physics (1980) 177-242. Solltons, Eds. R. Bullough and P. Caudrey, Springer-Verlag.

1,

1.

x,

This Page Intentionally Left Blank

Nonlinear Problems: Present and Future Bishop,D.K. Campbell,B. Nicolaenko (eds.) 0 North-HollandPublishing Company,1982 A.R.

91

NUMERICAL METHODS FOR NONLINEAR DIFFERENTIAL EQUATIONS James M. Hyman Center for Nonlinear Studies Theoretical Division, MS 610 Los Alamos National Laboratory Los Alamos, New Mexico 87544 U.S.A.

New and better methods for the numerical solution of partial differential equations are being developed at an ever-in-creasing rate. In this paper, directed to scientists trained in mathematics but not necessarily in numerical analysis, we try to unify and simplify the underlying crucial points in this development. Most of the new methods can be understood and classified according to how space, time, and boundary conditions are discretized and by how nonlinear algebraic equations that arise in the solution process are solved. We will discuss each point and present numerical examples showing how a simple linear analysis can fail. INTRODUCTION A numerical algorithm for the solution of nonlinear partial differential equations (PDEs) can be a highly complicated and problem-dependent process. A method developed for a particular test problem may not work for similar problems. Methods that work well in one space dimension may not be easily extended to two or three dimensions. Linear analysis can rarely ensure accuracy with nonlinear methods or highly nonlinear equations. Each year significant and powerful algorithms are discovered, but most of these methods have a similar underlying structure. To better predict when a method, which is almost always developed for relatively simple test problems, will extend to more complicated situations, we must understand this underlying structure. First, we will simplify the structure of these methods to study the general flow of the algorithms, understand their similar patterns of interconnections, and unify them in a single theory. In this synthesis, new properties and common features among seemingly different methods sometimes emerge that were not evident when analyzing a specific method for a specific set of equations. We do not imply that methods tailored to specific equations should be abandoned. Many excellent methods have' come from specialized analysis of a specific set of equations. But, by understanding the general patterns found in all the methods, we may gain a better view of how and why the algorithms work as they do. The prototype system of PDEs studied here can be written as ut = f(X,t,U),

u(x,o)

= uo

,

(1)

where the solution u(x,t) lies in some function space, x is in some domain fl, and f is a nonlinear differential operator. We use the notation ut and u to represent partial differentiation with respect to time and space.

92

J.M. HYMAN

On the boundary of R, the solution is constrained to satisfy the boundary condition b(x,t,u(x,t))

= 0, x

E

aR

,

(2)

where b is a nonlinear spatial differential operator. A discrete numerical method approximates u by an element U in some finite dimensional space whose components are the values of u at a discrete set of

mesh points. The differential operators f and b are replaced by discrete operations F and B operating on U. The discretized approximation to Eqs. (1) and (2) is a constrained system of ordinary differential equations (ODES), Ut = F(U),

B(U)

= 0

,

(3)

which are then integrated numerically. Evaluating F(U) and integrating Eq. ( 3 ) often requires solving large sparse systems of algebraic equations. The methods used to solve these equations often determine the success or failure of any numerical approach. This paper is organized so that the crucial choices for methods to discretize space, boundary conditions, time, and methods to solve the algebraic systems are analyzed independently. The numerical examples are in only one space dimension for simplicity. We avoid taking undue advantage of this or any linearity that may be present so that the general conclusions may be extended to more complicated problems in higher dimensions. SPACE DISCRETIZATION The numerical approximation of the spatial derivatives and the distribution of the mesh points determine how well the spatial operator f(u) and the solution u will be approximated. We describe some typical methods to approximate spatial derivatives and then we describe how the errors in a calculation are related to the order of accuracy of the method. Often, important properties of the solution behavior originate at the boundary and the numerical differentiation procedure must take the boundary conditions into account. For simplicity, we will not incorporate the influence of the boundary conditions into the discrete operator until the next section. Numerical Differentiation The guiding principle in choosing a numerical method to approximate the spatial operator is that the resulting discrete model should retain as closely as possible all the crucial properties of the original differential operator. For instance, for a hyperbolic PDE, the operator f is antisymmetric, so we try to approximate f by an antisymmetric discrete operator F. For a parabolic PDE when f is dissipative, we approximate f with a dissipative discrete operator. If f is in conservation form, we also choose a conservation form of F. All spatial differentiation methods we describe follow the same algorithmic flow. At time t during a calculation, we are given the approximate solution vector U at a discrete set of mesh points X and must generate a numerical approximation F(U) of f(u) at these mesh points. When f(u) is a nonlinear spatial operator, it will have terms such as g(u,x,t) or [s(u,x) g(u,x,t) ] . x x

NUMERICAL METHODS FOR NONLINEAR DIFFERENTIAL EQUATIONS

First, all nonlinear functions are evaluated to generate, say, the vectors G and S . Next, G, X, and S are input for a black-box-type subroutine package in the computer library. This subroutine then returns the approximate vectors G and (SG ) of derivatives at the mesh points as output. x x Thus, the spatial differentiation is totally divorced from the nonlinearities of the PDE. This modularity also reduces the redundancy of programming the same approximation to the spatial derivatives each time they appear in an equation. These differentiation routines are debugged and optimized for a particular machine only once--with no specific PDE in mind. The numerical differentiation methods we describe fall into three categories: finite difference methods, implicit methods, and transformation methods. The simplest and oldest general technique, the finite difference method, is described first. Finite Differences

In a finite difference approximation, the function g is expanded at mesh points near x. in a Taylor series about xi. These expansions are added and subtracted to givk

ax

I G (xi)

x

where bx = +(xi+l

= -Gi+2

-

+ 8Gi+l

-

12Ax

8Gi-1 + Gi-2

+ 0(h4)

xi-l) and

Here O ( A x ’ ) stands for a quantity bounded by some constant times Ax’ as Ax goes to zero. The formula f o r (sg ) uses the harmonic rather than the arithmetic to insure fl?!xXcontinuity in the solution 111. This crucial average for S . property of differential operator should be retained in the discrete approximation. The vector G of derivative values can be written as G a banded sparse m&rix.

= DG. Here, D is

Implicit Methods The implicit or Pad6 methods are a generalization of the finite difference methods. Equation ( 4 ) can be rewritten as

93

94

J. M. H YMAN

Approximating the second derivative operation with Eq. ( 6 ) and using matrix notation, we have

D2Gx = DIG +

4

O(Ax )

.

Here, D and D1 are tridiagonal matrices generated by Eqs. ( 4 ) and ( 6 ) . similar 2procedure can be used to approximate g xx'

A

Often, time discretization methods require solving implicitly defined equations such as Eq. ( 9 ) . In special cases the implicitly defined space and time systems can be solved simultaneously leading to an exceptionally high-order efficient method. These are called operator compact implicit methods [2]. Transformation Methods

In the transformation or pseudospectral method, G is first mapped by a transformation of the form m g = TG = 1 a.$.(x) j=1 J J into nondiscrete function space. The-lbasis functions 0.(x) and the transformation are chosen so that T and T are fast (5 m log ;b operations), and so that differentiation D is simple in the transform space. The derivative approximation can then be written as

G~ = T-~DTG

(11)

Some common transforms are based on the fast Fourier transform where the 0. are trigonometric functions, Tchebyshev, or Legendre polynomials. Choosing the $i as piecewise polynomials with compact support, such as the B splines, is another good choice. Often the transformation is chosen to incorporate some crucial property such as the periodicity or symmetry of g into the calculation. This can greatly improve the accuracy of G X'

Grid Resolution The accuracy of the numerical solution is largely determined by how well F(U) approximates f(u) and how well U resolves the solution u. These errors depend on the accuracy of the derivative approximation and the algorithm used to locally refine the mesh. Because the accuracy of the method strongly influences the grid needed to resolve the solution, we first describe these relationships. Accuracy Discretizing with a highly accurate difference scheme keeps the computer time and storage to a reasonable leveI in multidimensional calculations. The analysis is linear, but in practice, qualitative results usually hold for nonlinear problems. Usually, the best we can do for nonlinear equations is to seek out basic relationships for linear equations that will be stable to small perturbations. These are the results that are most often retained. The truncation error in (3) is the amount by which the mesh point values of the true solution to the differential equation (1) fail to satisfy the difference

NUMERICAL METHODS FOR NONLINEAR DIFFERENTIAL EQUA TIONS

e q u a t i o n (3). These e r r o r s a r e t y p i c a l l y O(AxJ) i n s i z e and j i s c a l l e d t h e o r d e r of t h e method. For any f i x e d accuracy c r i t e r i a , t h e number o f mesh p o i n t s M. needed i n a j t h - o r d e r l i n e a r c a l c u l a t i o n i s r e l a t e d t o t h e number of mesh p o i n t s needed by o t h e r methods according t o t h e r e l a t i o n s h i p

M 1 = C 2M22 = C 4M 4 = C M6 4 6 6

*

,

For t h e p e r i o d i c u n i d i r e c t i o n a l wave e q u a t i o n u = vu t h e phase e r r o r i n t r o duced w i l l be t h e same u s i n g second-, f o u r t h - , t o r s?xth-order d i f f e r e n c e s i f t h e number o f mesh p o i n t s i n t h e c a l c u l a t i o n s s a t i s f y [ 3 ]

M2

3

2 0.36 M4

3 0.12 M6

.

Table 1 compares t h e number o f p o i n t s p e r wavelength n e c e s s a r y t o o b t a i n a given phase e r r o r e i n t h e k t h F o u r i e r mode o f t h e p e r i o d i c s o l u t i o n t o ut = vu a t t i m e t u s i n g second-, f o u r t h - , and s i x t h - o r d e r s p a t i a l c e n t e r e d d i f f e r e k e s i n (3). 2nd o r d e r M2

4 t h o r d e r M4

6 t h o r d e r M6

4 5

3 4 5

4 8 16 32 64 128 256

7

Table 1.

10

7

14 19 27

8

2.6 0.65 0.16 0.04 0.01 0.0025 0.0006

10 13

P o i n t s p e r wavelength f o r second-, e n c e s t o have t h e same a c c u r a c y .

Accuracy e l (vkt)

fourth-,

and s i x t h - o r d e r d i f f e r -

I n a c a l c u l a t i o n where the s o l u t i o n c o n t a i n s many d i f f e r e n t f r e q u e n c i e s , t h e high modes (2-5 p o i n t s p e r wavelength) are approximated e q u a l l y p o o r l y w i t h a l l t h e methods. The middle modes (6-16 p o i n t s p e r wavelength) a r e computed much more a c c u r a t e l y w i t h t h e f o u r t h - and s i x t h - o r d e r d i f f e r e n c e s t h a n w i t h t h e second-order method. The s i x t h - o r d e r d i f f e r e n c e s are more a c c u r a t e f o r t h e lower modes t h a n e i t h e r second- o r f o u r t h - o r d e r d i f f e r e n c e s , b u t t h i s g a i n i s o f t e n l o s t because o f e r r o r s i n t r o d u c e d i n t h e approximation o f t h e boundary conditions. The r e l a t i o n s h i p o f t h e a c c u r a c i e s of t h e d i f f e r e n t methods compared t o t h e number of p o i n t s p e r wavelength i s even more i m p r e s s i v e i n h i g h e r dimensions. I n two space dimensions, t h e number i n Table 1 s h o u l d b e s q u a r e d , and i n t h r e e dimensions, cubed. The corresponding r e l a t i o n s h i p f o r t h e damping e r r o r i n Ut = Uxx f o u r t h - o r d e r f i n i t e d i f f e r e n c e s is [3]

M 2 = 0 . 4 4 M42

f o r second- v s

.

The q u a l i t a t i v e r e s u l t s from t h i s l i n e a r a n a l y s i s h o l d t r u e f o r many n o n l i n e a r e q u a t i o n s , a s shown i n F i g . 1 where t h e d e n s i t y i s p l o t t e d i n t o s o l u t i o n s of

95

J. M. H YMAN

96

the Euler equations of gas dynamics for a Riemann problem. (A complete description of these calculations is in Ref. 4.) Second-order finite differences were used in Fig. la and fourth-order in Fig. lb, otherwise they are the same. Note that the higher order differences resolve the solution better.

ROrltoOD" .m

-

cmmct dlKO"l\WY

05'

I

-

K

-

OO

E

j

I

Figure la Second-order

-

OO

Figure lb Fourth-order

Figure 1 Numerical Solutions to the Euler Equations for the Riemann Problem on a Grid of 50 Mesh Points The oscillations near the shock are caused by errors in the high frequencies where all the methods do poorly. To efficiently resolve the gradients in this problem, it would be best to adaptively refine the mesh around the discontinuities [ S ] . These methods either continuously redistribute a fixed number of mesh points to approximate the solution in some near optimal way as the solution changes in time [6] or they add and remove mesh points as needed to minimize the work required to approximate the solution within some prescribed accuracy [ 7 ] . This is done by equidistributing a mesh function that reflects the errors in a calculation on a given grid. BOUNDARY CONDITIONS Before calculating the solution to any differential equation, boundary conditions should be consistent to form a well-posed problem. A numerical method cannot generate reasonable results for a problem that does not have a well-defined reasonable solution. The importance of proper boundary conditions cannot be overstressed: boundary conditions exert one of the strongest influences on the behavior of the solution. Also, the errors introduced into the calculation from improper boundary conditions persist even as the mesh spacing tends to zero. A common error in prescribing boundary conditions for hyperbolic equations is

to over- or underspecify the number of boundary conditions. Overspecification usually causes nonsmooth solutions with mesh oscillations near the boundary. Underspecification does not ensure a unique solution, and the numerical solution may tend to wander in steady-state calculations. In either case, the results are not accurate and one should be skeptical of even the qualitative behavior of the solution. It should be noted that the way in which boundary conditions are specified for the difference equations can change a well-posed continuous problem into an ill-posed (unstable) discrete problem. However, our experience with the suggested implementations of following technique has been favorable.

NUMERICAL METHODS FOR NONLINEAR DIFFERENTIAL EOUATIONS

97

Two of the most common methods used to incorporate boundary conditions into discrete equations are the extrapolation and uncentered difference methods [ 4 ] . Extrapolation Method In this method, the domain of the problem is extended and the solution is extrapolated to fictitious points outside the integration region. The nonphysical solution at these points is defined so that the discrete equations are consistent with as many relationships as can be derived from the boundary conditions and differential equations. The extrapolation formula can do this best by incorporating the discrete boundary conditions [Eq. ( 3 ) ] into the extrapolant. Additional relations can be generated by differentiating Eq. ( 2 ) with respect to time, replacing all time derivatives by space derivatives using Eq. (l), and discretizing the resulting equations. For example, use a centered five-point formula to approximate the spatial derivative in the equation

with the boundary conditions B(u(x,t),t)

= 0, x = 0 , l

.

(16)

The five-point formula at x = Ax uses the nonexistent mesh point x - ~= -Ax. 1 is . constructed explicitly with an extrapolation The value U(x- ) needed here, formula as foliows; Eq. (16) is differentiated with respect to time to give aB

au - (U(O,t),t)Ut

+

aB

(U(O,t),t)

= 0

Discretizing Eq. (17) with Eq. (6) and rearranging yields the extrapolation formula

4

U-l = known data + O ( A x )

.

Uncentered Differences The second approach is to extend the number of boundary conditions so that all components of the solution are defined at the boundary. Again, these additional boundary conditions must be consistent with the original problem and as many relationships as can be derived from it. An uncentered difference approximation is then used to approximate the spatial derivatives at the mesh points nearest the boundary. This method is described for the linear hyperbolic system of M equations

98

J.M. HYMAN

with the boundary conditions su

= b(t),

x = x

.

(20)

Difficulties arise in defining the solution at the boundary when 0 < Rank(s) < Rank(h) = M, and there is no unique solution uo of Eq. (20). If Rank(s) = 0, all the characteristics are outgoing, and using either uncentered differences at the points near the boundary or straightforward polynomial extrapolation to the fictitious points gives accurate results. When Rank(s) = M, all the characteristics are entering the boundary and all the solution components can be solved for on the boundary. Uncentered spatial differences can then be used at the points near the boundary and will result in an accurate approximation of the boundary conditions. When Rank(s) is greater than zero but less than M, then by differentiating Eq. (20) with respect to time and replacing ut from Eq. (191, we have sh(x)ux

= b’(t),

.

x = x

(21)

Approximating U by second-order one-sided differences gives us SHoUo = [SHo(4U1

-

U2)

-

2Axbn(t)]/3

+

O(Ax3)

,

(22)

where H = H(xo). Equation (22) gives additional information about the boundary con8itions that is consistent with both the original boundary conditions of If we still do not have enough Eq. (20) and the differential Eq. (19). boundary conditions to solve for U uniquely, we can continue by differentiating Eq. (21) with respect to ttme and using Eq. (19) again. Often, H is nonlinear and the above procedure must be iterated. Usually one or two iterations will supply a stable accurate boundary approximation. Once U has been found, we can use uncentered finite differences to approximate the spttial derivatives at the mesh points near the boundary or we can extrapolate the solution to fictitious points outside the integration region by replacing the derivatives in Eq. (21) with second-order centered differences and solving for U,l. TIME DISCRETIZATION The numerical solution of Eq. (2) is advanced in time in discrete steps that vary depending on the local behavior of the solution; that is, the length of the time steps depends on whether the solution is evolving on a slow or fast time scale. The methods that approximate the time derivatives, like those that approximate the space derivatives, are based on Taylor series. The major difference between time and space differentiation is that time has a direction. This time flow allows savings in computer storage, but introduces questions about the time stability of the difference equations relative to the stability of the differential equation. The integration methods discussed in this section are Runge-Kutta multistep methods and can be written n+a

+ i=O

k2 At 2

i=O

k

pl,iFn-i+ At 2 i=l

n+a i yl,iF

called

k-cycle

NUMERICAL METHODS FOR NONLINEAR DIFFERENTIAL EOUATIONS

99

(23)

n+a

u

un+ 1

kl

= Z "k,i i=O

k

kg

,,n- i +

pk,iFn-i + At

At i=O

Z

n+a . Y~,~F

i=l

n+a k

=u

,,"+a Here, and Fn+a = F(Un+a) are approximations to the solution at time t = t + a(tn+l tn) = tn + aAt, where a is a scalar. n+a n

-

The method is explicit if y . = 0 for j L i and implicit otherwise. Implicit methods require solving one idd more algebraic systems on each time step. The extra work required to solve these systems is often rewarded by a substantially larger stability region than a similar explicit method might have. Explicit Methods The simplest integration method, called the forward Euler method Un+' = Un

+

At F(Un)

(24)

is linearly stable if At is chosen so that M t lies within the stability region shown in Fig. 2 . Here, A is any of the eigenvalues of the linearized Jacobian matrix of F.

Figure 2 Stability Region for the Forward Euler Method Eq. ( 2 4 ) If F in Eq. ( 2 5 ) is a second-order finite difference approximation [Eq. (611 to the parabolic equation Ut = suxx

(25)

with periodic boundary conditions and an equally spaced mesh between [-n,n],

where e takes on discrete values between 0 and 2n 131. From Eq. ( 2 6 ) and

100

J.M. HYMAN

F i g . 2 , it f o l l o w s t h a t t h e s t a b i l i t y r e s t r i c t i o n for A t i s l A t Amax!

5

2

,

or

+ < Ax

1 - 2

.

T h i s r e s t r i c t i o n i s c a l l e d t h e Courant-Friedricks-Lewy o r CFL c o n d i t i o n . We can v a r y A t t o r e t a i n a s t a b l e numerical s o l u t i o n as l o n g a s Eq. (28) i s n o t violated. As i n t h e s p a t i a l Equation (24) i s a f i r s t - o r d e r i n t e g r a t i o n method. d i s c r e t i z a t i o n s e c t i o n , h i g h e r z q r d e r methods a r e o f t e n more c o s t - e f f e c t i v e . If , t h e n t h i s v a l u e c a n be c o r r e c t e d t o second Eq. (24) i s used t o p r e d i c t o r d e r u s i n g t h e improved E u l e r c o r r e c t o r method.

ff

U n + l = Un + & 2

(pl+ Fn)

e x c e l l e n t e x p l i c i t method l e a p - f r o g p r e d i c t o r method

An

.

(29)

f o r hyperbolic equations is the

second-order

combined w i t h t h e t h i r d - o r d e r l e a p - f r o g c o r r e c t i o n method [ 4 ]

The s t a b i l i t y r e g i o n shown i n F i g . 3 i s v e r y good a l o n g t h e imaginary a x i s . Note t h a t i f one s t o p s a f t e r t h e p r e d i c t o r , t h e method i s u n s t a b l e when t h e A have nonzero r e a l p a r t s , making it u n s u i t a b l e f o r p a r a b o l i c e q u a t i o n s .

/ii -0

-I Re (161)

0

Figure 3 S t a b i l i t y Regions f o r Leap-frog P r e d i c t o r C o r r e c t o r Method The s t a b i l i t y f o r a c e n t e r e d f i n i t e d i f f e r e n c e approximation t o Ut= vux

NUMERICAL METHODS FOR NONLINEAR DIFFERENTIAL EOUATIONS

101

with periodic boundary conditions is determined by [3]

A2= i sinB/Ax

(33)

or A4 = i(8sinB

-

sin28)/6Ax

,

(34)

where A2 and A4 are the eigenvalues associated with the fourth-order finite difference approximation Eqs. (4) and ( 5 ) .

second- and

The resulting stability restrictions for the second-order space difference are

2v 5 1 2v 5

predictor stability [Eq. (30)]

(35)

predictor-corrector stability [Eqs. (30) and (31)]

(36)

For the fourth-order space difference [Eq. (511 they are &v<3

predictor stability [Eq.

-4 & v < -9 -8

(so)]

(37)

predictor-corrector stability [Eqs. (30) and (31)]

(38)

The leap-frog predictor is unstable for systems of equations with eigenvalues having a nonzero real part. Therefore, when artificial dissipation is added in shock calculations [4], or when the boundary conditions shift the spectrum of the discretized equation, the leap-frog method cannot be used without the corrector cycle. The first corrector application extends the limit on the maximum time step by 50% and improves the method to third order. Another difficulty of the leap-frog predictor is a unique type of error caused by time and space mesh decoupling. The odd and even points of a mesh are only weakly coupled for even spatial derivatives; errors with frequency 2Ax or 2At can degrade the accuracy of the solution with high-frequency noise. The corrector cycle couples the mesh points among the three time levels and prevents this instability. A relatively new explicit method, method IS], can be written as

u

n+a. l = U

n+a i-1

Here, a. = 1 for i

+ Atci[F

n+a i-1

-

called the

n+a

F

i-2]

.

iterated multistep

(IMS)

(39)

2 3.

An explicit predictor-corrector is used to start the iteration and the ci are chosen so that Eq. (39) increases the order of the method on each iteration for linear autonomous systems. For example, if Eqs. (24) and (29) start the iteration, then c . = l/i, i = 3,4,.... If the leap-frog predictor-corrector Eqs. (30) and- (31) start the iteration, then c3 = 3/10, c4 = 7/30, c = 4/21,.... 5

102

J.M. HYMAN

In general, the stability of the IMS methods increases on every iteration, as can be seen in Fig. 4. Another advantage is that IMS methods allow for local improvements in the stability and accuracy of the calculation. Only a single time level is used in the iteration, so only those ODE components that have failed to pass some accuracy test need be iterated. That is, by iterating locally in regions of rapid changes such as shock fronts, boundary layers, or regions with a refined mesh, the stability and accuracy of the calculation are improved precisely where needed.

n*eI A x h

Figure 4a Stability region for the Improved Euler IMS

Stability region for the Leap-Frog IMS

Implicit Methods Hany problems occur when the solution changes on a slow time scale but the stability criteria limit the time step far below that needed to retain accuracy. In these cases, it is often best to use a more stable implicit method. Two of the best implicit methods are the second-order trapezoidal rule

and the second-order backward difference (BDF) formula

Figure 5a Trapezoidal Method

Figure 5b Second-order BDF

Figure 5 Stability Regions for Trapezoidal Method Eq. (40) and Second-Order BDF Eq. (41) Method

NUMERICAL METHODS FOR NONLINEAR DIFFER EN TIAL EQUATIONS

103

On each time step of a one-cycle implicit method, we must solve a nonlinear algebraic system of the form

"n+ 1

+ Aty F(Un+l) = known quantities

.

(42)

Several iterative methods, discussed in the next section, show how Eq. (42) might be solved. A good first guess can often be made by using an explicit or extrapolation method. ALGEBRAIC SYSTEMS Iterative Methods In the discussions on space and time discretization, it became necessary to solve large sparse algebraic systems of equations, which can be written A(v)

-

b = 0

,

(43)

where A is a nonlinear discrete operator, b is a known vector, and the discrete solution vector is v. Often the solution of Eq. (43) is difficult to obtain directly, but the residual error r = A(w)

-

(44)

b

for an approximate solution w is easy to evaluate. P(w)

-

If there is a related system

b = 0

(45)

that approximates Eq. (43) and is easier to solve, the defect correction algorithm may be appropriate. Given a guess v near a root v of Eq. (43), we can expand Eq. (43) using n+l Taylor series tonget

-

0 = A(v~+~) b

-

vn. The defect correction iteration is any where E = v to Eq. (46)n++he simplest such iteration is

P(V,+~) = P(vn)

- A(vn)

+ b

.

O(E)

approximation

(471

This iteration will converge if v and J the Jacobian of P, are near enough to v and JA, respectively. Table" 2 lises some of the more common applications oP+Aefect corrections.

104

J.M. HYMAN

The iteration (47) can often be speeded up by using a one-step acceleration parameter w to give

These methods include successive overrelaxation, dynamic alternating direction implicit methods, and damped Newton. Often a two-step acceleration method

can speed up the convergence even more. These Tchebyshev [12] and conjugate-gradient 1111 methods.

methods

include

the

NONLINEAR TROUBLES When using linear methods to approximate the solution of well-posed linear equations, one can have some confidence in the results, thanks to Lax's equivalence theorem. This theorem states that, for these problems, a stable consistent approximation is necessary and sufficient for the numerical approximation to converge to the true solution as the mesh is refined. P(vn+l) = A(vn)

+

-

JA(~n)(~n+l

Method Newton [9]

vn)

Jacobi [9]

Diagonal of J A Lower triangular part of J

Gauss-Siedel 191

Lower triangular part of J

Line Gauss-Siedel [9]

A

+

A

first upper off-diagonal

Coarse grid operator + relax

Multigrid [lo]

using one of the above Concus-Golub-O'Leary Ill]

Symmetric part of J

A

If A = (I + At Lx + At L 1, Y then P = (I + At Lx)(I + At L 1,

[gl

~

.

Y

where L = linearized lower order approximation to L . LU where

L = lower triangular matrix

Incomplete LU method [91

U = upper triangular matrix Table 2 . Common examples of the defect correction iteration. The theorem is false for general nonlinear equations or nonlinear methods, shown in the following two examples.

as

NUMERICAL METHODS FOR NONLINEAR DIFFERENTIAL EQUATIONS

Nonlinear Equation Example Consider the nonlinear diffusion equation

The solution to this equation is a wave front traveling to the right Equation ( 5 0 ) can be rewritten as Ut

= 3u(u

uxx

+

2 2ux )

The solution U is discretized and the space derivatives are approximated by any method in the Space Discretization section. The time variable is integrated with any stable method from the Time Discretization section. Note that U (x ) = 0 , a discrete version of Eq. ( 5 1 ) , is zero for all t when t ix . > 0 since U. - 0 at these mesh points. This implies that the wave can never piopagate to the right of zero, no matter how fine a mesh is used. Thus, we have a stable consistent approximation that can never converge to the true solution. If Eq. ( 5 0 ) is differenced directly using Eq. (6) with G = U3 then the resulting numerical solution will converge to the correct answer. Nonlinear Method Example Consider the unidirectional wave equation

with the solution u(x,t) = u(x+t,O). After discretizing, we approximate the space derivative using a nonlinear transformation method (Space Discretization, Grid Resolution). The transform interpolates the discrete solution with a monotonicity preserving interpolant The derivative of this that has a continuous first derivative 1131. interpolant is then used at the grid points to define the ODES [Eq. (3)].

We know that the true solution to Eq. (52) is monotone and we might expect that an interpolant that preserves this monotonicity will greatly improve the accuracy of the calculation. However, since the interpolant is monotone, U (xi) = 0, we have Ut (x i) = 0 at all the grid points. The solution never cKanges!

SUMMARY We have used a modular approach to develop accurate and robust methods for the numerical solution of PDEs. The methods to discretize the spatial operator, the boundary conditions, and the time variable, and solve any algebraic system that may arise are combined when writing a code to solve the PDE system. As the last two examples show, special care must be taken when solving a nonlinear equation or when using a nonlinear method. This means that the code must be field tested.

105

J.M. HYMAN

106

The field test is to check the reliability of the method on a particular nonlinear system of PDEs. The numerical results should be insensitive to reformulations of the equations, small changes in the initial conditions, the mesh orientation and refinement, and the choice of a stable accurate discretization method. Another excellent analysis tool is verification that any auxiliary relationships (such as conservation laws) hold for the numerically generated solution. These checks should be made even if one is absolutely, positively sure that the numerical solution and coding are correct.

-

When the above checks are made, the methods given here can be combined to give reliable efficient methods for solving a fairly large class of nonlinear PDEs. Also, by using the modular approach presented here, these codes can evolve efficiently since new methods can be quickly incorporated into the program. ACKNOWLEDGEMENT

I am grateful to Joe Dendy, Blair Swartz, and Burton Wendroff for providing much welcome advice in our many discussions during this work. REFERENCES [l] Wachspress, E. L., Iterative solution of elliptic systems (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1966).

Leventhal, S., and Weinberg, B., "The operator implicit method for parabolic equations," J. Comp. Phys.

[2] Ciment, M.,

compact (1978)

135-166. [3] Hyman, J. M.,

The method of lines solution of partial differential equations, Courant Institute of Mathematical Sciences report COO-3077-139 (1976).

[ 4 ] Hyman, J. M., A method of lines approach to the numerical solution of

-

conservation laws, Adv. in Comp. Methods for PDEs 111, Vichnevetsky, R., and Stepleman, R. S., (eds) Publ. IMACS (1979) 313-321. [5] Hyman, J. M., "The numerical solution of time dependent PDEs on an adaptive mesh,'' Los Alamos Scientific Laboratory LA-UR-80-3702 (1980). [6] Gelinas, R. J., Doss, S. K., and Miller, K., "The moving finite element

method: applications to general partial differential equations with multiple large gradients," J. Comp. Phys. 40 (1981) 202-249. [7] Berger, M., Gropp, W., and Oliger, J., Grid generation for time dependent

problems: criteria and methods, numerical grid generation techniques, NASA Conf. Proc. Publication 2166 (1980) 181-188. [8]

Hyman, J. M., Explicit A-stable methods for the solution of differential equations, Los Alamos Scientific Laboratory LA-UR-79-29 (1979).

[9] Young, D. M., Iterative solution of large linear systems (Academic Press, New York, NY, 1971). [lo] Brandt, A., Multi-level adaptive solutions to boundary value problems, Math. Comp. 2 (1977) 333-390.

NUMERICAL METHODS FOR NONLINEAR DIFFERENTIAL EOUATIONS

[ll] Concus, P., Golub, G. H., and O'Leary, D. P., Numerical solution of nonlinear elliptic partial differential equations by the generalized conjugate gradient method, Computing 19 (1978) 321-339.

[12] Manteuffel, T. A . , The Tchebychev iteration systems, Numer. Math. (1977) 307-327.

a

for nonsymmetric linear

[13] Hyman, J. M., Accurate monotonicity preserving cubic interpolation, Los Alamos National Laboratory report LA-8796-MS (1981).

107

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) @ North-Holland Publishing Company, 1982

109

LIMIT ANALYSIS OF PHYSICAL MODELS M. FREMOND Laboratoire Central des Ponts et Chaussees 5 8 bld Lefebvre 75732 PARIS Cedex 15 - France

-

A. F R I A ~ Ecole Nationale d'IngEnieurs de Tunis B.P. n"37, BelvBdSre TUNIS - Tunisie

-

-

-

M. RAUPP Centro Brasileiro de Pesquisas Fisicas avenida Wenceslau-Braz 71, CEP 2000, RIO DE JANEIRO

- Bresil -

Let us consider a physical model where some internal variables must belong to a given set in order for the model to be valid or not to breakdown. Our aim is to determine the external actions which achieve this condition. We present two examples. The first one describes a simple case in limit analysis of structures in mechanics. The second describes the breakdown analysis of an electrical circuit. In conclusion a common formalism is presented which can be applied to other situations [2]. AN EXAMPLE IN MECHANICS [2]

Let us consider the beam described on figure 1,

A

h

m A

n

n

n

+I

X

D

Figure The beam loaded by

he torque Af.

Two equal and opposite torques, with intensity A , are applied at its ends. The interior forces of the beam are the bending moment at any point x of the beam. We assume the beam is made of a material which cannot support moments greater than k. This threshold k can have several physical meanings, elastic limit, yielding, ... We look for the external loads, caracterized by the number A , such that there exist a moment which equilibrate them and which are lower than the threshold k. Let us recall several notions, most of them being classical in the strengh of material theory. The set of kinematically admissible displacements v(x)

is

110

M. FREMOND, A. F R I A ~M. RAUPP

v = {v I vE"**'(-R,+!2)

I

; v(-i) =v(O) -v(i)

=o}

+!?.

with the norm IlvII=

Iv"(x)ldx.

-!2

We use throughout this text the Sobolev spaces W Z s 1 , H 1, H2 , regularity of the functions involved in the equations [ I ] .

... which

describe the

The set of the external loads is V'the dual space of V. The applied torques Af are, indeed, elements of V' because *Af,v%*

:1

X(V'(&)-V'(-~))

=X

v"(x)dx,

(the double brackets denote the duality application between V and V'). The set of the strain rates D and the set of the interior forces S are respectively

s =.D' = L-(-I,+L).

D = ~'(-i,+i), The deformation operator ned by

E,

which is linear and continuous from V into D, is defiE(V)

Let us call

t

E

= v",

the transposed operator of

E.

The mechanical power developped by the moment M and the strain rate d is

If,

-<M,d>=-

M(x)d(x)dx.

The set 1 of statically admissible bending moments, i.e. the set of M which equilibrate an exterior load Xf, is I = { M E S 13XElR ; W v E V , -<M,~(v)>+*Af,v*= = { M Es

I M"=O

on ] - ~ , o [ u ]o,+L[

0 ; or

, M is continuous at o ; M(-E) =M(L)

Let us remark that W M E S , V v E V , <:M,E(v)>=*Af,v*= This relation defines two linear applications

-L(M)

-

t ~(V)=hf}

-

A},

A(v'(i)-v'(-i)},

zE.C(lJR) , LEL(VJR)

by

-

X and L(v) = v ' ( i ) v' ( 4 ) .

The set G of the moments which can be supported is G={Ml MES

j

IM(x)l
a.e,}.

The problem we want to solve is : Find KCIR such that X E K

* 3 M E G n l such that C(M)

=A.

It turns out that K=Z(GnK) and that under reasonable assumptions K is a closed convex set (51. This set K describes the loads which have the possibility to be supported by the structure or for which the structure can be safe, The loads which are not in K cannot be supported Within the framework of plasticity the boundary of K is the set of the limit loads (71 ,

111

LIMITANAL YSlS OF PHYSICAL MODELS

The convex set K is of fundamental interest in practice. There are two ways to compute it : the static and kinematic methods. The s t a t i c method. It uses-the fact that the set K is convex, If it is possible to find M E G n I , the line [O,L(M)] is contained in K (figure 3 ) . Here this method leads to K = [-k,+k] Unfortunately in many practical applications it is impossible to find elements of G n I (for instance in 2 or 3 dimensions problems) and this method fails,

.

The kinematic method. It uses convex analysis results. Let us consider the support function 11G o f the convex set G :

-

and the function QEIR

I VEV}

qK1(9)= SUP{Q-L(V) -~,(E(v)) +a =

Sup{Q/

=

+a

v"(x)dx-k/

-a

Iv"(x)Idx

I vEV}.

-9,

It is easy to show that

J, (Q) is either equal to +m if Q > k or to 0 if Q G k . In K1 general $ is the indicator function of a convex set K1. Under reasonable assumpK1 tions K equals K1 [ 2 ] , [3] , [5].

Let us notice that KlC{Q

I Q*L(v)

-nG(~(v))
for any

V.

This is the key point of the kinematic method (figure 4) and we have Kl={QI QG1nf{nG(E(v))

{Q I QGInf{llG(E(v))

Iv€v;L(v)=~}~ I vEV

...

; L(v) = - I ] } ,

The practical steps of the method are the computation of It and minimisation pro.G blems. In mechanics this kinematie method-is oft'en more efficient. AN EXAMPLE IN ELECTRICITY 121 141.

Let us consider the electrical circuit described on the figure constitutive law is shown on figure 2 is connected in parallel This cell blows if the potential difference v2 between its two the given threshold v At the time t = O , the circuit being at

.

2 . A cell whose with the condenser. connections exceeds rest (the conden-

ser is discharged and there is no current 'through it) and electromotive force N

is applied. The problem is to find the controls X = (A,,

... ,AN)

such that the cell

does not blow during the time interval [O,T]. This problem is related to the preceeding one but i s n o longer static. Let us define, v1 the potential difference between the connections of the resistance and inductance block, j1 the current intensity inside this block, j2 the

112

M. FREMOND. A. F R I A ~ " M. , RAUPP

current intensity sum of the current intensities a2 and b2 which pass through the condenser and the cell, i the current intensity which passes through the electromotive force.

R

L

e, i

Figure 2 The circuit and the constitutive law of the cell. We have the relations [61 N

I: A e

=v1+v2,

n = ~

v1 = R j l + L

dj 1 , dt

--

dV2 a2=C-, dt

(3)

where a$ is the subdifferential of the indicator function of [-vc,+vc], i = j1 = j, =

a2

+ bp,

dv2 v2(0) = 0 , ( 0 ) = 0, N Our problem is to find the set KC'IR of the controls X which can be supported by the circuit : i.e. for which the cell does not blow, XEK,

t/tE [O,T]

, b2(t)

PO,

Let us define some sets which are useful to describe K :

I

=

H2(0,T),the

E = H'(O,T),the

set of the current intensities inside the electromotive force, set of the electromotive forces,

the electrical work developped by e and' i is

J1 = J p = H2(0,T),

the sets of current intensities j, and j2, dv2 V1 = H1(O,T), V 2 = { u z € H 3 ( 0 , T ) ; v 2 ( 0 ) = t (0) = O}, the sets of the difference of potential v1 and v2

,

113

LIMITANALYSISOF PHYSICAL MODELS

J = J l X J 2 , V

VlXV2.

the electrical work developped by j = (jl ,jz) and v = (vl,v2) is =

:1

(j lvl + j2vZ)dt.

The first Kirchoff law (equilibrium of the nodes) is (jl,jz) = (i,i) or The operator

E

E(i)

=

j,with E(i) = (i,i).

is linear and continuous from I into J.

The second Kirchoff law (equilibrium of the loops) is y i E I , where

'E

=(e,i%

is the transposed operator of

We define the set

1 of

or e = i l + i ? or

t

E(v)=~,

E.

the potential differences v which equilibrate a control X

and the set G of the potential differences which can be supported by the elements of the circuit :

One can remark that N

N N This relation defines two linear applications K € L ( I , l R ) and L€l(I,E ) by

-L(v)

=

X and L(i) = ( <en,iS ) ,

It turns out that

K = ^i(Gn 1). The formulation of both electrical and mechanical problems are exactly the same. It is then possible to use the same static and kinematic methods to find K. The " s t a t i c " method Let us choose XERN and solve the set of equations (1,2,3,5,6) with bg=O. If the solution is such that Ivg(t) I
Let us define the support function JIG of G and compute II,(E(i))

= vclTli-CR o $+$ldt+vc

LCIg(T)I

The convex set K1 is defined by its indicator function

if i(T)=O,

114

M. FREMOND. A

eK1 ( A )

FRIAA, M. RAUPP

= sup{A*L(i) -nG(c(i))

I i E I}.

Under reasonnable assumptions [ 3 ] , we have K = K1. The key point of the method is then K=KlC{A

I X*L(i)

-IIG(c(i))}

for any i.

N Let us choose a direction q oflR , we have

KC{AI A.q-Inf{IIG(E(i))

1

i E V ; L(i)=q}}.

Figure 3 Figure 4 The static method approaches K by the interior and the kinematic method by the exterior.

The two step of this methods are again the computation of IIG and minimisation problems. An example

Let us assume that R = O , N = 1 , Vt, e(t) = 1 , and define w - l/V'rC. It is possible to show by the kinematic method (i.e. without solving a differential equation) that R = [-XO,+XO]

with X O = - VC if wT>n and 1 - VC coswT if wT
It is natural that the limit electromotive force to be big for small T because the circuit is at rest at the time t = O and the potential difference v2 is continuous. FORMALISM 121 , [51, 171

.

The electrical and mechanical problem are both illustration of a general formalism based on the classical framework of physics and continuum mechanics. Let us consider a structure which is described by interior variables d E D and

115

LlMlTANALYSlSOF PHYSICAL MODELS

S E S = D' (dual space of D ) which develop the power (or work) and by exterior variables v E V and f E F = V' (dual space of V) which develop the power (or work) is
WV'vEV,

WSGS,

We say that the interior variable

s

>=4v,

<E(V),S

t

equilibrates f if

E(S)

E

*.

t

E(s)=~.

We assume that the structure can support only the interior variables which are inside a convex set G C S and that it is submitted to a loading process [71 defined by a finite dimension subspace -Fn of F (f E F n

-

f=

Xifi ; (Xi) = XEIRn) i= 1 The practical problem is to find the set K of IRn such that for any XEK, the exn ternal load 1 hifi is equilibrated by an internal variable s which is in G. i= 1 We define the set of the internal variables which equilibrate an external load of

We have

n

1 Xi*fi,V * . i= This relation defines two applications L and L by WEV,

wsE1,

< E(V),S>

=

1

-

L ( s ) = A and L(v) = ( t E

i'

v *),

It turns out that K =

t(Gn1)

and that under reasonable assumptions both static and kinematic methods can be used to compute K [ 3 1 . The static method requires the knowledge of elements of G n l . The kinematic method requires the computation of the support function II of G G and the minimisation of a function, Let us also remark that the static method is an approach of K by the interior the kinematic method is an approach by the exterior (figures 3 and 4). REFERENCES [ I ] Adams, R., Sobolev Spaces (Academic Press, London, 1 9 7 5 ) . [2] FrCrnond, M . , MCthodes variationnelles en calcul des structures (Cours de D.E.A. Ecole Nationale des Ponts et Chaussses, Paris, 1 9 7 9 ) . [3] FrCmond, M., Friaf, A., Comptes Rendus de 1'AcadBmie des Sciences, 286, SBrie A, 1978, p. 107.

[ 4 ] Frgmond, M., Friaf, A., Raupp, M., Comptes Rendus de l'Acad6mie des Sciences, 292, SCrie 11, 1981, p. 2 1 . [51 Friaf, A., Le matdriau de Norton-Hoff gBnCralisB en BlasticitC et viscoplasticit6. Thsse. UniversitB Pierre et Marie Curie, Paris, 1979 I61 Raupp, M., Baiocchi, D., Bol. SOC. Mat, Brasil (to appear). (71 Salenson, J., Calcul B la rupture et plasticit6 (Cours de D.E.A. Ecole Nationale des Ponts et ChaussCes, Paris, 1 9 7 8 ) .

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North-Holland Publishing Company, 1982

117

V I S C O S I T Y SOLUTIONS OF HAMILTON-JACOB1 EOUATIONS

Michael G. C r a n d a l l * Department o f Mathematics and Mathematics Research Center U n i v e r s i t y o f Wisconsin-Madison Madison, Wisconsin INTRODUCTION

T h i s paper concerns problems o f t h e forms H(x,u,Du) U(X)

= 0

= z(x)

,

for

x

E

Q

for

x

E

an

,

and for

t

>

0, x

E

n ,

= z(x,t)

for

t

>

0, x

e

an ,

U(X10) = u o ( x )

for

x

E

ut t H(x,t,u,Du) u(x,t)

= 0

;

which we r e f e r t o as, r e s p e c t i v e l y , t h e D i r i c h l e t and Cauchy problems f o r Hamilton-Jacobi equations. I n (DP) and (CP), n i s an open subset o f RN, asl i s t h e boundary o f n, t h e unknown u and t h e H a m i l t o n i a n H a r e r e a l - v a l u e d f u n c t i o n s o f t h e r e l e v a n t arguments and Du = (uX1.".,uX ) indicates the N s p a t i a l g r a d i e n t o f u. Equations o f t h i s s o r t a r i s e i n many c o n t e x t s , i n c l u d i n g mechanics, c o n t r o l t h e o r y , c a l c u l u s o f v a r i a t i o n s , d i f f e r e n t i a l games, and t h e a p p r o x i m a t i o n o f s o l u t i o n s o f l i n e a r h y p e r b o l i c equations. A n a l y s i s o f (DP) o r (CP) by t h e c l a s s i c a l method o f c h a r a c t e r i s t i c s (see below) i s l i m i t e d t o l o c a l c o n s i d e r a t i o n s owing t o t h e f a c t t h a t c h a r a c t e r i s t i c s which h e r e "cross". T h i s phenomena l e a d s t o t h e development o f "shocks" r o u g h l y means t h a t d e r i v a t i v e s develop d i s c o n t i n u i t i e s and t h e n t h e a n a l y s i s by c h a r a c t e r i s t i c s i s no l o n g e r v a l i d . A g l o b a l t h e o r y o f (DP) o r (CP) t h u s r e q u i r e s a n o t i o n o f s o l u t i o n broad enough t o encompass d i s c o n t i n u i t i e s i n d e r i v a t i v e s . To s e t t h e stage f o r t h e n o t i o n i n t r o d u c e d h e r e i n , we f i r s t r e c a l l t h e c l a s s i c a l method i n a very s i m p l e case and make some remarks about t h e inadequacy o f t h e most obvious way t o d e f i n e g e n e r a l i z e d s o l u t i o n s o f (DP), (CP).

-

-

L e t us g i v e t h e c l a s s i c a l s o l u t i o n of (CP) i n t h e case where n = R and H = H(ux) depends o n l y on ux. Consider a ( s u f f i c i e n t l y smooth) f u n c t i o n u(x,t) which s a t i s f i e s ut + H(ux) = 0 f o r 0 < t < T, x E R. D e f i n e X(s,xo)

for

0 < s < T, xo

E

R by

*Sponsored by t h e U n i t e d S t a t e s Army under C o n t r a c t No. DAAG29-80-C-0041.

118

M.G.

CRANDALL

We w i l l prove s h o r t l y t h a t

where

po = ux(xO,O).

Moreover

and we conclude from (0.2)-(0.4)

These formulae determine

that

and

u

ux

from the i n i t i a l data

( x t tH'(ux(x,O),t) the transformation (x,t) ( i ) says ux i s constant on t h e l i n e s x = xo +

t

u(x,O)

so long as

i s well-behaved. However, (0.5) tH'(ux(xo,O)) and i f two o f

these cross (as they must unless H'(u,(xo,O)) i s monotone i n x o ) , we conclude t h a t uX must develop a d i s c o n t i n u i t y a t t h e f i r s t such crossing. O f course, (0.5) ( i i ) also y i e l d s c o n t r a d i c t o r y i n f o r m a t i o n about the values o f u when c h a r a c t e r i s t i c s cross, but d i s c o n t i n u i t i e s develop i n ux f i r s t . (Roughly, u n t i l c h a r a c t e r i s t i c s cross ux remains bounded and u t h e r e f o r e cannot develop "jumps".) It remains t o v e r i f y (0.2). This f o l l o w s from

d

U,(X(S,X,),S)

= uXxX' t U x t = uXXH' t U x t = ( U t t H(u,)),

= 0

.

We now have r e c a l l e d t h a t i n order t o have a global time theory f o r = 0,

t

>

0,

u~(x),

x

E

R

U t t H(Ux)

u(x,O)

=

X E

R ,

,

which extends t h e c l a s s i c a l theory, we must a l l o w d i s c o n t i n u i t i e s i n ut and ux. The simplest way t o do t h i s i s j u s t t o r e q u i r e t h a t u be ( l o c a l l y ) L i p s c h i t z continuous (so u i s d i f f e r e n t i a b l e a t almost every (x,t)) and then ask the equation be s a t i s f i e d almost everywhere. This simple approach i s q u i t e Indeed inadequate, f o r one w i l l not then have uniqueness o f s o l u t i o n s o f (0.6). consider the problem ut

-

u(x,O)

(u,)' = 0,

= 0,

t

>

0, x

E

R , (0.7

X E

R.

Clearly, u I 0 i s a s o l u t i o n o f (0.7). Another s o l u t i o n , i n t h e almost everywhere sense, i s u = 0 f o r 1x1 > t and u = t 1x1 f o r t 1x1. can one d i s t i n g u i s h between these solutions?

-

HOW

Let us motivate t h e answer t o t h i s question given herein. To "solve" (0.6) one frequently uses t h e approximation o f i t by t h e regularized problem

119

VISCOSITY SOLUTIONS OF HAMlL TONJACOB1 EOUA TIONS

vt

H(vX)

t

V(X,O)

-

t

c v X X = 0,

uo(x),

)

X E

0, x

e

R ,

R ,

i n which E > 0 and c o n s i d e r s t h e l i m i t o f t h e s o l u t i o n o f (0.6) as, Assum6 T h i s i s an example o f t h e method o f " v a n i s h i n g v i s c o s i t y " . on

-

Hb,)

V t +

R 0 =

EVXX

1

7 (VVt)

U(R x ( O , C - ) ) ~

IP E

( t h e nonnegative

+

0.

C i functions

We have, on t h e s u p p o r t o f 7 ,

(0,m)).

x

Let

= 0.

E

1

+ H(;IPV~)

-

1

E

7qVXx

and (9v)t

9Vt =

9vx = ( 4 ,

- 9tV

-

9vxx = (9Vlxx

'pXV

-

I

y

2PXVX

- VXXV , 0

= (9Vlxx

29;

v - 9xxv - 76 ( 9 V l x + 9 *@

X

so t h a t

and (vv),,(x0,t0) t o conclude

- -9t 9

at E

+

V

+ H(-

2": - V ) - "(-2. v - -V ) 9X 9

9

qxx 9

(xo,tO). Assuming t h a t t h e s o l u t i o n 0 we t h e r e f o r e expect t h a t

- -" u 9

t

H(-

@X u) 9

on t h e s e t where

qu

< 0,

so we can

< 0 v

o f (0.6), t e n d s t o a l i m i t

< 0

u

as

(0.8)

assumes i t s maximum value.

I n t h e n e x t s e c t i o n t h e n o t i o n o f s o l u t i o n suggested by t h e s e c o n s i d e r a t ons i s presented i n g e n e r a l i t y t o g e t h e r with a few b a s i c r e s u l t s . Subsequently, n S e c t i o n 2, t h e r e l a t e d e x i s t e n c e and uniqueness t h e o r y i s i l l u s t r a t e d i n mode problems. D e t a i l s and p r o o f s a r e n o t g i v e n i n t h i s b r i e f i n t r o d u c t o r y e x p o s i t i o n . They may be found i n t h e paper [l] by t h e a u t h o r and P. L. L i o n s which i s t h e o r i g i n o f t h i s t h e o r y . We a l s o r e l y on [l] and P. L. L i o n s [6] o r t h e i r r e f e r e n c e s t o t h e v e r y s u b s t a n t i a l p r e v i o u s l i t e r a t u r e on t h e s u b j e c t o f Hamilton-Jacobi equations. Here we o n l y mention t h a t much o f t h i s l i t e r a t u r e i s concerned w i t h convex H a m i l t o n i a n s and o u r n o t i o n o f s o l u t i o n i n t h e g e n e r a l nonconvex case appears c o m p l e t e l y new. There i s some p a r a l l e l , however, w i t h t h e t h e o r y o f a s i n g l e c o n s e r v a t i o n law, e.g. [5], [7]. We s h o u l d a l s o m e n t i o n t h e works o f Fleming [S] and Friedman [4] i n t h e nonconvex case, wherein i t i s shown t h a t t h e v a n i s h i n g v i s c o s i t y method converges. When t h i s method converges, one

120

M.G. CRANDALL

has s i n g l e d o u t a p a r t i c u l a r s o l u t i o n and t h i s can be adopted as a ( n o n i n t r i n s i c ) d e f i n i t i o n o f a s o l u t i o n . F i n a l l y we mention [2] i n which e r r o r e s t i m a t e s a r e g i v e n which e s t a b l i s h t h e convergence o f t h e v a n i s h i n g v i s c o s i t y method as w e l l as a c l a s s o f d i f f e r e n c e a p p r o x i m a t i o n s t o o u r v i s c o s i t y s o l u t i o n s . SECTION 1.

VISCOSITY SOLUTIONS

L e t 0 be an open subset o f RM and F : 0 x R x RM .+ R be c o n t i n u o u s . u ). The e q u a t i o n We denote p o i n t s o f 0 by y and Du = (uyl. YM

...,

F(y,u,Du)

= 0

(1.1)

includes t h e equations

H(x,u,Du)

= 0

and

ut

H(x,t,u,Du)

t

= 0

by t a k i n g

M x R x R ) M denote t h e c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s on 0 and on 0 x R x R r e s p e c t i v e l y and convergence i n t h e s e spaces means u n i f o r m convergence on compact

y = x

i n t h e f i r s t case and

i n t h e second. C(0)

y = (x,t)

C"

s e t s . U(0)' denotes t h e nonnegative on compact subsets o f 0. If

Q

C(0)

E

C(O

0 which a r e s u p p o r t e d

we s e t

= Iy

E

0 : Q(y) = max J, 0

>

01

E-(Q) = { y

E

0 : ~ ( y )= min

<

01

E,(Q)

f u n c t i o n s on

and

and J,

0 If Q

0,

does n o t assume a p o s i t i v e maximum v a l u e on

D e f i n i t i o n 1. If 7

E

A function

U(O)',

k

E

then there i s a y F(YSU(Y),

u

E

R and E

E,(q(u

I f IP

E

C(0)

E,(Q)

=

Q, e t c .

which s a t i s f i e s

-

E+(~(u k))

-

then

k))

f

6,

f o r which

-

i s a v i s c o s i t y subsolution o f

1

.

U(0)'. k

E

then there i s a y

R E

and E-(q(u

F = 0.

E-(V(U

-

k))

S i m i l a r l y , when

-

k))

#

u

satisfies

#,

f o r which

i t i s a v i s c o s i t su e r s o l u t i o n o f F = 0. I f (1.3), i d s h o n o f F = 0.

(1.3)

u

s a t i s f i e s b o t h (1.2)

and

We sometimes use " v i s c o s i t y s o l u t i o n o f F < 0" i n p l a c e o f " v i s c o s i t y s u b s o l u t i o n o f F = 0", e t c . The l i n e of argument which l e d f r o m (0.6) t o (0.7) (when a p p l i e d t o u k ) a l s o l e a d s f r o m sub- and s u p e r s o l u t i o n s o f (Of6) t o t h e c o r r e s p o n d i n g " v i s c o s i t y " n o t i o n s . We c a u t i o n t h e r e a d e r h e r e t h a t t h g v i s c o s i t y s o l u t i o n s o f F = 0 a r e not, i n g e n e r a l , t h e v i s c o s i t y s o l u t i o n s o f -F = 0. However, u i s a v i s c o s i t y s o l u t i o n o f F < 0 i f and o n l y i f v = -u i s a v i s c o s i t y s o l u t i o n o f -F(x,-v,-Dv) > 0. I f u E C1(0), t h e n

-

121

VISCOSITY SOLUTIONS OF HAMIL TONJACOBI EOUATIONS

- k))

D(v(u

-

v

= 0

on

E+(v(u

np =

nu

on

-

so

k))

,

~ + ( v ( u- k))

e t c . It f o l l o w s t h a t c l a s s i c a l s o l u t i o n s o f F < 0, F > 0 and F = 0 a r e a l s o v i s c o s i t y s o l u t i o n s . It i s a l s o t r u e t h a t smooth v i s c o s i t y s o l u t i o n s a r e c l a s s i c a l s o l u t i o n s , as we see l a t e r . To s e t t h e stage f o r t h e next r e s u l t , c o n s i d e r t h e f o l l o w i n g problem: (ux)2

-

1 = 0,

- l < x < l

~ ( - 1 )= ~ ( i= )n

.

(1.4)

-

T h i s problem has a l a r g e s t L i p s c h i t z c o n t i n u o u s s o l u t i o n u = 1 1x1 and a s m a l l e s t L i p s c h i t z continuous s o l u t i o n u = 1x1 1 which s a t i s f y t h e equation I t has many o t h e r s o l u t i o n s , e.g. a.e.

-

un(-l) =

n,

u;(x)

= (-1)j

-

j = 0,...,4n

defines a s o l u t i o n

un

1

on

-

~ ( j2n)/2n,

(j

t

-

1

2n)/2n[,

,

such t h a t

0 < un < 1/2n

for

un + 0 u n i f o r m l y , b u t u z 0 i s n o t a s o l u t i o n o f c o n t r a s t , f o r v i s c o s i t y s o l u t i o n s we have:

n = 1,2,...

.

Clearly

(u,)~ = 1 anywhere.

In

( S t a b i l i t y o f t h e c l a s s o f v i s c o s i t y s o l u t i o n s ) . L e t {F,) be a M sequence i n C(O x R x R ) and u, be a v i s c o s i t y s o l u t i o n o f FQ = 0. L e t F, F i n C(O x R x R M ) and u, + u i n C(O x R x RM ). Then u i s a v i s c o s i t y s o l u t i o n o f F = 0.

Theorem 1.

+

once.

We p r o v e t h i s r e s u l t t o i l l u s t r a t e t h e n a t u r e o f D e f i n i t i o n 1 a t l e a s t For p r o o f s of subsequent r e s u l t s t h e reader i s r e f e r r e d t o [ll.

-

L e t q E -D(O)+, k E R and 9 E E+(v(u k)). Then v(Y)(u(Y) - k ) > 0 v(j)(u,(y) k) > 0 f o r l a r g e R. Thus E+(v(u, k ) ) f d f o r large and, by assumption, t h e r e e x i s t s yQ such t h a t

Proof.

-

ando

Passing t o a subsequence i f necessary, we assume y + i n (1.5) and (1.6) shows t h a t E Et(v(u -'k)) an% F(jsu(j),

-

Dv($)) < 0 p(i)

-

?E

0.

Taking t h e l i m i t s

.

Hence u i s a v i s c o s i t y s o l u t i o n o f F < 0. S i m i l a r l y , u i s a v i s c o s i t y s o l u t i o n o f F > 0, and hence a v i s c o s i t y s o l u t i o n o f F = 0. I n t h e above pcoof we used t h e m i l d requirements o f D e f i n i t i o n 1 i n t h a t k)) # @ we concluded o n l y t h e e x i s t e n c e o f from E+(v(u E E (v(u k)) f o r which t h e d e s i r e d i n e q u a l i t y held. However, one can prove D e f i n i t t o n 1 c o u l d

-

-

122

M.G. CRANDALL

be s t r e n g t h e n e d a g r e a t deal w i t h o u t a l t e r i n g t h e n o t i o n d e f i n e d . The n e x t r e s u l t o u t l i n e s t h e e x t e n t t o which t h i s i s t r u e . F o r $ E C(0) we l e t d($) = {y where

0 : D$(y)

E

"D$(y)

Theorem 2. F(Y,U,

exists)

e x i s t s " means

Let

-

IP

-

Ip

U,IP,$

II, i s ( F r e c h e t ) d i f f e r e n t i a b l e a t y.

C(0)

E

and

u

be a v i s c o s i t y s o l u t i o n o f

DIP + D$) c 0 everywhere on

F = 0.

Then

E,(V(U

-

$))

E-(IP(u

-

$ ) ) n d(lp) n d ( $ )

d(V) n d($)

and F(Y,u,

DIP

+

D$) > 0

everywhere on

.

Using Theorem 2 (and t h e p r o o f o f Theorem 2) one can p r o v e : Theorem 3.

If

(Consistency).

F(Y,u(Y),Du(Y))

=

u

i s a viscosity solution o f

F = 0,

then

0 on d ( u )

I n p a r t i c u l a r , a L i p s c h i t z continuous v i s c o s i t y s o l u t i o n o f s a t i s f i e s t h e e q u a t i o n p o i n t w i s e almost everywhere.

F = 0

As a f i n a l example o f t h e many r e s u l t s o f [l, S e c t i o n I], we c o n s i d e r t h e s i t u a t i o n i n which 0 i s d i v i d e d i n t o two open p a r t s 0- and 0, by a s u r f a c e

0.,

r.

i ( y ) be t h e normal t o r a t y u E C(0) be u- i n 0- and u,

Let Let

continuously d i f f e r e n t i a b l e i n

0,

E

r in

and assume i ( y ) p o i n t s i n t o 0, and assume u, is

respectively.

Let

u+

be c l a s s i c a l

F = 0 in 0 We ask how t h e jump i n nu -across r i s f' r e s t r i c t e d i n o r d e r t h a t u be a v i s c o s i t y s o l u t i o n o f F = 0 i n 0 ( i t i s assumed t h a t Du, extend c o n t i n u o u s l y t o r). The answer i s t h a t u i s a

solutions o f

viscosity solution i n

where

I

0

i f and o n l y i f :

i s t h e i n t e r v a l w i t h endpoints

means t h e i n n e r - p r o d u c t o f

a

and

b.

Du,(y)*;(y) and

pTDu*(y)

and

Du-(y)*i(y),

a-b

means t h e ( e q u a l ) p r o j e c -

t i o n o f Du,(y) and Du-(y) on t h e t a n g e n t space t o r a t y. T h i s c r i t e r i o n 1x1) o f (0.7) i n t h e i n t r o d u c t i o n and a l l s o l u r u l e s o u t t h e s o l u t i o n max(0,t 1x1, as t h e r e a d e r can v e r i f y . t i o n s o f (1.4) mentioned i n t h e t e x t except 1

-

SECTION 2.

-

MODEL EXISTENCE AND UNIQUENESS RESULTS

We now i n d i c a t e , i n model problems, some o f t h e v e r y g e n e r a l s t a t e m e n t s c o n c e r n i n g e x i s t e n c e and uniqueness o f s o l u t i o n s o f (DP) and (CP) w h i c h a r e v a l i d f o r v i s c o s i t y s o l u t i o n s . L e t us c o n s i d e r t h e Cauchy problem ut t H(Du) = 0, u(x,O)

t

>O,

X E

RN

= un(X)

where H E C(RN). L e t BUC(n) f u n c t i o n s on n. We have:

denote t h e bounded and u n i f o r m l y c o n t i n u o u s

VISCOSITY SOLUTIONS OF HAMIL TONJACOBI EOUATIONS

Theorem 4. u

E

C(RN

x

Let

(iii)

BUC(RN),

E

i s a viscosity solution of

u ( x , ~ ) = uo(x) u

Then t h e r e i s e x a c t l y one f u n c t i o n

with the following properties:

LO,-))

(i)u

(ii)

uo

BUC(RN

E

123

for

H ( D ~ )=

o

on

RN

H z 0 and

uo

x

(o,-).

x e R ~ .

[O,T])

x

+

ut

f o r each

T h i s r e s u l t i s c o m p l e t e l y new.

T

>

0.

I f , e.g.,

i s nowhere

d i f f e r e n t i a b l e , then u(x,t) u o ( x ) i s nowhere d i f f e r e n t i a b l e i n x. The a b i l i t y t o f o r m u l a t e an e x i s t e n c e and uniqueness r e s u l t f o r (2.1) i n o u r g e n e r a l i t y thus requires t h e a b i l i t y t o discuss n o n d i f f e r e n t i a b l e s o l u t i o n s o f t h e equation, and t h i s has n o t been p o s s i b l e before. However, t h e e x i s t e n c e a s s e r t i o n s can be proved by expanding on t h e arguments i n t h e i n t r o d u c t i o n concerning t h e r e l a t i o n s h i p o f t h e v a n i s h i n g v i s c o s i t y method and t h e n o t i o n o f v i s c o s i t y s o l u t i o n s , so we can adapt known methods here. The uniqueness i s t h e n t h e main new p o i n t . Theorem 4 a l l o w s one t o d e f i n e a f a m i l y o f mappings S ( t ) : BUC(RN)

+

BUC(RN) by l e t t i n g

S(t)S(T) = S ( t +

IS(t)uo(x

+

Y)

S(t)uo(*)

One can a l s o show t h a t f o r

s o l u t i o n o f (2.1).

be t,t

u(*,t)

where

> 0 and u o y v o

u f

i s the BUC(RN)

,

T)

- S(t)uoo()l

<

SUP

+

lU&

N

Y)

-

- UO(4I

ZER

The f i r s t two p r o p e r t i e s show t h a t nonexpansi ve mappings on BUC(RN).

S(t)

i s a semigroup o f o r d e r - p r e s e r v i n g

The more complex problem ut + H(x,t,u,Du) U(X,O)

,

= 0

= U"(X)

H t o have t h e v a l i d i t y o f t h e uniqueness a s s e r t i o n .

r e q u i r e s r e s t r i c t i o n s on For example, i f

i s nondecreasing i n

(i) H(x,t,r,p)

(ii)

H(x,t,r,p) Rn

x

,

r

i s u n i f o r m l y c o n t i n u o u s on

[O,T]

[-R,R]

x

x

Ip : lp( < R I

-

H(y,t,r,p)l

f o r each

R

>

0

,

(2.3)

and (iii)

l i m supI(H(x,t,r,p) a+0

(x

- yI

f o r each

< R

a,

>

0

(x

- yllpl

:

< R, 0 < t < T,

Irl

< RI

= 0

,

t h e n bounded v i s c o s i t y s o l u t i o n s o f (2.2) which assume t h e i r i n i t i a l - v a l u e s u n i f o r m l y a r e unique. Note t h a t (2.3) ( i ) r u l e s o u t c o n s e r v a t i o n l a w f o r m f o r (2.2). Assumptions on H may be weakened i n exchange f o r s t r i c t e r requirements

M.G. CRANDALL

124

on s o l u t i o n s . See [l]. The n e c e s s i t y o f some requirement l i k e (2.3) ( i i i ) can be seen by c o n s i d e r i n g t h e l i n e a r H a m i l t o n i a n H(x,t,u,p) i n which = b(x)*p, case (2.3) ( i i i ) reduces t o L i p s c h i t z c o n t i n u i t y o f b ( x ) . From c l a s s i c a l c o n s i d e r a t i o n s one expects problems l i k e (2.2) t o e x h i b i t f i n i t e p r o p a g a t i o n speeds under s u i t a b l e assumptions. T h i s s t i l l h o l d s t r u e i n t h e g e n e r a l i t y o f v i s c o s i t y solutions. F o r example, we have: Theorem 5.

Let

s a t i s f y (2.3)

H

ut + H(x,t,u,Du)

solutions o f

(i),

= 0

L,R,T

on

OT

ql

for

=

>

0

and

RN

x

[O,T].

u,v

be v i s c o s i t y

Assume t h a t

a r e f i n i t e and

-

IH(x,t,r,p) IxlCR-Lt Then

u(x,O)

< LIP

H(x,t,r.p)l and

-

lpl.lql

< C.

Irl

< m ,

O < t < T .

< v(x,O)

1x1 < R i m p l i e s

on

u < v

H

O f course, no such r e s u l t h o l d s i f

on

1x1 < R

-

L t , 0 < t < T.

i s merely continuous i n

p.

W h i l e we have, so f a r , focused on t h e p u r e Cauchy problem, t h e r e s u l t s o f [ I ] cover b o t h (CP) and (DP). Moreover, t h e y a r e more p r e c i s e and g e n e r a l i n t h e cases we have considered. We conclude by g i v i n g a uniqueness and c o n t i n u o u s dependence p r o o f i n t h e s i m p l e s t s i t u a t i o n t o i l l u s t r a t e t h e b a s i c argument w i t h o u t t e c h n i c a l c o m p l e x i t i e s . L e t H E C(RN), and u.v,m,n where

E

N CO(R 1

Co(RN)

(2.4)

denotes t h e c o n t i n u o u s f u n c t i o n s on

RN

vanishing a t i n f i n i t y .

Assume t h a t u + H(Du) < m

,

v + H(Dv) > n

,

i n t h e v i s c o s i t y sense. u(x)

We p r o v e t h a t

-

We may assume t h a t

M = max ( u ( x )

-

v(x))

>

0

.

RN Choose

cp c

q(0) = 1,

Set

D(RN)+ 0 <

such t h a t

v <

cpa(x) = cp(x/a)

cp,

and l e t

- v(Y,))

Ma = ~,(x,-Y,)(u(x,)

The e x i s t e n c e o f

1 and ~ ( x )= 0

xy,,

has compact s u p p o r t .

xy,,

E

1x1 > 1

for

RN

.

(2.7)

satisfy

> v,(x-Y)(u(x)

- a(Y))

for

f o l l o w s from t h e assumptions t h a t Putting y = x

i n (2.8)

X,Y

u,v

E +

and u s i n g q ,

RN 0

at

(2.8) and

1 we f i n d

VISCOSITY SOLUTIONS OF HAMIL TONJACOBI €QUA TIONS

-

M < Ma < u(x,) Since

xae E+(p(*

125

(2.9)

v(Y,)

- y,)(u(*)

-

and (2.5)

v(y,))

holds, Theorem 2 g i v e s (2.10)

S i m i 1a r l y ,

(2.11) where we used t h a t yields Ub,)

Now pn

- V(Y,) - yaI

(Dp)(x

< mb,)

-

- y)

= -Dyp(x

- n(xa)

n(Ya) = mh,)

-

y),

>

i s t h e modulus o f c o n t i n u i t y o f

n.

(xu

< a since

pa(xa

and t h e r e s u l t f o l l o w s upon l e t t i n g

- y).

a

f

0.

S u b t r a c t i n g (2.11)

+

Thus

n(x,) In(x,)

- n(y.1 - n(Ya)l

T h i s w i t h (2.12)

and (2.9)

f r o m (2.10) (2.12)

<

pn(a)

where

give

0.

REFERENCES

1. C r a n d a l l , M. G. and L i o n s , P. L.,

V i s c o s i t y s l u t i o n s o f H a m i l t n-Jacobi equations, Mathematics Research Center Technical Summary Report 82259, U n i v e r s i t y o f Wisconsin-Madison, 1981.

2.

C r a n d a l l , M. G . and Lions, P. L., Approximation o f v i s c o s i t y s o l u t i o n s o f Hamilton-Jacobi equations, i n p r e p a r a t i o n .

3.

Fleming, W. H., N o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s P r o b a b i l i s t i c and game t h e o r e t i c methods, i n : Problems i n N o n l i n e a r Analysis, CIME, Ed. Cremonese, Rome, (1971).

4.

Friedman, A., The Cauchy problem f o r f i r s t o r d e r p a r t i a l d i f f e r e n t i a l equations, I n d i a n a Univ. Math. J. 23 (1973) 27-40.

5.

Krufkov, S . N., F i r s t o r d e r q u a s i l i n e a r equations w i t h s e v e r a l space v a r i a b l e s , Math. USSR-Sb. 10 (1970) 217-243.

6.

Lions, P. L.,

7.

V o l ' p e r t , A. I., The spaces (1967) 225-267.

-

Generalized s o l u t i o n s o f Hamilton-Jacobi e q u a t i o n s , t o appear.

BV and q u a s i l i n e a r equations, Math. USSR-Sb. 2

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) @ North-Holland Publishing Company. 1982

127

BIFURCATION OF STATIONARY VORTEX CONFIGURATIONS Julian I. Palmore University of Illinois Department of Mathematics 1409 West Green Street Urbana, Illinois 61801 U.S.A.

We examine the dynamics problem of finitely many vortices in a circular disk and study stationary configurations of the vortices. For any fixed number of two or more vortices we prove that there are families of stationary vortex configurations in which bifurcation occurs. Sharp lower bounds on the numbers of stationary configurations are obtained by topological methods. INTRODUCTION An old problem of the dynamics of finitely many vortices in the plane is restated for vortices interacting in a circular disk. Our purpose here is to demonstrate the way in which topological and analytical methods can be applied to the problem of vortices in a disk to yield new sharp results on the bifurcation of solutions within families of stationary configurations. We examine the vortex problem as a Hamiltonian dynamical system and use critical point theory to study the stationary configurations. Recall that a stationary configuration of vortices is a configuration such that each vortex moves uniformily in a circle so as to maintain the same relative configuration. Thus, there is a uniformily rotating coordinate system in which the configuration of vortices is fixed. We begin the discussion by stating a main result on bifurcation within families of stationary configurations. Theorem A.

For any n

2 and for any choice of positive circulations K1,

for vortices in a disk, there are n!/2 which bifurcations occur,

...,

K n’ families of stationary configurations in

In order to prove Theorem A we use a principle which is well-known in critical point theory. We select families of stationary configurations from the planar problem and continue these solutions into the problem of vortices in a disk. The index of each stationary configuration is known from the planar problem. By following the continuation of the solutions toward the boundary of the disk we prove that an index change occurs at a degeneracy within the family. The principle employed is that a degeneracy in a family of solutions acccompanied by a change in the index implies bifurcation. The interplay of ideas from celestial mechanics and dynamical systems is evident in the study of the vortex problem. Compare the methods and results of [l]. In [ 2 ] we announced the idea of using topological methods, useful in studying celestial mechanics and the relative equilibria solutions, to study the vortex problem.

120

J.I. PALMORE

A second result is on the classification of stationary vortex configurations.

In [ 3 ] and [ 4 ] we state several results on the classification of stationary configurations in Kirchhoff's problem. Here, the classification for the vortices in the plane carries over to this setting and is modified where bifurcations occur.

of Theorem A we state the following theorem on the existence of particular stationary configurations.

As a starting point for the proof

Theorem B.

For any

n

1. 2 and for any choice of positive circulations

for vortices in a disk, there exist configurations.

n!/2

K1,

..., n' K

families of collinear stationary

STATIONARY VORTEX CONFIGURATIONS AND THE HAMILTONIAN We identify stationary vortex configurations with critical points of the restricted Hamiltonian. Let E2 denote the Euclidean plane with inner product ( , ) and norm 11 11. We write all functions and their derivatives using vector notation for the convenience of the analysis. Let D 2 C E 2 denote the open unit disk, D2 = {x E E2I1IxII < 1). The configuration space of the n-vortex problem in the disk with positive circulations K1,...,K is given by

M

=

(D2)"

-

A

where the diagonal A is the set of all configurations such that two or more vortices coincide. Thus, A = U A and A = {(xl,...,x ) E (D2)"[ xi = xj}. ij Here the union is over all i < j.ij The Hamiltonian function H is defined on M; thus, M is also the phase space of the problem. Hamilton's equations are not used directly; w e write the Hamiltonian in vector notation as

Singularities in H arise on the boundary of the disk and on the diagonal. Those on the boundary appear in the terms in the second sum for which i = j. Those singularities on the diagonal appear in the first sum. The Hamiltonian H(x) is not translation invariant. Therefore, the center of vorticity need not be constant. The Hamiltonian is invariant under rotations of E2. Thus, the function I : M * IR defined by I(x) = 4 C K.IIXil12 is an integral of Hamilton's equation. Let S denote the set CI

sCI

= {(x,,

...,x,)

E

(D~>"\I(X)= a } .

Then S, n M is invariant under the flow defined by the dynamical system. As is the case for Kirchhoff's problem, for positive circulations of the vortices, collisions between vortices in a disk cannot occur for ~1 sufficiently small. Let Ha

denote the restriction of H

to

sCI.

129

BIFURCATION OF STATIONARY VORTEX CONFIGURATIONS

Theorem 1. A configuration (xl,...,xn) the vortices if and only if

(xl,

...,x )

E

Sa is a stationary configuration of is a critical point of

This statement gives us a critical point criterion by which bifurcation within families of stationary configurations can be detected. DETECTING BIFURCATION In order to detect bifurcation in a family of stationary configurations of vortices we examine the index of critical points as an analytic criterion. As an example B of a one parameter family of critical points we consider the function ft : W2 +

defined by

f (x,y) = tx2 t

+

(1

-

t)x4

+ y2.

The derivative of

ft is dft(x,y)

(2tx + 4(1 - t)x3,2y). For all t E W (x,y) = (0,O) is a critical For t 2 0, ( 0 , O ) is the only critical point. For t < 0, there are critical points: ( 0 , O ) ((-t/2(1 t))%,O) and (-(-t/2(1 t))+,O). second derivative is dbft(x,y) = diag(2t + 12(1 - t)x2,2). For t 2 0

=

-

origin is a minimum of

ft,

-

nondegenerate for t > 0. t E R,

of the origin as a critical point of

=

point. three The the

By observing the index

we observe that the index

ft, 0) to 1 (t < 0). The index of a fixed neighborhood of changes from 0 (t theoriginmust be preserved when all critical points are nondegenerate. Thus, for t < 0 there must be at least two additional critical points in a neighborhood of the origin provided that ft has nondegenerate critical points.

The procedure used above to detect a change in the critical points of

ft is the

one we u s e to detect bifurcation within families of stationary configurations. As an example of detecting bifurcation within a family of stationary configurations we consider the two vortex problem with unit circulation. We observe that the symmetric configuration is always a stationary configuration. The vortices are located on a line at positions x1 = (-a,O) and x2 = (a,O) for 0 < a < 1. The second derivative of H, evaluated in the transverse translation direction v = (0,1), i = 1,2 is i 2

D‘H,(x)

(v,v)

2

= 2

[S]

s o

for all 0 < c1 < 1 and v = (v1 ,v 2). Therefore the transverse translation direction is not a null direction. In the longitudinal transverse direction v = (vl,v2), vi = (l,O), second derivative is evaluated as 4 4 D ~a H (x) (v,v) = - -2 - -2 8a2 z + 2x l+a 1-a (14)

i = 1,2,

the

where 2X = 2/a2 (1+3c1”)/(1-a4). The value of a 2 for which the derivative is zero is the positive root of the polynomial equation p(a) = 1 5a+ 3a2 - 7a3 = 0. We find a = = .2137403’ and = .4623206* = a*.

-

The two translation directions are directions in which the Hessian can be decomposed. As the ize of the configuration is increased by increasing a , we find that for a < a3 , the Hessian D2Ha(x)(v,v) is positive so that D2H,(x) is positive definite for < a*. For c1 = a*, the Hessian is degenerate with rank equal to 1. For c1 > a*, the stationary configuration is a saddle point of Ha. In order to maintain the consistancy of the topological type of a fixed neighborhood of the symmetric stationary configuration, we must have two additional critical points, both of them minima with index 0. These critical points are the translates in the direction v of the symmetric configuration,

130

J.I. PALMORE

MAIN THEOREMS In this section we state two results and prove that degeneracy in a translation mode always occurs along a family of collinear vortex configurations. Thus, we prove Theorem A. Several results are needed. Compare [ 4 ] . Theorem 2. For any n 2 and for any choice of positive circulations K~,...,K n' for the vortex problem in the plane, there are n!/2 collinear-stationary configurations, A s critical points of the reduced Hamiltonian H these configurations are nondegenerate with index equal to n - 2. -,

Here H denotes the Hamiltonian _function for Kirchhoff's problem induced on complex projective space Pn-2(k) - An-2 which arises as the reduced phase space in the vortex problem as well as the reduced configuration space of relative equilibria in the planar n-body problem of celestial mechanics. Compare [ l ] . is the maximal dimension of a subspace of The index of a critical point x E S, the tangent space of Sa on which the Hessian D2h(x) is negative definite.

Theorem 2 enables us to prove Theorem B by using the implicit function theorem and the observation that for the n-vortex problem in the disk those stationary configurations for CL sufficiently small must approximate stationary vortex configurations in the plane. Let Tx denote the tangent space of

Sa

the normal space. The Hessian D2Ha(x)

IT^

.

:TI

D~H~(x) e D~H~(x)

n

at x E Sa

1

Let

...,xn)

(xL,

denote

splits on TxSa as D2Ha(x) =

Let v = (l,...,l) E (El)" and v = (ll,...,l ) E and transverse translation directions, respectively. Theorem 3.

and let Tk

i n n S,

E (E )

denote the longitudinal

be a collinear stationary configuration

..., .

of the vortices with positive circulations K ~ , K n non null on the normal subspace of the tangent space.

The Hessian D2G (x)

a

is

A s in the example of two vortices we search for bifurcations within the collinear

configurations in the longitudinal translation direction. Let Fa(xl,

...,x )

parameterized by a , tex configuration.

denote the family of collinear stationary configurations, which contains

Theorem 4. The Hessian D2H (x ),

(xl,

...,xn)

E

...,

xa E Fa(xl, xn), longitudinal translation direction for some a > 0. a

Proof.

a

The Hessian Dz~a(xa)(v,v) D2Ha(Xa)(V,V)

=

1 isj -2

is evaluated as

0

Sa,

a stationary vor-

is degenerate in the

131

BIFURCATION OF STATIONARY VORTEX CONFIGURATIONS

Here A

is the Lagrange multiplier determined by

Within the

the terns io i = j contribute

sum,

For a sufficiently large, this sum is negative as x1 and xn approach the boundary of the disk D2. The sum over i # j reduces to

+ 4(xi' xj) i i X i-xj1 2

-

((l-iix ii2)(1-nx u 2 ) i j Let

IIx 11 = a

i

and

Ilx.ll J

=

2(iixiii2

+

+ n x i-"j

iiX

12

j 11~)~.

-

2uxiu2iix

11211

j

B and examine the function

f(a,B) = (-2+a2+B2)((1-a2)(1-B2) + (a-BI2)

+ 4aB(a-B)2 -

2(a2+B2-2a2B2)

on the unit square 0 z a , B 5 1. The only zero of f(a,B) is at (a,B) = (1,l). Thus, there are no zeros of f in the interior and f(a,B) < 0 f o r 0 < a,B < 1. Consequently, each term in the sum over i # j is negative. Our analysis shows that for any collinear stationary configuration the first sum is negative. As IIxlll or IIx 11 approaches 1 the Hessian is eventually negative. The index of the critical point has changed by configurations.

1 (at least) along the family of stationary

ACKNOWLEDGEMENT This research was supported in part by a grant from the National Science Foundation, U.S.A. REFERENCES J. Palmore, Measure of degenerate relative equilibria, I, Annals of Math. (1976), p. 421-429.

104

3. Palmore, Relative equilibria of interacting vortices in two dimensions, Abstract 768-58-4, Notices of the her. Math. SOC. (1979), p. A-482.

J. Palmore, Central configurations, CW complexes and the homology of projective spaces, in Classical Mechanics and Dynamical Systems, ed. by R. Devaney and Z . Nitecki, Marcel-Dekker, NY (1981), p. 225-237. J. Palmore, Relative equilibria of interacting vortices in two dimensions (1981), to appear in Proc. Nat. Acad. of Sciences.

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North-Holland Publishing Company, 1982

133

EXACT INVARIANTS FOR TIME-DEPENDENT NONLINEAR HAMILTONIAN SYSTEMS H. R. Lewis

Los A1 amos National Laboratory P. 0. Box 1663 Los- A1 amos, NM 87545 U.S.A. and P. G. L. Leach

Department o f Appl i e d Mathematics La Trobe U n i v e r s i t y Bundoora 3083 Australia

Two methods, one based on canonical based on

an assumed

invariants

H

1 p2

for

certain

V(q,t). c l a s s o f nonlinear, =

former

transformations

and

Hamil tonians

of

the

type

I n v a r i a n t s are found e x p l i c i t l y time-dependent p o t e n t i a l s V(q,t).

t

one

structure, a r e used t o determine exact for a The

method i s then developed t o f i n d exact i n v a r i a n t s f o r

Hamiltonians o f t h e form H = H(q,p,p(t),;(t)). I. INTRODUCTION The search f o r exact i n v a r i a n t s f o r nonautonomous Hamil tonian systems has been prompted by t h e o r e t i c a l studies i n plasma physics and quantum mechanics where such systems play an important r o l e . fields,

our

results

are

Apart also

from a p p l i c a t i o n s

of

interest

in

the

in

these

theory

and other

o f canonical

transformations i n a n a l y t i c a l dynamics. Exact i n v a r i a n t s f o r nonautonomous recent years

by

using

a

1inear

systems

have

been

found

during

v a r i e t y o f methods Cll, and i n v a r i a n t s have a l s o been

found f o r some nonlinear systems

C21.

The work t h a t i s reported b r i e f l y here has

as i t s aim the determination o f exact i n v a r i a n t s f o r much broader classes o f nonautonomous systems. I n Section 11, we o u t l i n e a method t h a t uses canonical transformations f o r Hamil tonians o f the form

(1.1)

134

H.R. LEWIS, P.G.L. LEACH

t o o b t a i n i n v a r i a n t s f o r a c e r t a i n c l a s s o f p o t e n t i a l s . The contents o f Section I 1 a r e described i n d e t a i l i n a forthcoming a r t i c l e [31. The i n v a r i a n t s obtained i n Section I 1 a r e a l l quadratic i n p.

I n Section 111, we begin with

the

ansatz

t h a t an i n v a r i a n t i s quadratic i n p,

and solve the equation

d1

Z

a'+ [I,H] at

directly,

= 0

(1.3)

where H has t h e form (1.1).

o f p o t e n t i a l s than i n Section 11.

The r e s u l t i s i n v a r i a n t s f o r a wider c l a s s

I n Section

I V we

return to

t h e method o f

Section 11, b u t do n o t r e s t r i c t the dependence o f the Hamiltonian on p t o be as i n (1.11. 11.

The d e t a i l s o f Sections 111 and I V w i l l be published elsewhere [4,51. CANONICAL TRANSFORMATION APPROACH

I n t h e conventional treatment o f canonical

transformations,

f u n c t i o n contains a mfxture o f new and o l d variables. unconventional type o f generating v a r i a b l e s only.

Q The

Q(q.p.t)

function,

which

the

generating

For t h i s work we employ an

is a

function o f the

old

Consider a canonical transformation from (q,p) t o (Q,P) where

,

P = P(q,p,t)

o r i g i n a l Hamiltonian H(q,p,t)

.

(2.1)

and the new Hamiltonian K(Q(q,p,t)

, P(q,p,t),t)

are r e l a t e d by

(2.2)

where F(q,p,t)

i s t h e generating f u n c t i o n o f

functions

(2.2)

in

c o e f f i c i e n t s o f dq

as

the

transformation.

Treating

functions o f the o l d coordinates only, we separate the

dp and the p a r t t h a t does n o t i n v o l v e e i t h e r t o o b t a i n

aT and aT

all

TIME-DEPENDENTNONLINEAR HAMlL TONIAN SYSTEMS

p -aQt - = aF o aP aP

135

,

(2.4)

H t p -aQ- K t - = aF O .

(2.5)

at

at

We can e l i m i n a t e F(q,p,t)

from (2.3)-(2.5)

i n two ways, viz.,

by

differentiating

(2.5) w i t h respect t o q and (2.3) w i t h respect t o t o r by d i f f e r e n t i a t i n g (2.5) w i t h respect t o p and ( 2 . 4 ) w i t h respect t o t.

So f a r

constraints

the

discussion

has been

quite

on

P

general.

t h a t w i l l enable a r e s u l t t o be obtained.

a u x i l i a r y function, p ( t ) , which i s such t h a t p dependence

We obtain, respectively,

k 0,

We

now

introduce

some

We introduce a n o n t r i v i a l

and we assume

that

the

time

o f the transformation i s expressed completely by dependence o f Q and P

and 15,

We take

and assume t h a t V(q,t) cannot be w r i t t e n w i t h i t s time dependence expressed e n t i r e l y through p ( t ) and 8 t ) . Because the new Hamiltonian, K, does n o t depend on Q, the new momentum, P, w i l l be an i n v a r i a n t .

136

H. R. LEWE, P. G. L. LEACH

With these constraints, (2.6) and (2.7) become

(2.10)

(2.111

i s defined as

i n which the bracket CP,Qla6

(2.12)

I n addition, we use the Poisson bracket r e l a t i o n f o r Q and P and the requirement o f consistency o f the time e v o l u t i o n o f Q i n t h e two coordinate systems, i.e.,

1

CQ.Plqp

,

(2.13)

(2.14)

The

latter

equation

i s also

a n a l y s i s o f (2.10), (2.111, detail

elsewhere C31.

a consequence o f (2.101,

(2.131, It

and (2.14)

(2.111,

and (2.13).

i s lengthy and w i l l

and

6.

We

can

illustrate

functional dependence o f P on consequence o f

(2.10)

in

i s based on examination o f the equations t o uncover

r e s t r i c t i o n s o f functional dependences o f t h e unknown functions on q,p,p,

The

be given

the

variables

type o f argument by d e s c r i b i n g how the

i t s arguments

and (2.11).

the

I n (2.111,

is

restricted

as

an

immediate

each q u a n t i t y except p m a n i f e s t l y

depends on t o n l y through e x p l i c i t dependence on p ( t ) and

d(t).

Therefore, e i t h e r

vanishes, o r p i s expressible completely i n terms o f p and 6, o r both. I f p were expressible completely i n terms o f P and d, then (2.10) would r e q u i r e t h a t

[P,Qlp;

the

dependence

of

av -

be expressible s o l e l y i n terms o f P and aq assumed t h i s n o t t o be t h e case. Therefore, we must have t

d;

b u t we have

(2.15)

TIME-DEPENDENTNONLINEAR HAMIL TONIAN SYSTEMS

Considering t h i s as a f i r s t - o r d e r p a r t i a l d i f f e r e n t i a l equation

137

for

P,

we

then

find

Where

r

i s an a r b i t r a r y function.

The follows

result

[31.

o f c a r r y i n g the

a n a l y s i s t o a conclusion can be summarized as

We have derived an e x p l i c i t i n v a r i a n t f o r any p o t e n t i a l o f the form

(2.17)

where a ( t ) i s any p a r t i c u l a r s o l u t i o n o f the d i f f e r e n t i a l equation

,

a = f,(a,t)

f, i s an a r b i t r a r y function

(2.18)

of

its

such t h a t c: + c: = 1,

constants

arguments, W(u)

c,,

c 1 and c 2 are a r b i t r a r y

i s an a r b i t r a r y f u n c t i o n o f

u, and u i s

defined by

q u =

-

c,(cl c2

+ c2a)

- cla

(2.19)

We have found a canonical transformation for which t h e new Hamiltonfan i s

(2.20)

The i n v a r i a n t i s the new momentum, which i s given by

(2.21)

I 313

H. R. LEWIS, P.G. L. LEACH

where

v = (c,

-

cla)p + d(c,q

-

.

c,)

(2.221

The new coordinate i s

(2.23)

where T i s an a r b i t r a r y function. These r e s u l t s may be i n t e r p r e t e d i n terms o f a transformation t o v a r i a b l e s under a general i t e d canonical transformation

action-angle

where Q and P a r e given by (2.231 and (2.211, respectively, and

T =

t

J (c2

- c,a(t'))'2

dt'

.

The new Hamiltonian i s then simply t h e i n v a r i a n t 111.

(2.24)

P.

INVARIANTS QUADRATIC I N THE MOMENTUM

Because t h e i n v a r i a n t s found i n Section I 1 a r e a l l quadratfc i n the momentum p, i t i s natural t o i n q u i r e whether t h e admissible c l a s s o f p o t e n t i a l s associated with

those i n v a r i a n t s contains a l l p o t e n t i a l s f o r which t h e r e e x i s t s an i n v a r i a n t

quadratic i n p.

H =

1

Thus, we assume a Hamiltonian

p 2 + V(q,t)

and make the ansatz t h a t there e x i s t s an i n v a r l a n t o f the form

(3.1)

TIME-DEPENDEN T NONLINEAR HAMI L TONlAN SYSTEMS

139

That i s , I i s t o s a t i s f y

(3.31

Substitution

of

(3.1) and (3.2) i n t o (3.3) gives the system o f

partial

d i f f e r e n t i a l equations

afO aq

- f

The

1

-2f

2

av a f l -+-=o aq

av a f O -+-=o aq

at

.

s o l u t i o n o f (3.4)

elsewhere [41.

(3.6)

at

(3.7)

-

(3.71 i s r e l a t i v e l y s t r a i g h t f o r w a r d and w i l l be described

The r e s u l t i s the following.

There e x i s t s an i n v a r i a n t quadratic

i n p i f , and only i f , the p o t e n t i a l i s o f the form

where F, a, and W are a r b i t r a r y functions, p 1 and p P are any p a r t i c u l a r s o l u t i o n s of

H. R. LEWIS, P.G. L. LEACH

(3.10)

and k i s an a r b i t r a r y constant.

1(q,p,t)

1

= +P1(P

-

b2)

The i n v a r i a n t quadratic i n p i 5

-

(3.11)

T h i s r e s u l t i s a g e n e r a l i z a t i o n o f the r e s u l t obtained i n Section

I 1 because

the p o t e n t i a l may now contain two independent functions o f time. FURTHER DEVELOPMENT OF THE CANONICAL TRANSFORMATION APPROACH

IV.

I n Section

11,

Hamiltonian equal t o to

our canonical transformation approach was a p p l i e d t o a 1 p 2 p l u s a p o t e n t i a l whose time dependence was assumed not

be expressible e n t i r e l y

through dependence on

p(t)

and b ( t ) .

We now assume

t h a t the time dependence o f t h e Hamiltonian i s expressible s o l e l y i n terms o f p ( t ) and

b ( t ) , and we a l l o w a general dependence of t h e Hamiltonian on p.

That i s , we

take

The new canonical variables Q and P again are taken t o be functions o f q, p,

b.

and

Equations (2.10) and (2.11) now read

I n (4.2) p

p

and (4.31,

a l l q u a n t i t i e s except P’ depend on t o n l y through dependence on

and 15. Therefore, we e i t h e r take

TIME-DEPENDENTNONLINEAR HAMlL TONIAN SYSTEMS

[Q,P1bq

,

= 0

= 0

[Q,PIp;

,

141

(4.41

o r r e q u i r e p t o s a t i s f y a d i f f e r e n t i a l equation o f the form

.

P = g(p,b)

(4.5)

k

(As before, we r e q u i r e p

0.) The former choice i s t o o r e s t r i c t i v e .

Adopting t h e

l a t t e r , we w r i t e

These two equations may be combined t o d i f f e r e n t i a l equation f o r P,

give

a

single

homogeneous p a r t i a l

The s o l u t i o n i s

where u(q,p,p)

i s r e l a t e d t o f , and f, by

(4.9)

and f i s an a r b i t r a r y function. Substituting

r

f o r P i n (4.2)

and

(4.31,

taking

two

combinations o f t h e

r e s u l t i n g equations, and r e w r i t i n g the Poisson bracket requirement on Q and P, we o b t a i n the fundamental equations

aK *=

[Q,Hl

-+

b - aQ -+ ap

P -aQ

a;

,

(4.10)

142

H.R. LEWIS, P.G.L. LEACH

T aK T ar --- - air - - ar au

’

ap

(4.11)

(4.12)

where

. .au + [u,H] P

(4.13)

a0

D e t a i l e d analysis o f r e s t r i c t i o n s on functional dependences r e q u i r e d by these equations f i n a l l y leads t o a r e s u l t . The e n t i r e d e r i v a t i o n w i l l be published elsewhere 151.

Here we summarize the r e s u l t s .

We f i n d t h a t Q may be w r i t t e n as a f u n c t i o n o f two canonical variables, u

(as

defined above) and v:

(4.14)

where

(4.15)

and

CU,Vl

= [u,II = 1

The functions

ad a;

-+

aS + a~ [u,S] -

a4

a;

(4.16)

u and v a l s o s a t i s f y

a] [u,irl

a;

.

= 0

,

(4.17)

= 0

.

(4.18)

TIME-DEPENDENTNONLINEAR HAMIL TONIAN SYSTEMS

The d o t denotes t o t a l time d i f f e r e n t i a t i o n , as i n (4.131,

143

and a bracket w i t h o u t

subscripts denotes the usual Poisson bracket w i t h respect t o q and p.

As

i n t h e problem considered i n Section 11, there i s no loss o f g e n e r a l i t y i n

t a k i n g K(P,p) t o be l i n e a r i n P and, i n fact, we take

where N(u.v) i s the invariant.

The permissible form o f the Hamiltonian

is

given

by

where

=

The functions ,,a,

al,

and a 2 are r e l a t e d t o

(=Dl by

u, v and g(p,b)

[u,a,l

6.

i s not l i n e a r i n

a2(u,p,6)

o , 2. a0 t

[usall

=

o ,

a1 , % + CI,all

[ I , a o l = G,(u,P)

(4.21)

= G,(u,P)

,

(4.22)

(4.23)

-

Equations (4.21) constitute

a

(4.23) together w i t h t h e Poisson bracket r e l a t i o n f o r u

consistent

set

and

v

o f equations i n themselves and are c o n s i s t e n t w i t h

(4.17) and (4.18). V.

DISCUSSION

The r e s u l t s o f the various analyses t h a t have been described b r i e f l y here i n d i c a t e t h a t i t i s possible t o find i n v a r i a n t s f o r a wide class o f tfme-dependent one-dimensional Hamiltonian systems.

Such r e s u l t s w i l l

have

generalizations

to

144

H.R. LEWIS, P.G.L. LEACH

higher

dimensions

as

has

been

found

by

Lewis

counterpart o f the problem described i n Section

[6] f o r the three-dimensional The

111.

results

reported

results.

I n t h e e a r l i e r work, canonical coordinates, here c a l l e d (u,v),

obtained

for

which

the

transformation

from

(q,p)

to

(u,v)

coordinate transformation q t o u w i t h the momentum transformation being

in

I V mark a fundamental change from those o f Section I 1 and I 1 1 and e a r l i e r

Section

induced

by

the coordinate transformation.

f o r more general transformations i s admitted.

have been

consisted from p

in a to

v

I n Section I V , t h e p o s s i b i l i t y

Indeed, one might l i s t the types o f

problem t h a t could be t r e a t e d i n terms o f t h e p dependence o f u.

I n Sections

I 1 and IV, the time dependence o f the canonical transformations i n Section I 1 1 we saw t h a t two

was through t h e s i n g l e function p ( t ) whereas independent

functions

may

arise.

Part

of

our

program o f

investigation o f

i n v a r i a n t s f o r Hamiltonian systems i s t o consider, using t h e method o f Sections I 1 and

IV,

the e f f e c t o f i n t r o d u c i n g several independent functions.

The o t h e r p a r t

i s t o apply t h e method o f Section I V t o a Hamiltonian whose time dependence i s n o t through p ( t ) and b ( t ) alone.

We hope t o r e p o r t on these matters soon.

ACKNOWLEDGMENTS Much o f

the work

described here was performed w h i l e one o f us (PGLL) was a

V i s i t i n g Scientist with the National Laboratory.

Center

for

Nonlinear

Studies

at

the

Los

Alamos

Without the support o f t h e Center and a t r a v e l grant from La

Trobe U n i v e r s i t y the v i s i t would n o t have been possible.

TIME-DEPENDEN T NONLINEAR HAMIL TONIAN SYSTEMS

145

REFERENCES

C11 Lewis,

H. R.,

Phys. Rev. L e t t . 18

Phys. Rev. L e t t . 18

1976-1986; Lewis, Bahar, L. Y., their

(1967)

(1967) 636; Lewis, H.

R.,

Phys. Rev.

Quadratic i n t e g r a l s

for

and

510-512,

J. Math. Phys.

H. R.,

Euratom,

9 (1968)

172 (1968) 1313-1315; Sarlet, W. and linear

nonconservative

systems

and

connection w i t h the inverse problem o f Lagrangian dynamics ( p r e p r i n t ,

RUGsam 80-16, September Rijksuniversiteit, Math. SOC.

Theoretische

Mechanica,

Gent, 8-9000 Gent, Belgium); Leach, P. G. L.,

1980, I n s t i t u u t voor

J. Austral.

208 (1977) 97-105; Leach,

P. G. L.,

J. Math. Phys.

21 (1980)

300-304. C2l S a r l e t ,

W.

and Bahar,

Leach, P. G. L.,

i n J. Math. Phys.; 20 (1979) 2054-2057; Ray,

H. R. and Leach,

time-dependent

I n t . J. Nonlinear Mech.

t o be published

J. L., J. Math. Phys. (1980) 4-6.

C31 Lewis,

L. Y.,

nonlinear

P. G. L.,

15 (1980) 133-146;

Ray,

J. R. and Reid,

J. R.,

Exact i n v a r i a n t s f o r Hamil t o n i a n systems, to be

Phys. L e t t . 78A

a

class published

of in

J. Math. Phys. C41 Lewis,

H. R. and Leach,

invariants

for

P. G. L.,

A d i r e c t approach t o f i n d i n g exact one-dimensional time-dependent c l a s s i c a l Hamil tonians, t o be

submitted t o J. Math Phys.

[51 Leach, P. G. L., Lewis, H. R. and Sarlet, W.,

Exact i n v a r i a n t s f o r a c l a s s o f time-dependent nonlinear Hamiltonian Systems, 111, t o be submitted t o J. Math

Phys.

C6l Lewis, H. R.,

I n v a r i a n t s quadratic i n the momentum f o r three-dimensional scalar and vector p o t e n t i a l s , t o be submitted t o J. Math. Phys.

This Page Intentionally Left Blank

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) @ North-Holland Publishing Company, 1982

147

ISOLATING INTEGRALS I N GALACTIC DYNAMICS AND THE CHARACTER OF STELLAR ORBITS W i l l i a m I . Newman Department o f Earth and Space Sciences University o f California Los Angoles, C a l i f o r n i a 90024

The necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f separable s o l u t i o n s t o t h e Hamilton-Jacobi equation a r e w e l l known. We show how t h e Hamilton-Lagrange equations o f motion can be employed t o o b t a i n t h e same i n t e g r a b i l i t y c o n d i t i o n s and i s o l a t i n g i n t e g r a l s o f motion. I n p a r t i c u l a r , f o r p o t e n t i a l s t h a t a r e used i n g a l a c t i c dynamics t h a t do n o t possess t h r e e i s o l a t i n g i n t e g r a l s , t h i s approach f a c i l i t a t e s t h e use o f p e r t u r b a t i o n methods and t h e understanding o f t h e s o - c a l l e d box, t u b e and s h e l l o r b i t s t h a t have been observed i n numerical studies. INTRODUCTION AND REVIEW Galaxies a r e ensembles o f $1011 s e l f - g r a v i t a t i n g s t a r s . I n d i v i d u a l s t e l l a r o r b i t s a r e e s s e n t i a l l y f r e e from strong b i n a r y i n t e r a c t i o n s (i.e. c o l l i s i o n times a r e s u b s t a n t i a l l y l o n g e r than t h e age o f t h e universe f o r t y p i c a l g a l a x i e s ) . Indeed, each s t a r appears t o move under t h e i n f l u e n c e o f a g r a v i t a t i o n a l p o t e n t i a l which describes t h e cumulative e f f e c t o f t h e mass d i s t r i b u t i o n o f t h e galaxy. The t i m e s c a l e f o r an o r b i t i s $108 years and a g i v e n s t a r w i l l have executed $100 o r b i t s moving a t a speed o f several hundred km/sec d u r i n g t h e g a l a x y ' s existence. Many g a l a x i e s show l i t t l e s i g n o f e v o l u t i o n and, we can surmise, t h e g r a v i t a t i o n a l p o t e n t i a l associated w i t h t h e i r mass d i s t r i b u t i o n s changes very l i t t l e w i t h time. I n i t s most p r i m i t i v e form, g a l a c t i c dynamics seeks t o e x p l a i n t h e q u a l i t a t i v e character o f s t e l l a r o r b i t s due t o t h e g a l a c t i c g r a v i t a t i o n a l p o t e n t i a l . Then, we can hope t o b e t t e r understand how s t e l l a r o r b i t s c o n t r i b u t e t o t h e s t a b i l i t y o f a galaxy t o g r a v i t a t i o n a l c o l l a p s e and t o t h e morphology o f t h e i r host. Since t h e s t a r s which c o n s t i t u t e a galaxy a r e so numerous and m a n i f e s t o n l y c o l l e c t i v e e f f e c t s , we t r e a t t h e g a l a c t i c mass d i s t r i b u t i o n as a f l u i d w i t h mass d e n s i t y p ( x ) a t s p a t i a l p o s i t i o n x. [For t h e reasons g i v e n above, we assume t h a t p i s independent o f time.] Corresponding t o t h i s mass d e n s i t y i s t h e g r a v i t a t i o n a l p o t e n t i a l V(x) which s a t i s f i e s t h e g r a v i t a t i o n a l Poisson's e q u a t i o n

2

V V(x) = 471Gp(5)

From observations o f galaxies, we a r e a b l e t o (approximately) deduce t h e mass density. I n p a r t i c u l a r , we w i l l consider t h e s o - c a l l e d " e l l i p t i c a l galaxies," f o r which surfaces o f equal mass d e n s i t y a r e w e l l - d e s c r i b e d b y e l l i p s o i d s w i t h unequal axes. By employing t h e Green's f u n c t i o n f o r Poisson's equation, t h e s o l u t i o n f o r t h e p o t e n t i a l i n (1) reduces t o a quadrature. The r e s u l t i n g potent i a l i s g e n e r a l l y q u i t e a s y n e t r i c b u t i s otherwise smooth and f r e e from o s c i l l a t i o n s . We wish, then, t o understand t h e c h a r a c t e r o f s t e l l a r o r b i t s under t h e i n f l u e n c e o f t h i s o b s e r v a t i o n a l ly-deduced p o t e n t i a l , I n a c o n s e r v a t i v e f o r c e f i e l d , t h e motion o f a p a r t i c l e (1.e. a s t a r ) has a t l e a s t one i n t e g r a l o f motion, t h e energy o f t h e p a r t i c l e . I f t h i s were t h e o n l y i n t e g r a l

148

W.I. NEWMAN

o f motion, t h e s t a r would be a b l e t o v i s i t all r e g i o n s o f phase space t h a t a r e e n e r g e t i c a l l y a v a i l a b l e t o i t g i v e n s u f f i c i e n t time. T h i s " e r g o d i c hypothesis" t o g e t h e r w i t h t h e random c h a r a c t e r o f t h e r e s u l t i n g s t e l l a r o r b i t s c o u l d undermine t h e g l o b a l s t a b i l i t y o f a galaxy. The e x i s t e n c e o f o t h e r i n t e g r a l s o f motion would reduce t h e r e g i o n o f phase space a c c e s s i b l e t o a s t a r and e l i m i n a t e t h e s t o c h a s t i c behavior t h a t would o t h e r w i s e r e s u l t . Should e x t r a i n t e g r a l s o f motion e x i s t , i t would be p o s s i b l e t o assemble a l i n e a r combination o f s t e l l a r o r b i t s which would reproduce t h e u n d e r l y i n g mass d e n s i t y d i s t r i b u t i o n , i.e. o b t a i n a s e l f - c o n s i s t e n t d e s c r i p t i o n o f dynamical e q u i l i b r i u m w i t h a d e n s i t y p r o f i l e c o r responding t o e l l i p t i c a l galaxies. Thus, we would l i k e t o determine t h e number (and n a t u r e ) o f i n t e g r a l s o f motion f o r a g i v e n g r a v i t a t i o n a l p o t e n t i a l . Loosely speaking, i n t e g r a l s o f motion correspond t o a q u a n t i t y t h a t i s conserved i n each o f i t s coordinates. For example, a three-dimensional simple harmonic o s c i l l a t o r preserves t h e energy o f v i b r a t i o n i n each o f i t s ( p r i n c i p a l ) c o o r d i n a t e axes. I n a d d i t i o n , i f t h e " s p r i n g constant" i s i s o t r o p i c (i.e. t h e r e s t o r i n g f o r c e depends o n l y upon displacement from e q u i l i b r i u m and n o t upon t h e d i r e c t i o n o f displacement), the frequency o f v i b r a t i o n along each o f t h e c o o r d i n a t e axes i s t h e same, which then c o n t r i b u t e s two more i n t e g r a l s o f t h e motion. From t h e s i x Lagrange o r Hamilton equations o f motion f o r a p a r t i c l e i n a time-independent, c o n s e r v a t i v e f o r c e f i e l d , we can have a t most f i v e i n t e g r a l s o f motion. However, o n l y t h e i s o t r o p i c harmonic o s c i l l a t o r ( d e s c r i b e d above) and t h e Kepler problem o f c e l e s t i a l mechanics possess f i v e i n t e g r a l s o f motion (Goldstein, 1980). I n those two cases, t h e f o u r t h and f i f t h i n t e g r a l s o f motion a r e a b i p r o d u c t o f t h e s p e c i a l symnetry and frequency degeneracy i n t h e problem. S p h e r i c a l l y symmetric p o t e n t i a l s possess,in general, f o u r i n t e g r a l s o f motion, t h e energy and t h e t h r e e components o f t h e v e c t o r a n g u l a r momentum. (The f o u r t h i n t e g r a l o f motion can again be considered t o be t h e r e s u l t o f a frequency degeneracy between t h e e and Q motions o f t h e p a r t i c l e , ) I n t h e absence o f s p h e r i c a l symmetry, a maximum o f t h r e e constants o f t h e motion can be obtained. The problem o f determining how many i n t e g r a l s o f motion e x i s t f o r a g i v e n potent i a l remains unsolved. I n 1891, StB'ckel determined t h e c o n d i t i o n s under which the s o l u t i o n t o Hamilton-Jacobi e q u a t i o n were a d d i t i v e l y separable, a s i t u a t i o n wherein t h r e e i n t e g r a l s o f t h e motion n a t u r a l l y emerge. I n t h i s case, t h 2 dynamical behavior i n each o f t h e t h r e e c o o r d i n a t e s would be p e r i o d i c w i t h a w e l l d e f i n e d frequency. Usually, t h e t h r e e frequencies would be d i s t i n c t and t h e behavior i n each degree o f freedom would be independent o f t h e o t h e r two. The i n t e g r a l s o f motion would d e s c r i b e t h e amount o f energy o r momentum t h a t c o u l d be associated w i t h each coordinate. Since t h e motion i n each c o o r d i n a t e i s p e r i o d i c , i t i s customary t o express each i n t e g r a l as an " a c t i o n v a r i a b l e , " a q u a n t i t y t h a t has t h e dimensions o f angular momentum ( o r o f energy times t h e p e r i o d ) . I t should be p o i n t e d o u t t h a t each a c t i o n v a r i a b l e e s t a b l i s h e s t h e frequency o f motion along t h e corresponding c o o r d i n a t e as w e l l as t h e s p a t i a l e x t e n t o f t h e o s c i l l a t i o n . I n t h a t sense, t h e i n t e g r a l s o f motion determine t h e " c l a s s i c a l t u r n i n g p o i n t s " a l o n g each o f t h e independent axes. C l e a r l y , t h i s k i n d o f c o o r d i n a t e s e p a r a b i l i t y i s h i g h l y geometry-dependent. StEckel (1891) showed t h a t t h e necessary and s u f f i c i e n t c o n d i t i o n s f o r s e p a r a b i l i t y ( i n t h e Hamil ton-Jacobi problem) r e q u i r e d t h a t t h e p o t e n t i a l possess c e r t a i n symnetries i n c o o r d i n a t e bases d e r i v e d from confocal quadrics (Morse and Feshbach, 1953; Pars, 1965),a s e t o f eleven c o o r d i n a t e systems o b t a i n e d from c o n i c s e c t i o n s which f i n d f r e q u e n t a p p l i c a t i o n i n mathematical physics. T h i s problem became fashiona b l e i n t h e 1920's and 1930's as p h y s i c i s t s sought t o understand t h e connection between c l a s s i c a l and quantum mechanics. E i s e n h a r t (1934) showed t h a t t h e o n l y orthogonal systems o f coordinates f o r which s e p a r a b i l i t y o f S c h r z d i n g e r ' s e q u a t i o n was p o s s i b l e were those i n which t h e Hamilton-Jacobi e q u a t i o n was separable. The t r a n s i t i o n from c l a s s i c a l t o quantum mechanics was f u r t h e r r e i n f o r c e d by t h e observation t h a t t h e a c t i o n v a r i a b l e s were "quantized" i n t h e process; i.e. t h e y became eigenvalues o f a separable s e t o f o r d i n a r y d i f f e r e n t i a l equations. LyndenB e l l (1962) r e v i v e d t h e i s s u e o f i n t e g r a l s o f t h e motion i n a p p l i c a t i o n t o s t e l l a r (i.e. g a l a c t i c ) dynamics. C a l l i n g t h e constants o f t h e motion " i s o l a t i n g

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i n t e g r a l s , " he proceeded t o o b t a i n i n t e g r a b i l i t y c o n d i t i o n s t h a t were q u i t e s i m i l a r t o StB'ckel 'S (1891) f o r orthogonal c o o r d i n a t e systems. These c o n d i t i o n s , however, are n o t e x c l u s i v e as t h e d i s c o v e r y by H6non (see Berry, 1978) o f a t h i r d i n t e g r a l f o r t h e Toda p o t e n t i a l has shown. Indeed, t h e overthrow o f t h e e r g o d i c hypothesis (Kolmogorov, 1954) and t h e emergence o f t h e Kolmogorov-Arnold-Moser theorem has r e v o l u t i o n i z e d o u r way o f t h i n k i n g about i n t e g r a b i l i t y and g a l a c t i c dynamics.

E l l i p t i c a l g a l a x i e s may be considered t o be spherically-symmetric o b j e c t s which have experienced some asymnetrizing i n f l u e n c e , I n p a r t i c u l a r , we may r e g a r d t h e p o t e n t i a l V ( 5 ) as being a s p h e r i c a l l y - s y m n e t r i c term w i t h an a d d i t i v e a s p h e r i c a l p e r t u r b a t i v e p o t e n t i a l . Thus, i n t h e sense o f Kolmogorov, Arnold and Moser, we might expect t h e p e r t u r b a t i o n t o preserve t h r e e i n t e g r a l s o f motion, although t h e i r exact c h a r a c t e r i s unknown. (The f o u r t h i n t e g r a l o f motion common t o s p h e r i c a l l y - s y m n e t r i c p o t e n t i a l s , an a r t i f a c t o f frequency degeneracy, i s deI t has been long-suspected by astronomers, n o t a b l y stroyed b y t h e p e r t u r b a t i o n . ) Contopoulos (1967), t h a t t h e i n t e g r a l s o f t h e motion ( a p a r t from t h e energy) are m o d i f i e d forms o f angular momentum. As a paradigm f o r t h i s problem, Contopoulos considered axisymnetric p e r t u r b a t i o n s t o an otherwise s p h e r i c a l l y - s y m n e t r i c p o t e n t i a l . Then, i t i s easy t o show t h a t t h e angular momentum component a l i g n e d along t h e symmetry a x i s i s e x a c t l y conserved, i n a d d i t i o n t o t h e energy. Were i t n o t f o r t h e p e r t u r b a t i o n , t h e " t h i r d i n t e g r a l " would b e associated w i t h t h e t o t a l angular momentum ( i .e. the magnitude o f t h e angular-momentum v e c t o r ) . Saaf (1968) and Innanen and Papp (1977), among others, have attempted t o show t h a t t h e t h i r d i n t e g r a l o f motion i n aspherical mass d i s t r i b u t i o n s i s t h e t o t a l a n g u l a r momentum. T h e i r work, however, d i d n o t accomnodate f u l l y t h e n o n l i n e a r c h a r a c t e r o f g a l a c t i c p o t e n t i a l s and d i d n o t completely r e s o l v e t h e c h a r a c t e r o f s t e l l a r o r b i t s and t h e n a t u r e o f t h e t h i r d , i s o l a t i n g i n t e g r a l o f motion. F o r t h i s reason, astronomers have employed numerical i n t e g r a t i o n s o f t h e equations o f motion t o b e t t e r understand t h e problem, Computer-generated s t e l l a r o r b i t s do n o t p r o v i d e d i r e c t i n f o r m a t i o n about t h e i n t e g r a l s o f motion. The energy o f a p a r t i c l e i s conserved and each angular momentum component undergoes complex u n d u l a t i o n s about some seemingly constant value. Astronomers have adopted two t e s t s f o r i n t e g r a b i l i t y or, conversely, I n t h e f i r s t , they p l o t t h e s t e l l a r o r b i t p r o j e c t e d o n t o an erogodicity. a r b i t r a r y surface, u s u a l l y a plane i n Cartesian geometry which passes through t h e coordinate o r i g i n . On t h i s f i g u r e , astronomers a l s o p l o t a curve t h a t demarcates t h e r e g i o n a c c e s s i b l e t o t h e p a r t i c l e . For example, t h i s c o u l d be a p r o j e c t i o n o f t h e " z e r o - v e l o c i t y s u r f a c e " (i.e. t h e s u r f a c e c o n t a i n i n g t h e volume i n which energy conservation provides t h e p a r t i c l e w i t h a p o s i t i v e k i n e t i c energy) onto t h e plane. I f t h e erogodic hypothesis were t o hold, a l l regions contained w i t h i n the z e r o - v e l o c i t y surface would be a c c e s s i b l e t o t h e s t a r . The p l o t s produced by numerical experiments show t h a t t h e s t e l l a r o r b i t s a r e u s u a l l y c o n f i n e d t o a s m a l l e r area on t h i s f i g u r e than t h e e r g o d i c hypothesis would suggest. Contopoulos (1967) has c a l l e d t h e c h a r a c t e r i s t i c shapes ( t h a t demarcate t h e area f i l l e d by s t e l l a r o r b i t s ) boxes, tubes o r s h e l l s . These generic forms o f o r b i t s can be e a s i l y understood i n terms o f t h e Stackel (1891) separable p o t e n t i a l s . I n t h e orthogonal coordinates c o n s t r u c t e d from confocal quadrics, we saw t h a t t h e motion along each c o o r d i n a t e a x i s was independent and had a p a i r o f c l a s s i c a l t u r n i n g p o i n t s . Thus, f o r example, i f we c o u l d express t h e p o t e n t i a l i n C a r t e s i a n coordinates as U l ( x ) + U 2 ( y ) +Ug(z) where U i , i = 1,2,3 a r e r e a l - v a l u e d f u n c t i o n s o f t h e associated coordinate, t h e s t e l l a r o r b i t would be bounded according t o

z a -< z c zb

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Here, Xa, Xb, etc. are the c l a s s i c a l t u r n i n g p o i n t s which a r e r e l a t e d t o t h e i s o l a t i n g i n t e g r a l s ( i n t h i s geometry, the energies o f v i b r a t i o n along each a x i s ) and t o the shape o f U1, U2, and u3. Equation ( 2 ) describes a "box o r b i t " since the c o n s t r a i n t s produce a rectangular parallelepiped. I n o t h e r separable geomet r i e s , notably s h e r i c a l geometry, t h e p i c t o r i a l association w i t h a box i s somewhat strained, [For s p h e r i c a l l y symmetric p o t e n t i a l s , r, 8, and $ have c o n s t r a i n t s o f the form ( 2 ) , i.e. a r e i n d i v i d u a l l y bounded.] A s h e l l o r b i t emerges i f t h e c o n s t r a i n t s i n (Z), f o r example,are such t h a t one o f t h e coordinates does n o t change appreciably. I n spherical geometry, c o n f i n i n g t h e r a d i u s t o a narrow range o f values would produce a s h e l l o r b i t . Tube o r b i t s , on t h e o t h e r hand, r e s u l t from n e a r l y resonant frequencies associated w i t h the separate coordinates. (Since o n l y a100 o r b i t a l periods c o n s t i t u t e a g a l a c t i c l i f e t i m e , near-resonance condit i o n s produce tube o r b i t s . I f s u b s t a n t i a l l y more o r b i t a l periods were involved, the tube o r b i t would assume a b o x - l i k e character,) These o r b i t s appear very much l i k e a "Lissajous figure." Although t h i s discussion o f o r b i t types i s s t r i c t l y c o r r e c t o n l y f o r separable systems, s i m i l a r behavior i s manifest i n o r b i t s p l o t t e d i n p r o j e c t i o n ( i n the manner described above) which do n o t d i s p l a y e r g o d i c i t y . This f i r s t t e s t , then, provides s u b s t a n t i a l i n s i g h t i n t o the i n t e g r a b i l i t y o f the problem. The second technique, o r i g i n a l l y suggested by Poincarb, involves a mapping which i s most r e a d i l y applied t o two-dimensional problems. I t can be a p p l i e d t o g a l a c t i c problems where the p o t e n t i a l i s azimuthally symmetric, i.e. does n o t depend on 0 (Richstone, 1981). Since t h e Lagrangian has one ignorable coordinate, the Hamiltonian s i m p l i f i e s t o one i n c o r p o r a t i n g two s p a t i a l coordinates, the c y l i n d r i c a l coordinates r and z. The mapping, known as the surface o f section, i s the i n t e r s e c t i o n o f the constant energy surface w i t h the e q u a t o r i a l o r g a l a c t i c plane ( z = 0) and has r and pr ( t h e momentum conjugate t o r) as i t s coordinates. Each time the s t a r crosses the g a l a c t i c plane i n the upgoing ( o r , a l t e r n a t i v e l y , the downgoing) d i r e c t i o n , a p o i n t i s made on t h e ( r , p r ) plane. This mapping process i s area preserving (a consequence o f L i o u v i l l e ' s Theorem) and reveals whether o r n o t the motion i s integrable. I f the system i s i n t e g r a b l e , the p o i n t s appear t o form a smooth, closed curve. Otherwise, the mapped p o i n t s explore a two-dimensional region o f the (r,pr) plane. I n concluding o u r discussion o f numerical t e s t s f o r i n t e g r a b i l i t y , i t i s important t o d i s t i n g u i s h between the mathematical d e f i n i t i o n o f i n t e g r a b i l i t y and t h a t adopted by the astrophysical c o n u n i t y . To the mathematician, an i n t e g r a l o f motion i s a q u a n t i t y t h a t never changes as t + To t h e a s t r o p h y s i c i s t , an i s o l a t i n g i n t e g r a l i s a q u a n t i t y t h a t has n o t p e r c e p t i v e l y changed during the l i f e t i m e o f the universe (%lo0 s t e l l a r o r b i t periods). I n producing p r o j e c t e d p l o t s o f o r b i t s and surfaces o f section, t h e a s t r o p h y s i c i s t w i l l cease the numerical i n t e g r a t i o n a f t e r 1010 years. I f by t h a t time no m a n i f e s t l y ergodic behavior i s present, he w i l l conclude t h a t the p o t e n t i a l i s i n t e g r a b l e !

-.

The numerical experiments o f Schwarzschild (1979) and Richstone (1980, 1981) have provided much i n s i g h t i n t o g a l a c t i c dynamics. Schwarzschild's work, i n p a r t i c u l a r , has shown t h a t i t i s possible t o superpose s t e l l a r o r b i t s so as t o reproduce the underlying mass d e n s i t y d i s t r i b u t i o n . I n o t h e r words, he obtained a consistent ( a l b e i t numerical) d e s c r i p t i o n o f dynamical e q u i l i b r i u m where the density p r o f i l e corresponded t o t h a t o f observed e l l i p t i c a l galaxies. Since Schwarzschild's'model galaxy had no symnetry axis, surfaces o f s e c t i o n were n o t a v a i l a b l e as a diagnostic t e s t o f i n t e g r a b i l i t y , Other numerical t e s t s t h a t he performed, however, provided a strong i n d i c a t i o n o f t h e existence o f a second and t h i r d i n t e g r a l o f motion. Richstone (1980, 1981) .chose a model galaxy possessing a symmetry a x i s so t h a t the p e r t u r b i n g p o t e n t i a l was a f u n c t i o n o f the g a l a c t i c l a t i t u d e ( e s s e n t i a l l y , n/2 e ) and n o t o f 0. He was, therefore, a b l e t o p l o t s t e l l a r o r b i t s i n p r o j e c t i o n as w e l l as surfaces o f section. O f the many cases t h a t he considered, n o t one was ergodic! I n t h e work o f Schwarzschild (1979) and Richstone (1980, 1981) , the angular momentum o s c i l l a t e d around a constant value suggesting t h a t t h e t o t a l angular momentum was t h e leading term i n a p e r t u r b a t i o n expression f o r the t h i r d i n t e g r a l o f motion.

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The a s s o c i a t i o n between a n g u l a r momentum and i n t e g r a l s o f t h e m o t i o n f o r s t e l l a r systems i n modest d e p a r t u r e from s p h e r i c a l symmetry p r o v i d e s a m o t i v a t i o n f o r e x p l o r i n g t h e r e l a t i o n s h i p between c o o r d i n a t e geometry and i s o l a t i n g i n t e g r a l s . The s t a n d a r d approach [see, f o r example, Landau and L i f s h i t z (1960) o r G o l d s t e i n ( 1 9 8 0 ) l employs an a n a l y s i s o f t h e Hamilton-Jacobi e q u a t i o n , We show h e r e t h a t t h e Hamilton-Langrange e q u a t i o n s o f m o t i o n can be employed i n o r d e r t o o b t a i n t h e same i n t e g r a b i l i t y c o n d i t i o n s , a l t h o u g h we s h a l l do so e x p l i c i t l y o n l y f o r s p h e r i c a l c o o r d i n a t e geometry ( t h e geometry b e s t s u i t e d t o e l l i p t i c a l g a l a x i e s ) . T h i s method o f a n a l y s i s i s e s p e c i a l l y v a l u a b l e s i n c e dynamical problems a r e most e a s i l y s o l v e d u s i n g t h e Lagrange o r H a m i l t o n e q u a t i o n s o f m o t i o n and f a c i l i t a t e s t h e use o f p e r t u r b a t i o n methods f o r e l l i p s o i d a l mass d i s t r i b u t i o n s . ISOLATING INTEGRALS AND STRETCHED COORDINATES L e t us b e g i n by c o n s i d e r i n g t h e m o t i o n o f a u n i t mass i n t h e p l a n e under t h e i n fluence of t h e p o t e n t i a l V(r,$). From i t s Lagrangian L = 91 F 2 t L2 r 2 i 2

-

V(r,+)

Y

(3)

;

(4)

c o n j u g a t e t o r and $; v i z .

we o b t a i n t h e momenta P r and p, p r = - La

aP

a =i, p =--;-L = r 2.b 4 a+

and t h e Lagrange e q u a t i o n s o f m o t i o n

[It i s assumed t h a t t h e p o t e n t i a l V(r,g) and i t s p a r t i a l d e r i v a t i v e s a r e e v e r y The a s s o c i a t e d Hamil t!nian where continuous.]

i s a f i r s t i n t e g r a l f o r t h e system o f e q u a t i o n s ( 4 ) and (5).

A second i n t e g r a l f o r t h e m o t i o n can be o b t a i n e d u s i n g Hamilton-Jacobi a s s o c i a t e d Instead, c o n s t r u c t a phase t r a j e c t o r y e q u a t i o n f r o m t h e e q u a t i o n s f o r yethod:. 4 and P, [ t h e second p a r t o f e q u a t i o n s ( 4 ) and (5)], namely

.

(7)

I n o r d e r t h a t a f i r s t i n t e g r a l t o (7) e x i s t , V(r,+) must be e x p r e s s i b l e as V(r,g) = U ( r )

+ r-‘w(,)

(8)

( t h e r e s u l t we would have o b t a i n e d had we analyzed t h e Hamilton-Jacobi e q u a t i o n ) . Then, t h e phase t r a j e c t o r y e q u a t i o n ( 7 ) becomes

w i t h the f i r s t integral

where pa i s a c o n s t a n t . We s h a l l r e g a r d pa as a (conserved) c a n o n i c a l momentum which i s c o n j u g a t e t o some a n g u l a r c o o r d i n a t e t h a t has y e t t o be i d e n t i f i e d . In p a r t i c u l a r , we employ ( 6 ) , ( 8 ) and (10) t o express a t r a n s f o r m e d H a m i l t o n i a n

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2

H

71 p 2r

=

t

1p, + --

U (r)

r2

Two of t h e Hamilton equations of motion, namely

confirm the constancy of p, and i s o l a t e t h e associated coordinate @ ( n o t t o be confused w i t h $ ) . From the equation ( 4 ) f o r 6 together with (10) and ( 1 2 ) , we construct the phase t r a j e c t o r y equation

a+ = %P, = Upon re-examining ( l o ) , i t should now be c l e a r t h a t the angular momentum p o s c i l l a t e s in some fashion [since w i s a periodic function of 4 w i t h a pertod of 2n] and t h a t pQ i s an exactly-conserved quantity w i t h the dimensions of angular momentum. Morever, po represents the energy contained i n the angular momentum and the angle-dependent p a r t of t h e potential. The canonically conjugate coordinate 0 corresponds t o a "stretched" version of the o r i g i n a l coordinate $ t h a t depends on both the angle-dependent p a r t of the potential and the i n i t i a l conditions [cf. equation (1311. By means of t h i s "coordinate s t r e t c h i n g " transformat i o n , we have been a b l e t o eliminate t h e angle-dependent p a r t of the potential from the Hamiltonian, a transformation we w i l l now extend t o t h r e e dimensions. Consider the motion of a u n i t mass under t h e influence of a p o t e n t i a l V(r,e,g). From i t s Lagrangian 1 .2 1 2.2 1 2 L = -2r t -2r e + -r - V(r,e,g) 3 (14) 2 sin'e;' we obtain the momenta P r y pe and pg conjugate t o r , e and $; namely, pr

=

$L

=;.

p = 7~ a = r2i 6

=

p$

ae a-L a;

2

= r sin

2 '

e$

and the Lagrange equations of motion

br

=

re

be

=

r2 sinecose+- 2

-

P$ -

a 2

-

r s i n 2 e$ 2

t

aV(r,e,$)

ar aV(r,e,$)

ae

.

(16)

aV(r,e,$)

a$

The associated Hamiltonian 2

1 2

1Pe

H = 7pr + 7 7+

2 Pe 7r sin e 1

+ v(r,e,$)

i s an integral of motion f o r the system of equations (15) and (16). A second i n t e g r a l of motion emerges from t h e phase t r a j e c t o r y equation which [the l a s t p a r t of equations (15) and r e s u l t s from t h e equations f o r $ and (16)], namely

(17)

153

ISOLATING INTEGRALS I N GALACTIC DYNAMICS

I n o r d e r t h a t an i n t e g r a l t o (18) e x i s t , V(r,e,g)

must be e x p r e s s i b l e as

The phase t r a j e c t o r y e q u a t i o n and i t s i n t e g r a l a r e t h e same as t h o s e encountered b e f o r e , i.e. equations ( 9 ) and ( 1 0 ) . D e f i n i n g p and 8 u s i n g e q u a t i o n s ( l o ) , ( 1 2 ) and ( 1 3 ) , t h e H a m i l t o n i a n ( 1 7 ) becomes [empToying ( 1 9 ) ]

1 2

2 1Pe

1

7-t

H = - 2p ' t -

2 V+

2 Po

r s i n e

Wr,e)

.

(20)

The c o o r d i n a t e s t r e t c h i n g a p p l i e d i n d e f i n i n g 8 and po have rendered t h e Hamiltonian c y c l i c i n the v a r i a b l e 8 . A t h i r d i n t e g r a l o f m o t i o n emerges f r o m t h e phaSe t r a j e c t o r y e q u a t i o n which r e s u l t s f r o m combining t h e e q u a t i o n s f o r 8 and pe [ t h e m i d d l e p a r t o f e q u a t i o n s ( 1 5 ) and ( 1 6 ) ] w i t h e q u a t i o n ( 1 9 ) . I n p a r t i c u l a r , we f i n d

r4 sinecoseg- 2 -dpe - de

-

a

r2 aW(r,e)

ae

w(+) sin2e

p+

which can be expressed u s i n g ( 1 0 ) and ( 1 5 ) as I)

Since p8 i s a c o n s t a n t o f t h e motion, a necessary and s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e o f a t h i r d i n t e g r a l o f t h e m o t i o n i s t h a t W(r,e) be e x p r e s s i b l e as w(r,e)

=

u(r)

t r-2v(e)

.

(23)

Combining t h i s r e s u l t w i t h ( 8 ) , we see t h a t

t h e i n t e g r a b i l i t y c o n d i t i o n we would have o b t a i n e d had we s o l v e d t h e HamiltonJacobi e q u a t i o n . Thus, t h e phase t r a j e c t o r y e q u a t i o n (22) now becomes

with the integral

where po i s a constant.

Note t h a t v ( e ) t

2 1 P8

2.2 sin

p a r t o f t h e p o t e n t i a l , which we s h a l l c a l l ?(e),

e

i s o l a t e s t h e latitude-dependent and i n t r o d u c e s l a t i t u d i n a l

W.I. NEWMAN

154

t u r n i n g p o i n t s i n t o t h e problem i f p, (a c o n s t a n t o f t h e motion associated w i t h t h e azimuthal angular momentum) i s non-zero. As before, we s h a l l r e g a r d po as a (conserved) canoni'cal momentum which i s conjugate t o some angular c o o r d i n a t e e ( n o t t o be confused w i t h e ) t h a t has y e t t o be i d e n t i f i e d . We employ equations ( 2 0 ) , (23) and (26) t o express a transformed Hamiltonian 2 H = -1p 2 t 1'0t U ( r ) 2 r 2 7

.

(27)

From t h i s Hamiltonian, we a s s o c i a t e po w i t h t h e t o t a l angular momentum (and a " c e n t r i p e t a l b a r r i e r " ) j u s t as we a s s o c i a t e pa [ v i a ( 2 6 ) l w i t h t h e azimuthal component o f t h e angular momentum. Two (new) Hamilton equations o f motion d e s c r i b e PO and o r namely (28) This again confirms t h e constancy o f po and p r o v i d e s a d e f i n i t i o n f o r 8 . From equation (15) f o r 6 t o g e t h e r w i t h (26) and (281, we c o n s t r u c t t h e phase t r a j e c t o r y equation * de = pe F- -

which provides a more u s e f u l d e f i n i t i o n o f 0 . From e q u a t i o n ( 2 6 ) , i t f o l l o w s t h a t t h e ( t r u e ) t o t a l angular momentum pe o s c i l l a t e s whereas .pn, which i s an e x a c t l y conserved q u a n t i t y w i t h t h e dimensions o f a n g u l a r momentum, does not. For t h e three-dimensional problem, po r e p r e s e n t s t h e energy contained i n a l l o f the angular momentum components @ t h e angle-dependent p a r t o f t h e p o t e n t i a l . The new s t r e t c h e d c o o r d i n a t e o has rendered t h e Hamiltonian ( 2 7 ) c y c l i c i n f l angular coordinates. The t h r e e i n t e g r a l s t h a t emerge.in s p h e r i c a l geometry a r e t h e energy [ o r Hamiltonian ( 2 7 ) ] and q u a n t i t i e s r e l a t e d t o t h e t o t a l angular momentum p

0

2. = r o

and i t s azimuthal component p,

2 . 2

= r sin

ei

[the l a t t e r i n c l u d e s t h e l a t i t u d i n a l dependence o f three-dimensional problems n o t g i v e n i n (12)]. Equations (10) and (26) show t h a t we can have independent t u r n i n g p o i n t s i n our two angular coordinates. For r e a l i s t i c g a l a c t i c p o t e n t i a l s , t h e r a d i a l p o t e n t i a l U ( r ) w i l l cause t h e Hamiltonian (27) t o produce two r a d i a l t u r n i n g p o i n t s (a c e n t r i p e t a l and p o t e n t i a l b a r r i e r , r e s p e c t i v e l y ) . Out o f these, t h e box, s h e l l and tube o r b i t s ( t h e l a t t e r c u r r e n t l y r e f e r r e d t o a "pipes") described e a r l ie r a r i s e n a t u r a l l y . The phase t r a j e c t o r y methods j u s t employed can a l s o be adapted t o o t h e r separable p o t e n t i a l s o f t h e S t l c k e l type, although we do n o t do so here. I n t h i s communic a t i o n , we have surveyed a few noteworthy developments i n s t e l l a r dynamics and have e x p l o i t e d t h e r e l a t i o n between dynamics and t h e geometry o f curved spaces. The most exhaustive i n v e s t i g a t i o n o f t h i s r e l a t i o n s h i p was due t o Synge (1926) who employed methods o f d i f f e r e n t i a l geometry and tensor c a l c u l u s . Indeed, t h e r e a r e grounds f o r cautious optimism t h a t t h e f u r t h e r a p p l i c a t i o n o f these techniques t o gether w i t h t h e t o p o l o g i c a l l y - o r i e n t e d methods o f t h e "new n o n l i n e a r dynamics" w i l l y i e l d a s u b s t a n t i a l i n c r e a s e i n o u r understanding o f g a l a c t i c dynamics and

ISOLATING INTEGRALS IN GALACTIC DYNAMICS

155

structure, This work was i n i t i a t e d w h i l e a t the I n s t i t u t e f o r Advanced Study, Princeton, N.J. I n preparing t h i s paper, t h e author has b e n e f i t t e d g r e a t l y from conversations w i t h F. Dyson, J.N. Bahcall, R.G. Newton, S.D. Tremaine, M. Schwarzschild, D.O. Richstone, W.M. Kaula and F.H. Busse. T h i s research was supported by a g r a n t from t h e C a l i f o r n i a Space I n s t i t u t e t o t h e Department o f E a r t h and Space Sciences, University o f California. REFERENCES Regular and I r r e g u l a r Motion, i n : S. Jorna (ed.), Topics i n Berry, M.V., Nonlinear Dynamics, American I n s t i t u t e o f Physics Conference Proceedings, No. 46 (1978). Contopoulos, G., Problems o f S t e l l a r Dynamics, i n : J. E h l e r s (ed.), R e l a t i v i t y and Astrophysics, 2. G a l a c t i c S t r u c t u r e (American Mathematical Society, Providence, R.I., 1967). Eisenhart, L.P., Goldstein, H.

Separable Systems o f Stllckel, Ann. Math. 35 (1934) 284-305.

, Classical

Mechanics (Addison-Wesley,

Reading, Mass.

, 1980).

Innanen, K.A. and Papp, K.A., P a r t i c l e Dynamics i n Spheroidal Mass D i s t r i b u t i o n s , t h e T h i r d I n t e g r a l o f Motion and Angular Momentum, Astronom. J. 82 (1977) 322-328. Kolmogorov, A.N. , The General Theory o f Dynamical Systems and C l a s s i c a l Mechanics, t e x t o f address t o t h e 1954 I n t e r n a t i o n a l Congress o f Mathema t i c i ans

.

Landau, L.D. and L i f s h i t z , E.M.

, Mechanics

(Oxford: Pergamon, 1960).

Lynden-Bell , D., S t e l l a r Dynamics P o t e n t i a l s w i t h I s o l a t i n g I n t e g r a l s , Monthly Notices Roy. Astronom. SOC. 124 (1962) 95-123. Morse, P.M. and Feshback, H., New York, 1953). Pars, L.A.,

Methods o f T h e o r e t i c a l Physics (McGraw-Hill

,

A T r e a t i s e on A n a l y t i c a l Dynamics (Wiley, New York, 1965).

Richstone, D.O., Scale-Free, Axisymnetric Galaxy Models w i t h L i t t l e Angular Momentum, Astrophys. J. 238 (1980) 103-109. Scale-Free Models o f Galaxies 11: A Complete Survey o f Richstone, D.O., O r b i t s , Department o f Astronomy p r e p r i n t , U n i v e r s i t y o f Michigan. Saaf, A.F., A Formal T h i r d I n t e g r a l o f Motion i n a Nearly S p h e r i c a l S t e l l a r System, Astrophys. J. 154 (1968) 483-498. S t l c k e l , P.G. , Ueber d i e I n t e g r a t i o n des Hamil ton-Jacobischen D i f f e r e n t i a l gleichunger M i t t e l s t separation der v a r i a b e l n , Habi 1it a t i o n s c h r i ft, Hal 1e (1891 )

.

Schwarzschild, M., A Numerical Model f o r a T r i a x i a l S t e l l a r System i n Dynamical E q u i l i b r i u m , Astrophys. J. 232 (1979) 236-247. Synge, J.L., On t h e Geometry o f Dynamics, Philos. Transactions (A) 226 (1927) 31-106.

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PART II Nonlinearity in Field Theories and Low Dimensional Solids

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nimlaenko (eds.) 0 North-Holland Publishing Company, 1982

159

PHYSICS I N FEW DIMENSIONS V.

J. Emer.y

Physics Department * Rrookhaven N a t i o n a l Laboratory Ilpton, New York 11973 T h i s a r t i c l e i s a q u a l i t a t i v e account o f some aspects o f Dhysics i n few dimensions, and i t s r e l a t i o n s h i p t o n o n l i n e a r f i e l d t h e o r i e s . A f t e r a survey o f m a t e r i a l s and some o f t h e models t h a t have been used t o d e s c r i b e them, t h e v a r i o u s methods o f s o l u t i o n a r e conpared and c o n t r a s t e d . The r o l e s o f exact r e s u l t s , o p e r a t o r r e p r e s e n t a t i o n s and t h e renormal iz a t i o n group t r a n s f o r m a t i o n a r e described, and a u n i f o r m p i c t u r e o f t h e b e h a v i o r o f low-dimensional systems i s presented.

I.

INTRODUCTION

Many o f t h e fundamental problems o f condensed m a t t e r p h y s i c s may be regarded as examples o f n o n l i n e a r f i e l d t h e o r i e s . T h i s p o i n t o f view has been advantageous f o r t h e quantum-mechanical many-body problem and f o r t h e modern t h e o r y o f phase t r a n s i t i o n s , l e a d i n g t o successful t h e o r i e s o f s i n g l e - p a r t i c l e and c o l l e c t i v e It i s t h e b a s i s f o r o u r phenomena i n a wide v a r i e t y o f p h y s i c a l systems. c u r r e n t understandinq o f s u p e r c o n d u c t i v i t y and s u p e r f l u i d i t y , and has l e d t o a deeper understanding o f t h e u n i v e r s a l c o o p e r a t i v e e f f e c t s which a r e observed i n t h e neiqhborhood o f a c r i t i c a l p o i n t . Most o f these developments have r e l i e d upon t h e techniques o f quantum f i e l d t h e o r y , and have made l i t t l e r e f e r e n c e t o concepts t h a t have a r i s e n i n a p u r e l y c l a s s i c a l context. More r e c e n t l y , however, t h i s s i t u a t i o n has chanqed, and some o f t h e approaches t o p h y s i c s i n one and two space dimensions have heen much c l o s e r i n s p i r i t t o t h e ideas of c l a s s i c a l n o n l i n e a r f i e l d the0r.y; r e f e r e n c e s t o t h e sine-Gordon e q u a t i o n , s o l i t o n s , h r e a t h e r s , i n v e r s e s c a t t e r i n q , etc., have become r e l a t i v e l y comnonplace i n t h e l i t e r a t u r e o f elementary p a r t i c l e and condensed m a t t e r phy s ic s

.

A f e e l i n g f o r t h e r e l e v a n c e o f these concepts and t h e i r supplementary r e l a t i o n s h i p t o t h e more " c o n v e n t i o n a l " approach may be o b t a i n e d by c o n s i d e r i n g several examples from condensed m a t t e r physics. I n what f o l l o w s , t h e r e w i l l be no r e f e r e n c e t o t h e t h e o r y o f n o n - i n t e g r a b l e systems, a l t h o u g h i t s i n f l u e n c e i s b e g i n n i n g t o make i t s e l f f e l t , f o r example, i n a t t f m p t s t o understand some aspects o f t h e t h e o r y o f incomnensurate s t r u c t u r e s

.

The systems o f i n t e r e s t a r e q u i t e d i v e r s e i n t h e i r p h y s i c a l c h a r a c t e r i s t i c s . Some a r e quantum mechanical and one dimensional, o t h e r s a r e c l a s s i c a l and two dimensional. But i t t u r n s out t h a t t h e r e i s a comnon t h r e a d t o t h e i r mathematical f o r m u l a t i o n : a remarkably l a r g e number o f models a r e e q u i v a l e n t t o each other, e i t h e r e x a c t l y o r i n t h e a s y m p t o t i c p r o p e r t i e s which govern t h e k i n d s o f long-ranged order t h a t might be e s t a b l i s h e d . A l l bear some r e l a t i o n s h i p t o t h e sine-Gordon equation. The impetus f o r these developments has come f r a n b o t h t h e o r y and experiment, from t h e d e s i r e f o r a u n i f i e d view o f two-dimensional d e l s as w e l l

160

KJ. EMERY

as a need t o understand a number o f unusual o b s e r v a t i o n s . The i n g e n u i t y o f t h e s y n t h e t i c chemist and t h e e x p e r i m e n t a l p h y s i c i s t has been a c o n t i n u a l d r i v i n g f o r c e i n t h e whole f i e l d , and i t i s a p p r o p r i a t e t o s t a r t out w i t h some b r i e f mention o f t h e m a t e r i a l s t h e y have i n v e s t i g a t e d , b e f o r e g o i n g on t o d e s c r i b e t h e mathematical models, and t h e methods t h a t have been d e v i s e d i n o r d e r t o s o l v e them.

I I.

MATER IALS

Low-dimensional b e h a v i o r a r i s e s i n two p r i n c i p a l ways. Some m a t e r i a l s a r e e x t r e m e l y a n i s o t r o p i c , c o n s i s t i n g o f atoms, molecules o r i o n s arranged i n c h a i n s o r i n l a y e r s , which a r e weakly coupled t o t h e i r environment, and a c t independently over a wide range o f temperatures. The s t r u c t u r e imposes i t s own c o n s t r a i n t s on t h e m o t i o n o f e l e c t r o n s , and t h i s l e a d s t o c h a r a c t e r i s t i c low-dimensional b e h a v i o r o f t h e e l e c t r i c a l p r o p e r t i e s . The a l t e r n a t i v e i s t o have a r e s t r i c t i v e geometry; a f r e e f i l m o r a f i l m adsorbed on a s u r f a c e a r e e s s e n t i a l l y two dimensional, whereas p a r t i c l e s c o n f i n e d t o narrow channels may have a one dimensional c h a r a c t e r . I n most cases t h e r e a r e circumstances i n which t h e t r u e t h r e e - d i m e n s i o n a l n a t u r e o f t h e system makes i t s e l f f e l t , and understandinq t h e c r o s s o v e r t o t h i s reqime i s p a r t o f t h e i n t e r e s t i n t h e problem. The purpose o f s t u d y i n q t h e s e systems i s t o l o o k f o r phenomena t h a t may n o t occur, o r may he d i f f i c u l t t o produce i n more i s o t r o p i c m a t e r i a l s : e f f e c t s o f d i s o r d e r a r e expected t o he more pronounced, and c e r t a i n k i n d s o f phase t r a n s i t i o n may t a k e p l a c e more r e a d i l y . It has l o n q been a hope t o f i n d s u p e r c o n d u c t i v i t y a t r e l a t i v e l y h i q h temperatures i n a n i s o t r o p i c o r q a n i c m a t e r i a l s , where s t r o n g e l e c t r o n - e l e c t r o n i n t e r a c t i o n s may be produced by an e x c i t o n i c mechani sm2.

A l l o f these e f f e c t s a r e l i k e l y t o i n v o l v e an i n t e r p l a y between t h e many degrees o f freedan which r e s i d e i n a m o l e c u l a r c r y s t a l - - t h e s p i n and t r a n s l a t i o n o f e l e c t r o n s o r t h e spin, o r i e n t a t i o n , v i b r a t i o n and t r a n s l a t i o n o f t h e molecules. For t h i s reason, i t i s o f t e n q u i t e d i f f i c u l t t o e x t r a c t , f r a n r a t h e r i n d i r e c t experimental i n f o r m a t i o n , t h e p r i m a r y mechanism which d r i v e s t h e b e h a v i o r o f a p a r t i c u l a r p h y s i c a l system. The common approach has been t o s o l v e a number o f s i m p l i f i e d m d e l s , i n o r d e r t o d i s c o v e r t h e p a r t i c u l a r e f f e c t s t o l o o k out f o r and t o l i m i t t h e p o s s i b l e range o f e x p l a n a t i o n s o f a g i v e n experiment. A few examples w i l l i l l u s t r a t e t h e n a t u r e o f t h e systems which have been investigaled. One-dimensional m a t e r i a l s f r e q u e n t l y c o n t a i n r a t h e r f l a t o r g a n i c molecules such as TTF ( t e t r a t h i a f u l v a l ene), TSeF ( t e t r a s e l e n o f u l v a l e n e ) , TCNQ (tetracyanoquinodimethane) and TMTSF (tetramethyltetraselenofulvalene), a1 1 o f which may be arranged i n c l o s e l y packed stacks. I n TTFCuBDT. long-ranged c o r r e l a t i o n s i n chains o f l o c a l i z e d s p i n s c o n s p i r e w i t h p e c u l i a r i t i e s o f t h e l a t t i c e v i b r a t i o n s t o produce a d i m e r i z e d s t a t e 4 ( s p i n - P e i e r l s t r a n s i t i o n ) . The o r g a n i c metals llF-TCNQ, TSeF-TCNQ and (TMTSF) 2PF6 a r e e l e c t r i c a l c o n d u c t o r s because charge t r a n s f e r from donor t o a c c e p t o r molecules l e a v e s a p a r t i a l l y f i l l e d band o f s t a t e s 3 . The c o n d u c t i v i t y of these systems i n c r e a s e s t o a q u i t e h i q h v a l u e as t h e temperature i s lowered, b u t u l t i m a t e l y t h i s i s r e v e r s e d hy a m e t a l - i n s u l a t o r t r a n s i t i o n . However, a t s u f f i c i e n t l y h i q h pressure, (TMTSF)2PF6 becomes a superconductor. An example o f a s o - c a l l e d m o l e c u l a r metal i s Hq3-6ASF6, which c o n s i s t s o f c h a i n s o f Hq i o n s arranqed i n planes and i n t e r s p e r s e d w i t h AsF6- ions. I t s p e c u l i a r p r o p e r t i e s a r e a consequence o f t h e l a c k of commensurability between t h e Hg c h a i n s and t h e AsF6- l a t t i c e ( 6 i s a f u n c t i o n o f temperature and i s about 0.2) A t room temperature, t h e weakly coupled Hg c h a i n s form one-dimensional l i q u i d s 5 , b u t t h e y f r e e z e a t about 120 K. T h i s t r a n s i t i o n i s unusual i n t h a t i t i s an example o f c o n t i n u o u s f r e e z i n g . A t much lower temperatures t h e m a t e r i a l becomes a superconductor6, b u t t h e o r i g i n and p r o p e r t i e s o f t h i s s t a t e a r e n o t f u l l y understood. An e n t i r e session o f t h i s conference i s concerned w i t h t h e p r o p e r t i e s o f p o l y a c e t y l e n e , a l i n e a r system which i s thought t o form a d i m e r i z e d c h a i n with s o l i t o n d i s l ocations.

PH YSlCS IN FEW DIMENSIONS

Much o f t h e r e c e n t i n t e r e s t i n two-dimensional m a t e r i a l s has been c e n t e r e d upon a c l a s s o f systems which have a phase t r a n s i t i o n b u t a r e unable t o e s t a b l i s h t h e r e l a t e d long-ranged o r d e r because o f t h e d e s t r u c t i v e e f f e c t o f thermal f l u c t u a t i o n s . They accomplish t h i s as t h e temperature i s decreased below a c e r t a i n value, by remaininq on t h e verqe o f a t r a n s i t i o n t o an ordered s t a t e : every p o i n t i s c r i t i c a l . S o l i d s , superconductors, s u p e r f l u i d s and some phases o f l i q u i d c r y s t a l s a r e expected t o have t h i s b e h a v i o r i n two dimensions, and v e r i f i c a t i o n has been souqht i n adsorbed l a y e r s o r freely-suspended f i l m s . Another i n t e r e s t i n q p r o p e r t y o f o v e r l a y e r s i s t h e e x i s t e n c e o f p e r i o d i c s t r u c t u r e s t h a t may be commensurate o r incommensurate w i t h t h e s u h s t r a t e l a t t i c e . T h i s prohlem has been i n v e s t i q a t e d by t h e s c a t t e r i n q o f neutrons, x-rays and e l e c t r o n s as w e l l as by photoemission experiments, and, a l t h o u q h a p i c t u r e o f what i s qoinq on has s t e a d i l y been developed, i t i s s t i l l f a r from complete. Other two-dimensional forms o f s p i n o r d e r i n q , d i s p l a c i v e phase t r a n s i t i o n s and charqe-density wave s t a t e s a r e t o be found i n b u l k m a t e r i a l s w i t h a l a y e r e d s t r u c t u r e . A more e x t e n s i v e r e v i e w o f t h i s whole s u b j e c t may be found i n t h e proceedinqs o f t h e 19879 Kyoto summer s c h o o l 7 and t h e 1980 Lake Geneva, Wisconsin conference

.

111. MODELS

A t f i r s t s i g h t it seems i n a p p r o p r i a t e t o pay so much a t t e n t i o n t o c y r s t a l l i n e s o l i d s and l a t t i c e models i n a d i s c u s s i o n o f n o n l i n e a r f i e l d t h e o r i e s , but t h e r e i s much t o be gained from r e l a t i n g one t o t h e o t h e r by t a k i n g t h e continuum limit. :he l g c a t i o n of a p o i n t i n a simple c u b i c l a t t i c e i s s p e c i f i e d by a v e c t o r r = sn, where n has i n t e g e r components and s i s t h e l a t t i c e spacing. C o r r e l a t i o n f u n c t i o n s have a c h a r a c t e r i s t i c l e n g t h s c a l e s5, where 5 , t h e coherence l e n g t h o f t h e l a t t i c e model, i s a pure number. A f i e l d t h e o r y i s obtained by t a k i n g t h e continuum l i m i t s+O, i n which f i n i t e d i f f e r e n c e s become d e r i v a t i v e s , and l a t t i c e sums become i n t e q r a l s . The advantaqe o f c o n s i d e r i n q t h i s l i m i t , which i s c l e a r l y f i c t i c i o u s f o r a r e a l s o l i d , i s t h a t i t focuses a t t e n t i o n on t h e asymptotic p r o p e r t i e s o f c o r r e l a t i o n f u n c t i o n s , and f o r c e s us t o c o n s i d e r b e h a v i o r a t a c r i t i c a l p o i n t . For i f t h e l e n g t h s c a l e sE, and p o s i t i o n v e c t o r if are t o remain f i n i t e as s+0, i t i s necessary t h a t 5- (which i s t h e case a t a c r i t i c a l p o i n t ) and (which i s t h e asymDtotic l i m i t ) . Equally, t h e continuum l i m i t i s u s e f u f o r a f i e l d t h e o r y when t h e c r i t i c a l p r o p e r t i e s o f t h e correspondinq l a t t i c e model a r e known.

I;[-*.

I n t h e p r e s e n t context, however, t h e r e i s a f u r t h e r i m p o r t a n t reason f o r t a k i n q t h e continuum l i m i t - - i t i s c r u c i a l f o r e s t a b l i s h i n q t h e r e l a t i o n s h i p between models and o b t a i n i n q a u n i f i e d p i c t u r e o f p h y s i c s i n two dimensions. Space does not p e r m i t an account o f t h e t e c h n i c a l developments necessary f o r t h e implementation o f t h i s proqram, but a survey o f some o f t h e i n t e r e s t i n q Hamiltonians should g i v e a t l e a s t some i d e a o f t h e k i n d s o f system under c o n s i d e r a t i o n . C l a s s i c a l , two-dimensional models w i l l be d e s c r i b e d f i r s t . P o t t s Modelsg

H

P

=-J

1 n.n.

6ss

ij

Here, t h e summation i s c a r r i e d over near-neighbor s i t e s i,j w i t h Si=1,2,...q. on a square l a t t i c e . The s t a t e o f lowest energy has a l l s p i n s equal and i s q - f o l d degenerate. T h i s H a m i l t o n i a n i s r e l e v a n t f o r adsorbed f i l m s g and f o r magnetic systems. (q.2 i s i d e n t i c a l t o t h e k i n g model)

161

162

V.J. EMERY

A s h k i n - T e l l e r Model l o HA =

-1

nn

{JlSiSj+J2TiTj*l~SiTiSjTj}

(2 1

T h i s i s a two-component l a t t i c e qas and, f o r J1=J2=0, where Si='l, Ti='l. i t i s i d e n t i c a l t o t h e 4 - s t a t e P o t t s model. I n t e r f a c e Rouqhening"

HR

=

-I n,n

f(hi-h.)

.l

(3)

...

w i t h hi=fl,?1,?2, , i s a c e l l model of c r y s t a l growth. The c o n s t i t u e n t s a r e assumed t o f i l l a column o f c e l l s t o a h e i q h t h. a t l a t t i c e s i t e i. F o r t h e p h y s i c a l model, f ( h ) = h ( s o l i d - o n - s o l i d model], b u t another case o f p a r t i c u l a r importance i s f ( h ) = h 2 ( d i s c r e t e Gaussian model) which i s d i r e c t l y r e l a t e d t o t h e Coulomb gas and t h e xy-model , as w i l l be seen. Vertex Model s i 2 It i s imagined t h a t e v e r y v e r t e x i n a square l a t t i c e i s connected t o i t s f o u r neighbors by a l i n k , which has a sense ( r i g h t o r l e f t , up o r down) s p e c i f i e d by an arrow. There a r e s i x t e e n d i f f e r e n t k i n d s o f v e r t e x ( f o u r l i n k s , each w i t h two senses) and each i s assigned a d i f f e r e n t weight. The problem i s t o sum over a l l c o n f i g u r a t i o n s o f l i n k s . The e i g h t - v e r t e x problem ( i n which each v e r t e x has an even number o f incoming and o u t g o i n g arrows) has been s o l v e d e x a c t l y by It i s a l s o o f i n t e r e s t t o c o n s i d e r a more general staggered v e r s i o n Baxter". o f t h i s model w i t h two s e t s o f weights, one f o r each o f two i n t e r p e n e t r a t i n g s u b l a t t i c e s . Such a model has een shown t o be e q u i v a l e n t t o t h e P o t t s models13 and t o t h e A s h k i n - T e l l e r model 18

.

The xy-Model l 4 which has a H a m i l t o n i a n q i v e n by H = - J I V*V i .i XY n.n.

+

where V i i s a u n i t , two-dimensional v e c t o r , d i f f e r s from t h e p r e c e d i n q models + + b,y h a v i n g a continuous v a r i a b l e a t e v e r y s i t e . T h i s i s c l e a r i f V i * V j i s r e w r i t t e n i n t h e form c o s ( 8 i e j ) where e i and e j a r e t h e p o l a r + + T h i s H a m i l t o n i a n d e s c r i b e s a magnetic system, b u t i t angles o f V i and V j . has a l s o been used f o r t h e s u p e r f l u i d t r a n s i t i o n i n He4 f i l m s , f o r which 8 i i s t h e phase o f t h e o r d e r parameter. The duali4 o f Hxy i s a s p e c i a l case o f HR

-

Coulomb Gas14

Here t h e summation over i and j extends t o a l l s i t e s ( n o t o n l y near n e i h b o r s ) , and t h e charges Qi have values 0,+1,+2,.... T h i s model i s e q ~ i v a l e n t ' ~ t,o~ ~ t h e d i s c r e t e Gaussian v e r s i o n o f 4 , as mentioned e a r l i e r .

PHYSICS IN FEW DIMENSIONS

163

It i s a remarkable f a c t t h a t these a p p a r e n t l y q u i t e d i f f e r e n t models a r e c l o s e l y r e l a t e d : t h e p a r t i t i o n f u n c t i o n o f one may he t r a n s f o r m e d i n t o t h e p a r t i t i o n f u n c t i o n of t h e o t h e r , w i t h an a p p r o p r i a t e r e d e f i n i t i o n o f parameters. I n some cases, t h e t r a n s f o r m a t i o n i s exact i n o t h e r s i t i s as.mptotical1.y c o r r e c t f o r t h e c r i t i c a l proper tie^^"^. It i s o f t e n p o s s i h l e t o f i n d f u r t h e r e q u i v a l e n c e s between c o r r e l a t i o n f u n c t i o n s . However, t h e t r a n s i t i o n t o f i e l d t h e o r y i s most d i r e c t l y made v i a t h e t r a n s f o r m a t i o n o f a l l o f these models i n t o one-dimensional quantum mechanical systems. The l i n k i s p r o v i d e d by t h e t r a n s f e r matrixi7 T.

The p a r t i t i o n f u n c t i o n Z i s a sum over a l l c o n f i g u r a t i o n s o f v a r i a b l e s on t h e l a t t i c e . The elements o f T c o n s i s t o f t h e c o n t r i b u t i o n s t o t h i s sum from p a i r s of c o n f i g u r a t i o n s o f two n e i g h b o r i n g rows o f t h e l a t t i c e , and

Z

N

(61

= T r T

where N i s t h e number o f rows, I n t h e thermodynamic l i m i t (N-), Z i s dominated by t h e l a r g e s t eigenvalue o f T. The c o n f i g u r a t i o n s o f a row a l s o be regarded as s t a t e s o f a one-dimensional quantum system, w i t h T p l a y i n g t h e r o l e o f a t r a n s i t i o n matrix. I n t h i s i n t e r p r e t a t i o n , i f T i s w r i t t e n i n the form’* exp(-H), t h e n H i s t h e corresponding quantum H a m i l t o n i a n , and i t s ground s t a t e g i v e s t h e l a r g e s t eigenvalue o f T. It may appear t h a t one o f t h e o r i g i n a l space dimensions has been l o s t , b u t a c t u a l l y it has been r e p l a c e d by ( i m a g i n a r y ) t i m e which, i m p l i c i t l y o r e x p l i c i t l y , p l a y s an unavoidable r o l e i n quantum mechanics. F o l l o w i n q t h i s procedure, e v e r y one o f t h e models l i s t e d above may he r e l a t e d t o t h e s p i n Hamiltonianig

m p

H = H + H 0

1

t H

2

(7)

where

and

0.Y and a j z a r e P a u l i matrices. The H a m i l t o n i a n Ho ! ~ ~ ~t { re ~ Heisenberg-Ising l ! ~ ~ model, H i g i v e s an a n i s o t r o p y i n t h e xy p l a n e o f s p i n space and H2 i s a d i m e r i z a t i o n t h a t i s r e l a t e d t o t h e s t a g g e r i n g o f w e i g h t s

i n t h e Baxter model. T h i s i s t h e c e n t r a l model o f t h e f i e l d t o r h i c h a l l o t h e r s may be reduced, a t l e a s t asymptoticallyig. The s p i n r e p r e s e n t a t i o n (8)-(10) i s o n l y one way o f w r i t i n g H. Other u s e f u l forms i n terms o f f e r m i o n o r boson v a r i a h l e s w i l l be i n t r o d u c e d l a t e r . The parameters, g, y, A, q’, hs a r e known f u n c t i o n s o f t h e temperature and t h e parameters (J, q, etc.) o f t h e o r i g i n a l models. It w i l l he seen t h a t H g i s t h e c r i t i c a l H a m i l t o n i a n w h i l e H, and H 2 g i v e thermal o r f i e l d p e r t u r h a t i o n s away from t h e c r i t i c a l p o i n t s o f t h e o r i g i n a l model s. It i s now p o s s i h l e t o s t a t e how t h e quantum models, mentioned i n S e c t i o n 2, may he f i t t e d i n t o t h e p i c t u r e . The H a m i l t o n i a n f o r a s p i n - P e i e r l s system4 i s Ho+H2, where H, r e f e r s t o l o c a l i z e d s p i n s i n a u n i f o r m l a t t i c e , and H2 d e s c r i b e s t h e e f f e c t s o f d i m e r i z a t i o n i n t h e low-temperature phase. The m o t i o n o f t h e mercury i o n s i n Hg3-6AsF6, and t h e s p i n o r charge degrees o f freedom o f e l e c t r o n s i n o r g a n i c conductors a r e r e l a t e d t o H o + H ~ , b u t , t o show more

184

V.J. EMERY

e x p l i c i t l y how t h i s comes about, i t i s necessary t o know something about t r a n s f o r m a t i o n s between spins, bosons and fermions. This i s described i n t h e next section.

IV.

METHODS OF SOLUTION

It i s a remarkable f e a t u r e o f one-dimensional p h y s i c s t h a t , f o r a number o f models, e i g e n s t a t e s and e i g e n v a l u e s a r e known e x a c t l y . They have been o b t a i n e d i n v a r i o u s e q u i v a l e n t forms--Bethe's amsatz f o r t h e w a v e f u n c t i o n (see Dr. quantum i n v e r s e s c a t t e r i n g 2 ' and t h e s e m i c l a s s i c a l method f o r Andrei's t a l i f i e l d theory T h i s whole approach i s c l o s e l y r e l a t e d t o t h e ideas o f c l a s s i c a l n o n l i n e a r physics, and i t works f o r systems which a r e e x a c t l y integrable.

Once an exact s o l u t i o n i s a v a i l a b l e , i t miqht seem t h a t t h e r e i s l i t t l e more t o be said. However i t i s n o t easy t o work w i t h t h e w a v e f u n c t i o n s and, with one exception22, i t has n o t heen p o s s i b l e t o e v a l u a t e c o r r e l a t i o n f u n c t i o n s which a r e r e q u i r e d i n o r d e r t o assess t h e p r o s p e c t s f o r v a r i o u s k i n d s o f l o n p r a n g e d order. Furthermore, t h e method has n o t so f a r succeeded f o r t h e most qeneral H a m i l t o n i a n o f Eqs. ( 6 ) - ( l o ) , i n c l u d i n g d i m e r i z a t i o n , and hence t h e r e i s e v e r y reason t o seek a l t e r n a t i v e approaches, even approximate ones, t h a t a r e n o t so s p e c i f i c . Two a r e o f p a r t i c u l a r i m p o r t a n c e - - o p e r a t o r r e p r e s e n t a t i o n s , and t h e r e n o r m a l i z a t i o n group method. The idea o f u s i n g o p e r a t o r r e p r e s e n t a t i o n s i s t h a t t h e r e a r e exact r e l a t i o n s h i p s between spin, f e r n i o n and boson o p e r a t o r s , and i t may happen t h a t a problem i s i n t r a c t a b l e i n one r e p r e s e n t a t i o n but may be e x a c t l y s o l u b l e i n an her. Perhaps t h e best-known example i s t h e Jordan-Wigner t r a n s f o r m a t i o n

$1

+

a

m

m- 1 t 1 c . c . ) ~t j=1 J J m

= exp(in

m-1 (I-

m

= exp(in

1

j=l

t

c.c.)~ J J m

where

amd C+m, c, a r e f e r m i o n c r e a t i o n and a n n i h i l a t i o n operators. This t r a n s f o r m a t i o n i s used i n s o l v i n q t h e two-dimensional I s i n q and i t may be used f o r any l a t t i c e model. On t h e o t h e r hand t h e h on r e p r e s e n t a t i o n s o f F o r fermions i n one s p i n o r f e r m i o n o p e r a t o r s 2 4 r e l y on t h e continuum l i m i t dimension, i t i s p o s s i b l e t o d i s t i n q u i s h between r i q h t - g o i n g and l e f t - g o i n g p a r t i c l e s , with f i e l d o p e r a t o r s $+(x) and $-(x) r e s p e c t i v e l y . Then $*(x) may be w r i t t e n 2 4

53 .

where +(x) i s a Bose F i e l d and n ( x ) i s i t s c o n j u g a t e momentum. T h i s t r a n s f o r m a t i o n i s p a r t i c u l a r l y u s e f u l , because i t i s r a t h e r easy t o e v a l u a t e c o r r e l t i o n f u n c t i o n s when t h e o p e r a t o r s c o n s i s t of e x p o n e n t i a l s o f Bose Fieldsq4. Equations (11)-(14) a r e g i v e n i n o r d e r t o show t h e form o f t h e t r a n s f o r m a t i o n s . More d e t a i l e d d i s c u s s i o n s and a p p l i c a t i o n s t o a number o f problems my be found i n t h e l i t e r a t u r e o r i n review^^^,^^,^^.

PHYSICS IN FEW DIMENSIONS

The o p e r a t o r r e p r e s e n t a t i o n s g i v e t h e c o n n e c t i o n between t h e s p i n c h a i n and o t h e r one dimensional m a t e r i a l s , and a l s o show how t h e sine-Gordon e q u a t i o n comes i n t o t h e p i c t u r e . I n t h e continuum l i m i t , t h e charge- and spin-deqrees o f freedom o f t h e conduction e l e c t r o n s are d e c o ~ p l e d ~and ~ , each may be regarded a s a s e t o f s p i n l e s s fermions which, i n t u r n , a r e r e l a t e d t o H, + H, by a Jordon-Wigner t r a n s f o r m a t i o n . The H a m i l t o n i a n c o n t a i n s p r o d u c t s o f $.+(x) and $ t t ( x ) and, i n t r o d u c i n g t h e boson r e p r e s e n t a t i o n (14), i t i s found t h a t t h e c o n t r i b u t i o n from t h e i n t e g r a l o f n(6) c a n c e l s out l e a v i n g a f a c t o r p r o p o r t i o n a l t o c ~ s [ ( n p ) ~ / * t $ ( x ) ] , which i s t h e p o t e n t i a l energy d e n s i t y o f t h e sine-Gordon system. Thus, a l l o f these problems, as w e l l s t h e ordered phase o f mercury i o n s i n Hg3-&F6 (another sine-Gordonn system ) are r e l a t e d t o t h e s p i n chain.

2!

The sine-Gordon equivalence suqqests t h a t t h e r e may he s o l i t o n and b r e a t h e r e x c i t a t i o n s . However, t h e f i e l d s a r e q u a n t i z e d and, i n c o n t r a s t t o t h e c l a s s i c a l case21y26, t h e s o l u t i o n depends on t h e value o f p. S o l i t o n s and b r e a t h e r s e x i s t when p i s l e s s than a c r i t i c a l value, and t h e n t h e mass o f t h e lowest e x c i t a t i o n i s a measure o f t h e coherence l e n q t h i n t h e d i s o r d e r e d phase o f t h e correspondinq two dimensional prohlem; s i n c e t h e mass qoverns t h e decay i n ( i m a q i n a r y ) t i m e o f a quantum-mechanical system. It i s i n t h i s r e q i o n t h a t t h e sine-Gordon p i c t u r e c o n t r i b u t e s most e f f e c t i v e l y t o t h e s o l u t i o n o f t h i s i s q r e a t e r t h a n t h e c r i t i c a l value, t h e r e a r e no qroup o f p r o h l e m ~ * ~ . I f s o l i t o n s and t h e e x c i t a t i o n s a r e massless. T h i s r e i o n corresponds t o t h e l i n e o f c r i t i c a l p o i n t s i n t h e two dimensional t h e o r i e s z q , and i s more e f f e c t i v e l y t a c k l e d by t h e r e n o r m a l i z a t i o n qroup method. One way o f p h r a s i n g t h e r e n o r m a l i z a t i o n group method" i s t o focus on some q u a n t i t y such as t h e coherence l e n g t h 6 , and t o study t h e v a r i a t i o n of From t h e parameters (such as J ) r e q u i r e d t o keep 6s f i x e d , as s varies. r e s u l t i n g f l o w equations, i t i s p o s s i b l e t o e v a l u a t e t h e c r i t i c a l exponents. U s u a l l y t h i s procedure cannot be c a r r i e d o u t e x a c t l y u n l e s s t h e r e i s a small parameter i n which an expansion may be made. It does, however, t e l l us which are t h e r e l e v a n t v a r i a b l e s , t h e ones which must be taken i n t o account i n o r d e r t o get a complete d e s c r i p t i o n . The method i s most u s e f u l i n t h e neighborhood o f a f i x e d p o i n t o f t h e t r a n s f o r m a t i o n , and i t i s u s u a l l y necessary t o r e s o r t t o numerical c a l c u l a t i o n s o r t o some o t h e r method o f c a l c u l a t i o n i f a more q l o b a l p i c t u r e i s r e q u i r e d . Nevertheless i t can be a p p l i e d i n a r e l a t i v e l y s t r a i g h t f o r w a r d way t o more complex Hamiltonians, and t o i n c l u d e t h e d i m e r i z a t i o n H, which i s q u i t e d i f f i c u l t t o deal with o t h e r ~ i s e ' ~ . C l e a r l y , a l l o f these approaches have t h e i r advantages and l i m i t a t i o n s , and i t i s necessary t o r e s o r t t o a combination o f a l l o f them, i n o r d e r t o b u i l d up a complete p i c t u r e o f a q i v e n problem. The Bethe ansatz o r t h e quantum v e r s i o n s o f t h e i n v e r s e s c a t t e r i n q method a r e d i r e c t l y r e l a t e d t o c l a s s i c a l n o n l i n e a r t h e o r i e s . They q i v e exact expressions f o r t h e enerqy spectrum, b u t i t i s q u i t e d i f f i c u l t t o e v a l u a t e c o r r e l a t i o n f u n c t i o n s . The t r a n s f o r m a t i o n s hetween spins, fermions and hosons may h e l p t o t u r n a prohlem i n t o a more e a s i l y s o l u b l e form h u t , w i t h o u t f u r t h e r help, t h e y do not always q i v e a mass spectrum. They a r e a t t h e i r h e s t near a c r i t i c a l p o i n t , where t h e boson form i n p a r t i c u l a r i s u s e f u l i n q i v i n q t h e a l q e h r a i c decay o f c o r r e l a t i o n f u n c t i o n s and expressions f o r t h e a s s o c i a t e d c r i t i c a l exponents24. The r e n o r m a l i z a t i o n qroup was o r i q i n a l l y a t e c h i q u e o f quantum f i e l d theory. It i s v e r s a t i l e b u t does n o t q i v e a complete a n a l y t i c a l s o l u t i o n t o a problem i f t h e r e i s no small parameter i n which t o expand, o r i f r e q i o n s f a r from a f i x e d p o i n t cannot be disreqarded. Nevertheless, by assembling t h e c o n t r i b u t i o n s o f a l l o f these techniques, we have come t o a u n i f i e d p i c t u r e o f a l a r g e c l a s s o f one-or two-dimensional

165

166

V.J. EMERY

models. The commmon f e a t u r e i s a l i n e o f c r i t i c a l p o i n t s , alonq which c o r r e l a t i o n f u n c t i o n s o f t h e c l a s s i c a l o r quantum models decay a l q e b r a i c a l l y ; w i t h exponents t h a t depend upon the p o s i t i o n alonq t h e l i n e . The hoson representations and r e n o r m a l i z a t i o n qroup equations l e a d t o r e l a t i o n s h i p s between c r i t i c a l exponents i n what amounts t o an extension o f the exact r e s u l t s t o problems f o r which no exact s o l u t i o n existsig. O f f the c r i t i c a l line, i t i s necessary t o know the mass spectrum o r t h e coherence l e n q t h and t o make use o f a l l o f t h e methods i n order t o o b t a i n a s o l u t i o n . A survey o f t h i s approach and a d e s c r i p t i o n o f recent work i s given i n reference 19.

A l l o f these developments have been described i n t h e c o n t e x t o f condensed m a t t e r physics, but many of t h e r e s u l t s have been discovered independently by elementary p a r t i c l e t h e o r i s t s . T h e i r o b j e c t i v e has been t o p r a c t i c e on models showing confinement and asymptotic freedom i n t h e hope t h a t techniques may be o f value f o r the more physical f o u r dimensional theories. By now i t has t h e appearance o f a mature f i e l d . But i t i s t o o much t o expect t h a t such a t i d y p i c t u r e o f low-dimensional physics w i l l p e r s i s t . Already a number o f models, t h a t cannot be solved i m n e d i a t e l y by these methods, a r e being i n v e s t i g a t e d - - t h a t i s a symptom o f a healthy f i e l d . REFERENCES

* [

Research supported by DOE under Contract No.DE-AC02-76CH00016.

11 Auhry,

S., i n S o l i t o n s and Condensed Matter Physics, E d i t e d by Bishop A. R. and Schneider, T. (Springer-Verlaq, B e r l i n , Heidelberg, New York, 1978). [ 23 L i t t l e , W. A., Phys. Rev. 134A (1964) 1416-1424. [ 3) See f o r example a r t i c l e s i n Chemistry and Physics o f One Dimensional Metals, Edited b.y K e l l e r , H. J. (Plenum, New York, 1977); Molecular Metals, E d i t e d by H a t f i e l d , W. E. (Plenum, New York, 1979). [ 41 Cross, M. C. and Fisher, n. S., Phys. Rev. R19 (1979) 402. [ 51 Emery, V. J. and Axe, J. D., Phys. Rev. L e t t . 40 (1978) 1507. [ 61 For a review see Heeqer, A. J. and Macniarmid, A. G. i n Molecular Metals, reference 3, p. 419. [ 71 Proceedings o f the Kyoto Summer I n s t i t u t e 1979-Physics o f Low-Dimensional Systems, e d i t e d by Nagaoka, Y. and Hikami. S. ( P u b l i c a t i o n O f f i c e , Progress o f Theoretical Physics, 1979). [ 81 Ordering i n Two Dimensions, E d i t e d by Sinha, S. K. (North Holland, New York, Amsterdam, Oxford, 1980). [ 91 Potts, R. P., Proc. Camb. P h i l . SOC. 48 (1952) 106. [ l o ] Wegner, F. J., J. Phys. C5 (1972) L131. [ll]See e.g. Emery, V. J. and Swendsen, R. H., Phys. Rev. L e t t . 39 (1977) 1414. [12] For a review see M. P. M. den N i j s , J. Phys. A12 (1979) 1857. [13] Temperley, H. N. V. and Lieb, E. H., Proc. Roy. SOC. London Ser. A322 (19711 251. [14] Josb,’J, V., Kadanoff, L. P., K i r k p a t r i c k , S. and Nelson, 0. R., Phys. Rev. 616 (1977) 1217. [ l 5 ] van Beijeren, H., Phys. Rev. L e t t . 38 (1977) 993. [161 Chui, S. T. and Weeks, J. D., Phys. Rev. 814 (1976) 4978. [17] This i s a very common method o f s o l v i n g problems i n two-dimensional s t a t i s t i c a l mechanics, s p e c i f i c examples are given i n references 13 and 23. [ I S ] Formally t h i s i s c o r r e c t i n t h e l i m i t o f s t r o n g c o u p l i n g i n one d i r e c t i o n and weak c o u p l i n q i n the other. It i s often used as an approximate procedure t o f i n d a Hamiltonian w i t h t h e same c r i t i c a l p r o p e r t i e s as t h e c l a s s i c a l model. A more d e t a i l e d d i s c u s s i o n i s qiven i n reference 19.

PHYSICS IN FEW DIMENSIONS

[19] Black, J. L. and Emery, V. J., Phys. Rev. B23 (1981) 429 and a r t i c l e t o be published. [20] For a review see Fowler, M. i n Physics i n One Dimension, E d i t e d by J. Bernasconi and Schneider, T., S p r i n g e r S e r i e s i n S o l i d - s t a t e Sciences ( S p r i n q e r , B e r l i n , Heidelberq, New York, 1980). [21] Dashen, R. F., Hasslacher, R. H. and Neveu, A., Phys. Rev. 011 (1975) 3424. [22] Johnson, J. n., K r i n s k y , S. and McCoy, R. M., Phys. Rev. A8 (1972) 2526. [23] Schultz, T. D., M a t t i s , D. C. and Lieh, E. H., Rev. Mod. Phys. 36 (1964) 856. [24] Emery, V. J. i n H i q h l y Conductinq One-Dimensional S o l i d s , p. 247, E d i t e d by de Vreese, J. T., Evvard, R. P. and van noren, V. E. (Plenum, New York 1979). [25] Emery, V. J. i n Proceedinqs o f t h e K.yoto Summer I n s t i t u t e 1979, p. 1 ( r e f e r e n c e 7). [26] Luther, A., Phys. Rev. 815 (1977) 403. [27] Moving along t h e l i n e o f c r i t i c a l p o i n t s may correspond t o changing t h e temperature, o r parameters o f t h e model such as v e r t e x weiqhts o r t h e number (4) o f s t a t e s o f t h e P o t t s model. Thus t h e v a r i o u s models may have d i f f e r e n t exponents because they a r e r e l a t e d t o d i f f e r e n t p o i n t s on t h e c r i t i c a l line. [28] See f o r example I n t r o d u c t i o n t o t h e R e n o r m a l i z a t i o n Group and C r i t i c a l Phenomena, Pfeuty, P. and Toulouse, G. (Wiley, N e w York 1977).

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (edr.) 0 North-Holland Publishing Company, 1982

169

SOLUTION OF THE KONDO PROBLEM

N. Andrei Department o f Physics, New York U n i v e r s i t y New York, New York 10003 USA and Department o f Physics & Astronomy, Rutgers U n i v e r s i t y Piscataway, New Jersey 08854 USA The Kondo Hamiltonian i s shown t o be e x a c t l y soluble. We c o n s t r u c t the spectrum, c a l c u l a t e the magnetization curve and present an analyti c a l determination o f Wilson’s number. I n t h i s t a l k I s h a l l present an exact s o l u t i o n f o r t h e Kondo Hamiltonian

J>O

Here $,(x) operator.

i s an e l e c t r o n f i e l d with s p i n components a = 24 and

7:

i s a spin

The Hamiltonian represents a gas o f electrons i n t e r a c t i n g v i a s p i n exchange w i t h an i m p u r i t y l o c a l i z e d a t x=O, and described by t h e s p i n operator

3.

It has been extensively studied i n the p a s t [l] and many o f i t s p r o p e r t i e s were

understood.

Let me summarize those aspects t h a t a r e r e l e v a n t t o our problem.

To s t a r t w i t h , we s h a l l concentrate o n l y on q u a n t i t i e s t h a t are independent o f the c u t - o f f D.

I n other words, as i s t h e case with every renormalizable f i e l d

theory, requires a c u t - o f f i n order t o be p r o p e r l y defined. There are v a r i ous ways o f i n t r o d u c i n g c u t - o f f s and as long as they are kept f i n i t e , t h e r e a r e p r o p e r t i e s t h a t do depend on them and there are p r o p e r t i e s t h a t do not.

We

s h a l l i n v e s t i g a t e only t h e l a t t e r which are’ c a l l e d universal q u a n t i t i e s .

To e f f e c t t h i s we s h a l l r e s t r i c t the temperature T and t h e magnetic f i e l d H t o be small compared w i t h t h e c u t - o f f 0, (T, H<
N . ANDRE1

170

where f i s universal, and the only place where the d e t a i l s o f the c o n s t r u c t i o n enter i s the form of To and i t s dependence on the c u t - o f f and coupling constant, which i s , u s u a l l y , of t h e form To = DeJ-'.

To keep To i n v a r i a n t t h e coupling

constant must depend on the c u t - o f f i n such a way t h a t i t becomes l o g a r i t h m i )-I and the manner i t c a l l y smaller w i t h an increasing c u t - o f f , J = ( l o g approaches zero depends on the choice o f To.

TI

This p r o p e r t y i s c a l l e d asymptotic

freedom[ 31 [ 41. I n asymptotically f r e e theories, however, one may apply p e r t u r b a t i o n theory a t high temperatures o r magnetic f i e l d s T, H>>To ( b u t s t i l l TI H<
[ 2 ] and the approximation schemes break down, impurity susceptibility

xi.

But when one

one reaches a strong coupling regime

For a f r e e s p i n

xi

<

To q u a n t i f y t h i s , consider t h e i = H=O = ( C u r i e ' s Law) and

one expects t o f i n d , even when i n t e r a c t i o n s a r e present, t h a t the s u s c e p t i b i l i t y approaches t h i s value.

Thus

log log

xi

GO+ [I- 7l o g1TK -

+

T r

21 l o g2 K K

+O($EJ]

where we see t h e l o g a r i t h m i c c o r r e c t i o n s t y p i c a l t o asymptotic freedom. a l s o introduced a scale TK which i s d e f i n e d by absorbing t h e ( l o g T t o the ( l o g term.

r)-'

5

We have

)-* term i n -

0

Crossing over t o zero temperature one f i n d s [ 21 t h a t t h e suscepti b i 1it y , r a t h e r than d i v e r g i n g l i k e

F, i s f i n i t e and defines a new scale To 2

x' f i n i t e i s screening (quenching) which i leads t o a spinless i m p u r i t y and hence t o a f i n i t e x .

The physical mechanism t h a t renders

SOLUTION OF THE KONDO PROBLEM

The r a t i o W = TK/To i s a pure c a l c u l a b l e number [4].

171

It characterizes t h e cross-

over i n the p r o p e r t i e s o f a system from a weak t o a strong coupling regime (simi l a r t o QCO),

and requires non-perturbative c a l c u l a t i o n .

I t was f i r s t found

numerically by Wilson [ l ] and we s h a l l derive [5] an a n a l y t i c expression, using an exact diagonalization [ 6 ] [7] o f the Hamiltonian t o which we now turn. The Hamiltonian ( l a ) conserves the number o f e l e c t r o n Ne = J @;Oadx,

thus we may

go over t o f i r s t quantization formalism and e q u i v a l e n t l y study the f o l l o w i n g Hami 1tonian Ne

Ne

2 -%+JJ 1

h= -i

.

6(xi)Gi i=l

i=1ax’

(1.b)

The Hamiltonian now bears some s i m i l a r i t y t o a well-known model i n quantum f i e l d theory c a l l e d the c h i r a l Gross-Neveu model [4]. To b r i n g out the s i m i l a r i t y [6] we introduce a new quantum number LY = 1 o r 0. CY w i l l be r e f e r r e d t o as p u r i t y and t h e Ne electrons each c a r r i e s a = l and t h e

and we may r e w r i t e h as f o l l o w s (N = Ne+l)

impurity PO,

N=l h = -i

c

N + J

aiai

i,j=l

i=1

The only d i f f e r e n c e w i t h the former h i s t h a t now the i m p u r i t y ( w i t h x1 t h a t corresponds t o cr’=O)

But as i t c a r r i e s no k i n e t i c energy

i s allowed t o move.

we may form p e r f e c t wave packets t h a t do n o t disperse and c o n s t r u c t a l o c a l i z e d i m p u r i t y state. interact.

The f a c t o r (LY.-cY.)* insures t h a t o n l y electrons and i m p u r i t y 1 J We, then, want t o solve the eigenvalue problem

hS(xi,

ui,

ail = ES(xi,

oi,

ai)

where B ( x , LY, a) i s the wave function, depending on a l l the p a r t i c l e s ’ coordinates. I t i s antisymmetrical since we a r e dealing w i t h fermions: S(

...

xiuiai

... x j c1j a j

...) =

-

...

S(

x.ci.a J J j

... xioiai

...)

N. ANDRE1

172

Since the Hamiltonian i s i n v a r i a n t under s p i n r o t a t i o n s we a r e able t o f a c t o r i z e the wave f u n c t i o n 9=F(xa)E(a) and characterize the s p i n wave f u n c t i o n s t ( a ) by a Young Tableau R having up t o two rows R = [N-M, MI where N = N e + l i s t h e number o f p a r t i c l e s i n t h e system and M t h e number o f down spins.

R completely

characterizes the p r o p e r t i e s o f the wave functions under t h e r o t a t i o n group and under t h e permutation group SN. We t u r n now t o the dynamics and show t h a t h indeed can be completely diagonalized.

We s h a l l use the Bethe Ansatz [8] technique as b r i l l i a n t l y developed by

Yang [9] and Gaudin [ l o ] . labeled by permutation Q 0 < x

<. . .< x

E

Thus, d i v i d e c o n f i g u r a t i o n space i n t o N! SN.

regions

I n the i n t e r i o r o f r e g i o n Q, defined by

< L t h e p a r t i c l e s a r e f r e e .and we w r i t e F as a superposition o f

Ql QN plane waves labeled by N momenta Ki

and p u r i t y i n d i c e s pi,

[pi=l,

i=l.. .Ne,

The corresponding energy eigenvalue i s obviously

Ne

The N!xN!

m a t r i x o f constant c o e f f i c i e n t s

s o l u t i o n s across the boundaries. t r o n and one i m p u r i t y , pl=l,

5P (Q)

i s determined by matching t h e

For example, consider t h e case o f one elec-

p2=0.

The wave f u n c t i o n i n t h e region I ( i . e . ,

x1<x2)

i s given by

From t h e Hamiltonian h, we o b t a i n the a 2x2 m a t r i x Y r e l a t i n g t h e c o e f f i c i e n t s ,

SOLUTION OF THE KONDO PROBLEM

This can, obviously, always be done. three p a r t i c l e s w i t h 3!=6

173

The c r u c i a l t e s t a r i s e s when we consider

regions n o t a l l o f which are now adjacent.

The

regions can be arranged as shown i n Figure 1.

Figure 1: Schematic representation o f t h e Y matrices connecting d i f f e r e n t regions i n the case o f 3 p a r t i c l e s . 123 where the region Q = (312), region (312) matrix Y13

f o r example,

i s denoted by (312).

Thus t o go t o

from region (132) which i s adjacent t o i t one needs t o use t h e

which f l i p s p a r t i c l e s 1 and 3.

When one deduces from t h e Hamiltonian

h the s e t o f matrices Y one may t h i n k t h a t the problem i s solved.

'

This i s rare-

l y the case, as the matrices have t o s a t i s f y s t r i n g e n t consistency conditions. Thus, column Y-matrices

p (321)

E(321) = '12'13'23

can be obtained from column

'(123)

( 123 1

by repeated a c t i o n o f t h e

'

b u t equally w e l l i t could have been obtained by a d i f f e r e n t path E(321) = '23'13'12

'(123)

'

Only i f we g e t the same answer can we claim t h a t t h e Ansatz (2) i s consistent. For N p a r t i c l e s one.may t h i n k t h a t one needs many more consistency conditions, b u t i t turns o u t t h a t the f o l l o w i n g are s u f f i c i e n t

N. ANDRE/

174

Yij

,

= Yji

Yi jYi kYjk

YijYka

YjkYi

= YkaYij

( i j k fi are d i f f e r e n t )

kYi

The c r u c i a l p o i n t i s t h a t t h e Y matrices deduced from the Hamiltonian h (eq. do indeed s a t i s f y the ( f a c t o r i z a t i o n ) c o n d i t i o n (4) and we thus

( l b ) and (1.c))

f i n d t h a t the Kondo Hamiltonian i s soluble. As a l l columns {

can be c o n s i s t e n t l y derived from any one o f them we may conP c e n t r a t e on a reference column [I and determine it. To do so we need t o impose boundary conditions which we choose t o be p e r i o d i c .

F( ... xi = 0

... ) = F( ...xi

Thus we r e q u i r e

= L ...)

(5)

Let me i l l u s t r a t e the procedure i n an extremely simple case: h = -i

a

+

2Jb(x)

and we t r y a s o l u t i o n F(x) = [Ae(-x) + Be(x)]eikX I n t e g r a t i n g across the boundary x=O, we f i n d , -i(B-A)

so

+ J(B+A) = 0

B = =i +=J e-

.

iQ

i s the 1 x 1 m a t r i x Y

,

r e l a t i n g the c o e f f i c i e n t s i n t h e two regions. ted) wave i s produced.

Note t h a t no r e f l e c t e d ( d i f f r a c -

This i s an example o f no-production phenomenon, charac-

t e r i s t i c o f Bethe Ansatz systems: namely t h e same K values appear i n both r e gions.

I f we had a second d e r i v a t i v e t h i s Ansatz would n o t be so c o n s i s t e n t and

a wave w i t h (-K)

would be necessary.

To determine t h e allowed K ' s we impose p e r i o d i c boundary c o n d i t i o n s , and f i n d

eiQ = ,ikL

r

so t h a t the energy eigenvalue i s given by E = K = 2n n +

t.

The e f f e c t o f the i n t e r a c t i o n i s thus given by t h e phase Q which s h i f t s the energy from i t s f r e e value. We s h a l l now w r i t e down the answer f o r t h e allowed K ' s o f the e l e c t r o n s o f Hamilt o n i a n (1). I t i s found, using Yang's method, t h a t

SOLUTION OF THE KONDO PROBLEM

K(electron)

i

--

2~ ni

M

C

+

[O(2Ay-2)-n]

Y=l where O(x) = - 2 tan-' A1...A

25

:,

c =

32 1-qJ .

M are M complex numbers found by s o l v i n g the f o l l o w i n g s e t o f equations

The integers ( n

I ] are t h e quantum numbers o f t h e eigenstates and each allowed

j' Y

We are i n t e r e s t e d i n the l i m i t where N, t h e

choice uniquely determines a state.

number o f p a r t i c l e s , and M, the number o f down spins, go t o i n f i n i t y .

In this

case one can f i n d the density o f s o l u t i o n s o f equation (7) and e a s i l y determine the p r o p e r t i e s o f t h e eigenstates (6). The ground s t a t e f o r example i s a s i n g l e t and given by s o l v i n g eq. (7) w i t h a

= I +1. consecutive c o n f i g u r a t i o n , IY+l

Y

The simplest e x c i t a t i o n i s a t r i p l e t and i s given by generating two "holes" i n the ground s t a t e c o n f i g u r a t i o n , i . e . , etc.

Denoting by A1

t h e r e are y1 and y2 such t h a t

Iyl+l = IY1+2,

and A2 the A's corresponding t o t h e omitted i n t e g e r s

we f i n d [ 6 ] t h e e x c i t a t i o n energy i s given by

We note t h a t there i s no mass gap, as A

i

can be a r b i t r a r i l y l a r g e and negative.

This way one may go on and c o n s t r u c t any number o f elementary e x c i t a t i o n s . We s h a l l t u r n now t o study the e f f e c t o f a magnetic f i e l d on t h e system.

We

add t o JC a magnetic term

has t h e same commute, the t o t a l Hamiltonian xT = JC + Jc mag mag eigenstates as before. However, the ground s t a t e i n t h e presence o f a magnetic Since

Jc

and JC

f i e l d w i l l have f i n i t e magnetization.

Indeed, each s p i n a l i g n i n g w i t h the mag-

n e t i c f i e l d w i l l gain magnetic energy given by AE = 2pH w h i l e i t w i l l lose an amount given by eq.' (8). Thus the i m p u r i t y magnetization Mi w i l l be determined by these competing processes.

I t i s found t o be

176

N. ANDRE1

We note t h a t M’ has t h e advertised behavior, namely asymptotic freedom f o r h i g h magnetic f i e l d s and a crossover t o strong coupling regime when t h e f i e l d i s deFor l a r g e f i e l d s , H>>TH, i t approaches the f r e e value

creased.

and we have determined a new scale TH by absorbing t h e (log)-’ ( l o g ) - l term.

term i n t o the

When H+O, on t h e o t h e r hand, we see t h a t Mi approaches zero w i t h

a f i n i t e slope ( i . e . ,

susceptibility)

i n d i c a t i n g screening o f the i m p u r i t y spin.

The r a t i o between TH and To i s a uni-

versal number and i s given by

We have, then, achieved our goal t o c a l c u l a t e a u n i v e r s a l number and may have rested here had n o t Wilson chosen t o c a l c u l a t e a d i f f e r e n t r a t i o TK/To; b u t i t i s a c t u a l l y easy t o convert t o h i s number as b o t h TK and TH c h a r a c t e r i z e regions where p e r t u r b a t i o n theory i s v a l i d so t h a t t h e r a t i o can be e a s i l y calculated. 1 We f i n d TK where fin p = J dx(l-x)2x(n2csc2nx-x-2) and any = E u l e r ’ s

Ti-

0

constant.

Combining i t w i t h (13) we f i n a l l y have

TK

-

2g&2e-9/4

w=5-

= 0.102676 x 471

177

SOLUTION OF THE KONDO PROBLEM

which i s i n agreement w i t h the value found numerically by Wilson = 0.1032+0.0005

.

To summarize:

The Kondo model can be completely diagonalized and i t s essential physics exposed. One can c a l c u l a t e various universal numbers a n a l y t i c a l l y using our c o n s t r u c t i o n and f i n d them t o be i n agreement w i t h other ways o f c o n s t r u c t i n g t h e model [l]. O f course, using the exact s o l u t i o n one has access t o new and even more i n t e r e s t i n g physics.

I would l i k e t o thank J. H. Lowenstein f o r a close and f r u i t f u l c o l l a b o r a t i o n . REFERENCES:

[l] Wilson, K., Rev. Mod. Phys. [2] Anderson, P. W., [3] Gross H. 0.

47, 773 (1975).

Yuval, G.,

and Hamann, D. R.,

Phys. Rev.

E,

4464 (1970).

D., and Wilczek, F., Phys. Rev. L e t t . 30, 1343 (1973); P o l i t z e r , Phys. Rev. L e t t . 30, 1346 (1973).

[4] Gross O . , and Neveu, A., Phys. Rev. D

lo, 3235 (1979).

[5] Andre , N . , and Lowenstein, J. H., Phys. Rev. L e t t . [6] Andre , N., Phys. Rev. L e t t .

46, 356

(1981).

5 , 379 (1980).

[7] Wiegmann, P. B., Pis'ma Zh. Eksp. Theor. Fiz. 3 l , 392 (1980). [8] Bethe, H., Z. Phys.

c,205 (1931).

[9] Yang, C. N., Phys. Rev. L e t t .

[lo] Gaudin, M., Phys. L e t t .

19, 1312

e,55 (1967).

(1967).

This Page Intentionally Left Blank

Nonlinear Problems: Present and Future

179

A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.)

0 North-Holland Publishing Company, 1982

KINKS OF FRACTIONAL CHARGE I N QUASI-ONE-DIMENSIONAL SYSTEMS J. R. S c h r i e f f e r

Department o f Physics and I n s t i t u t e f o r Theoretical Physics University o f California USA Santa Barbara. CA 93106 Recent t h e o r e t i c a l studies have shown t h a t i n quasi-one-dimensional conductors having a P e i e r l s d i s t o r t i o n o f commensurability index n ( r a t i o o f the d i s t o r t i o n p e r i o d t o l a t t i c e spacing), t h e r e e x i s t excit a t i o n s whose charge i s Q = f 2e/n o r f 2e/n f e. These e x c i t a t i o n s are kinks i n the order parameter I$ describing t h e l a t t i c e d i s t o r t i o n . I n trans polyacetylene, n = 2 and Q = 0, f e with s p i n S = 4 0 respectively. For n = 3 (e.g., TTF-TCNQ a t 19Kb) Q = f 8 e, f e, f 4 e w i t h s p i n S = 0, 3, 0 respectively. Electronic states localized a t the k i n k have energies i n t h e P e i e r l s gaps. Properties o f these stable f r a c t i o n a l l y charged objects are discussed.

4

INTRODUCTION

F r a c t i o n a l l y charged e x c i t a t i o n s have o f t e n been considered i n t h e past. Rec e n t l y , however, i t has been shown how a theory c o n t a i n i n g o n l y fundamental p a r t i c l e s o f integer charge Q can lead t o e x c i t a t i o n s having charge q which i s a f r a c t i o n o f Q. I n a continuum model o f charge d e n s i t y waves i n one-dimensional metals i t was shown by Rice, Bishop, Krumhansl, and T r u l l i n g e r ' t h a t commensurability e f f e c t s can s t a b i l i z e kinks which c a r r y a charge q = Ae/n, where A0 = 2n/n, w i t h n being the commensurability, e.g., n = 2 f o r a dimerized system, n = 3 f o r a t r i m e r i z e d system. Independently, Jackiw and Rebbi2 studied a r e l a t i v i s t i c f i e l d theory model i n one dimension i n which a spinless Dirac f i e l d i s coupled t o a s e l f - i n t e r a c t i n g Bose f i e l d having two degenerate mean f i e l d ground states. They found t h a t the model possessed s o l i t o n e x c i t a t i o n s which correspond t o a change i n t h e number o f Dirac fermions i n the v i c i n i t y o f t h e k i n k being 4. A c-number Fermion s t a t e a t zero energy was found whose wavefunction was l o c a l i z e d about t h e s o l i ton. Depending on whether t h i s s t a t e i s occupied o r not, t h e s o l i t o n s would c a r r y Fermion number f 4, corresponding t o charge k e/2. Independently, J. Hubbard3 studied a one-dimensional t i g h t b i n d i n g chain w i t h very l a r g e o n - s i t e repulsions and weaker nearest neighbor i n t e r a c t i o n s . In t h i s l i m i t , the problem s p l i t s i n t o a s e t o f spinless Fermions and a s e t o f Heisensberg spins. Hubbard showed t h a t f o r one e l e c t r o n p e r two s i t e s on average (quarter f i l l e d band), an i n j e c t e d e l e c t r o n would s p l i t i n t o two kinks, each o f charge - e/2. A r e l a t e d problem w i l l be discussed by M. J. Rice i n t h e f o l l o w i n g paper. Motivated by experiments on trans (CH) , Su, S c h r i e f f e r , and Heeger (SSH)4 studied a one-dimensional model o f e l e c t r o n s hopping along a chain, w i t h t h e hopping i n t e g r a l l i n e a r l y modulated by l a t t i c e displacements. S i m i l a r studies were c a r r i e d o u t by Rice.6 Because o f t h e P e i e r l s d i s t o r t i o n , t h e mean f i e l d ground s t a t e i s degenerate. For the h a l f - f i l l e d band (one e l e c t r o n p e r s i t e on average) appropriate t o undoped (CH) , the ground s t a t e i s t w o f o l d degenerate. I n t h i s model, the symmetry b r e a k i n g ( l o s s o f i n v e r s i o n symmetry i n t h e phonon coordinates) i s dynamically generated by the electron-phonon coup1 i n g as opposed t o the imposed broken symmetry i n the r e l a t i v i s t i c model o f Jackiw and Rebbi.

J. R. SCHRIEFFER

180

For t h e (CH) model, i t was found t h a t t h e r e are low energy e x c i t e d s t a t e s correspondingXto s o l i t o n s which a c t as moving w a l l s separating domains having d i f f e r e n t ground states, A and 13. For each i s o l a t e d s o l i t o n , an e l e c t r o n i c s t a t e occurs i n the center o f the P e i e r l s gap w i t h a wavefunction centered about the s o l i t o n , as i n the Jackiw and Rebbi mode1.316 An important d i f f e r ence, however, i s t h a t i n (CH) e l e c t r o n s o f both s p i n o r i e n t a t i o n enter symm e t r i c a l l y i n t h e Fermi sea so & a t t h e charge q = f e/2 i s " s p i n masked," being doubled t o q = f e. These s t a t e s a r e o f s p i n zero, w h i l e f o r (CH) t h e r e i s an added charge s t a t e o f the s o l i t o n , q = 0 w i t h s p i n 4. Thus, i n s b a d o f f r a c t i o n a l charge being d i r e c t l y observable i n (CHI , t h e e f f e c t appears as an i n terchange o f t h e conventional charge-spin r e l a t i o n s o f e l e c t r o n i c e x c i t a t i o n s i n s o l i d s ; instead o f q = f e, S = 4 f o r conventional e l e c t r o n s and holes, one has q = f e, S = 0 f o r charged s o l i t o n s and q = 0, S = 4 f o r n e u t r a l s o l i t o n s . This breaking o f the charge-spin r e l a t i o n s o f t h e u n d e r l y i n g e l e c t r o n f i e l d i s c h a r a c t e r i s t i c o f the s o l i t o n e x c i t a t i o n s i n a degenerate ground s t a t e system. The o n e - t h i r d f i l l e d band ( t r i m e r i z e d chain) was s t u d i e d by Su and S c h r i e f f e r ' using t h e SSH hamiltonian. A consequence o f t h e t h r e e f o l d degeneracy o f the ground s t a t e i s t h e existence o f two types o f kinks, K1 and K, each having t h r e e charge states: 2e/3, -e/3, -4e/3 f o r K1 and -2e/3, e/3, and 4e/3 f o r K., The s p i n o f t h e f e/3 kinks i s S = 4 w h i l e t h e o t h e r kinks have S = 0. Thus, s p i n no longer masks f r a c t i o n a l charge f o r t r i m e r i z e d chains. Below I discuss the o r i g i n o f these r e s u l t s derived i n c o l l a b o r a t i o n w i t h W. P. Su.

TRIMERIZED CHAIN We consider the model Hamiltonian

H =

-

-u ) I ( c ~ + ~ , ~ c ~ , ~ + H . c+. )'2 K 2 (Un+l-un)2 n

t [to-u(un+l ns

+

'K2

2 un -2

(1)

wbere u i s t h e displacement o f the nth u n i t tfirom i t s e q u i l i b r i u m p o s i t i o n , un t and M i s the mass o f one i s &e n e l e c t r o n c r e a t i o n operator on t h e n c uRSt o f the chain. K i s an e f f e c t i v e s p r i n g constant d e s c r i b i n g an harmonic approximation t o the u bond energy. For t h e p e r f e c t l y t r i m e r i z e d chain, we w r i t e i n t h e ( a d i a b a t i c ) mean f i e l d approximation t h e most general f u n c t i o n having t h e t r a n s l a t i o n symmetry An = 3,

un = u

2n

C O S ( ~n

-e)

I n Fig. 1 the energy p e r s i t e i s p l o t t e d as a f u n c t i o n o f 0 f o r t h r e e values o f the amplitude u = 0, 0.0411 and 0.07x where, as an example, we have chosen t h e (CH) parameters t = 2.5 eV, 01 = 4.81 eV/h, K = 17.4 eV/X2. The energy i s minAs one can ' 0, f 2 d 3 , f 4n/3, mod f 271,. and uo 0.078. imizgd by 0 n/6 % see, t h e minimum energy path t o go between these minima i s by changing t h e phase angle e w i t h t h e amplitude e s s e n t i a l l y f i x e d . This leads t o a small b a r r i e r ( 2 t /A)Za. For t h e dimerized height o f order A 3 / t 2 and a s o l i t o n w i d t h E3 system, o n l y t h e ampfitude v a r i a b l e e x i s t s , and t h e eondensation energy passes through zero a t the center o f the k i n k , g i v i n g a b a r r i e r h e i g h t of order A2/t per s i t e , and a s o l i t o n w i d t h t2 ( 2 t /A) a. Hence, the w i d t h o f t h e t r i m e r e i z e d s o l i t o n i s l a r g e r than t h a t o? t h e dimerized system by t h e f a c t o r (2to/A) >> 1.

-

..

-

-

-

KINKS OF FRACTIONAL CHARGE IN QUASI-ONE DIMENSIONAL SYSTEMS

181

-2.75

-2

5Y

W

-2.76

-2.76

Figure 1: Total energy per s i t e p l o t t e d as a f u n c t i o n o f the phase angle 6 f o r t h r e e d i f f e r e n t values o f the amplitude o f t r i m e r i z a t i o n u.

To determine t h e charge and s p i n o f the kinks i n the o n e - t h i r d f i l l e d band case, one cannot use the charge conjugation symmetry d e r i v a t i o n which has been used f o r the dimerized system since t h i s symmetry gives no s i g n i f i c a n t i n f o r m a t i o n i n t h i s case. Rather, one can use 1) a Green f u n c t i o n method t o determine the charge and s p i n density o f the s o l i t o n or, 2) a simple b u t general argument based on t r a n s l a t i o n a l invariance and charge conservation. It i s t h i s l a t t e r argument we present below.

-

-

Consider an i n f i n i t e l y long chain, < n < m, w i t h 0 n/6 = 0, i . e . , the A ground state. Suppose t h a t 0 i s a d i a b a t i c a l l y defopmed so t h a t the chain remains i n the A s t a t e f o r an< n < nl, i s i n t h e B phase (0 n/6 = 2n/3) between nl and n2, i s i n the C phase (e n/6 = Zn). The change o f e near nl, n2, and n, need n o t be abrupt, b u t the t r a n s i t i o n region ( s o l i t o n wid!h) is assumed t o be small compared t o the spacings n2 nl , and n, n2. There w i l l be a c e r t a i n w i d t h 5, which w i l l minimize t h e energy. As a r e s u l t o f t h i s phase deformation, one has created 3 type I k i n k s ( i . e . , A-43, B-4, C+A).

-

-

-

-

-

To determine the charge and spin o f t h e kinks, suppose t h a t one constructs two surfaces, one a t n f a r t o the l e f t o f nl and another a t n f a r t o the r i g h t o f n,. From global c8nservation o f charge, we know t h a t t h e r t o t a l charge passing through these surfaces when 0 i s slowly deformed i s the negative o f t h e sum o f the charges o f the three s o l i t a n s , AQ, m

and j are the e l e c t r i c c u r r e n t d e n s i t i e s a t t h e two surfaces. Since n/& remaiis zero near n , no c u r r e n t flows a t t h i s surface, w h i l e t h e

where j 8

-

cRarge passing the right-hand %urface i s the e l e c t r o n i c charge i n one u n i t c e l l of s i z e 3a o f the t r i m e r i z e d chain, i.e., -2e and

AQ = 2e

.

(4)

182

J.R. SCHRIEFFER

However, from the invariance o f H under t r a n s l a t i o n s f. a, i t f o l l o w s t h a t a l l p r o p e r t i e s o f the t h r e e kinks must be t h e same. Thus, t h e fundamental charge o f type I kinks KO i s

AQ = 39

= 2e

(5)

KO

or

QKo

_- -2e 3

a

A s i m i l a r argument holds i f t h e phase angle 0 i s decreased as one moves t o t h e r i g h t so t h a t one passes from A t o C t o B t o A moving from l e f t t o r i g h t , w i t h 0 n/6 = 0, - 2n/3, 4n/3, 2n. The e l e c t r i c c u r r e n t i s no_w i n t h e reverse d i r e c t i o n , and the fundamental charge o f these type I1 kinks KO ( a n t i k i n k s ) i s

-

-

-

Since s p i n up and s p i n down s t a t e s a r e i d e n t i c a l l y occupied d u r i n g t h e a d i a b a t i c deformation, t h e s p i n o f KO and KO i s zero,

s

=s- = o

.

KO

Figure 2: The three degenerate ground s t a t e s o f a p e r f e c t l y t r i m e r i z e d chain.

KINKS OF FRACTIONAL CHARGE IN OUASI-ONEDIMENSIONAL SYSTEMS

183

As i n the dimerized case, l o c a l i z e d s t a t e s ar e expected i n the e l e c t r o n i c energy gap connected w i t h the s o l i t o n s since st at es are missing from the valence band, leading t o t h e f r a c t i o n a l charge. Using a Green f u n c t i o n method one can determine t h e electronic-spectrum i n t h e presence o f kinks. I f one imposes a sharp kinks K o r a n t i k i n k K on the system, one f i n d s t h e spectrum shown i n Fig. 3. For the kink, there are two gap states Q and Q l ocate d i n the upper h a l f o f t he lower gap and symmetrically i n the l%wer ha# o f the upper gap. D i r e c t c a l c u l a t i o n o f t h e f r a c t i o n a l parentage coe f f i c i e n t s o f these st at es shows t h a t f o r K, Q i s derived 1/3 from t h e bottom band and 2/3 from t he middle band, w h i l e I) conks 1/3 from t h e t o p band and 2/3 f r o m the middle band. Since 1/3 o f a s t a t d i s removed from the bottom band by K and a l l valence band st at es remain f i l l e d as K i s created, i t f o l l o w s t h a t i f I) and I) are unoccupied (K ), then the charge o f K i s Q = 2/3 e, i n agreem%nt w i d (6) and (8). Howher, i f I), i s s i n g l y octupied &en

and i f 6, i s doubly occupied, then 4e

Q K 2 = - 3

'

K2 = O

Figure 3: Gap sta tes associated w i t h (a) a sharp t y p e - I k i n k and (b) a sharp type- II kink.

J. R. SCHRIEFFER

184

S i m i l a r l y , the a n t i k i n k i corresponds t o having no holes i n I),, trons, so t h a t i t s c h a r g e y s

i.e.,

2 elec-

I n (ll), the f a c t o r o f two m u l t i p l y i n g 2e/3 a r i s e s from i n agreement w i t h (7). two s p i n o r i e n t a t i o n s i n accounting f o r the missing charge i n t h e valence band due t o the removal o f 2/3 o f a s t a t e per gpin by an a n t i k i n k from t h e lowest band. Since a l l s t a t e s are s p i n p a i r e d f o r KO, i t f o l l o w s t h a t

F i n a l l y , i f I), i s s i n g l y occupied i n

kI, then

and i f I), i s empty, then

~ i = + : e 2 We see t h a t K, charge.

,s- = o K2 and i(, a c t as a n t i p a r t i c l e s , having t h e same s p i n b u t reversed

As i n t h e dimerized case, t h e r e are c o n s t r a i n t s on c r e a t i n g s o l i t o n s . I f the chain i s t o be unaffected a t i n f i n i t y , t o p o l o g i c a l constrai-Ecs r e q u i r e t h a t kinks must be created i n p a i r s (KK o r KK), t r i p l e t s (KKK o r KKK), o r combinat i o n o f these allowed sets. From t h e k i n k quantum numbers (Q,S),

one can see t h a t f r a c t i o n a l charge and s p i n cannot be created g l o b a l l y b u t o n l y l o c a l l y . Hence, one would n o t be able t o observe t h i s type o f f r a c t i o n a l charge by measurements sampling the t o t a l charge o f t h e system. Rather, t h e f r a c t i o n a l charge r e f l e c t s an i n t e r n a l d i s t r i b u t i o n o f an i n t e g e r t o t a l charge o f t h e system. One method f o r observing f r a c t i o n a l charge i n polymers i s through t h e shot noise voltage f l u c t u a t i o n spectrum. For nth order commensurability, t h e p r i m a t i v e k i n k quantum numbers a r e

KINKS OF FRACTIONAL CHARGE IN QUASI-ONE DIMENSIONAL SYSTEMS

QK,

-- 2- e = _ n

Q and i0 S

As above, KO and states.

= S-

KO

KO

=

0

185

.

KO

can be decorated by electrons and holes i n the . J c a l i z e d

REAL OR APPARENT FRACTIONAL CHARGE? The question has been r a i s e d whether t h e f r a c t i o n a l charge discussed above i s a quantum mechanically sharp observable o r simply t h e quantum average o f two o r more i n t e g e r charges, t h e r e s u l t o f which i s f r a c t i o n a l . 8 I n other words, i f one can d e f i n e a quantum operator Q which measures t h e charge o f the s o l i t o n , then i s the p r o b a b i l i t y d i s t r i b u t i o W ) P(Q) o f t h i s operator i n t h e one s o l i t o n sector a d e l t a f u n c t i o n a t the f r a c t i o n a l k i n k charge QK

o r i s i t given by

where

91, Q,,

.. .

are integers,

and o n l y the expected charge i s f r a c t i o n a l

?

= 1 aiQi = QK i

(19)

An exapple of apparent f r a c t i o n a l charge, (18) and (19), i s given by considering the H, ion. Suppose the e l e c t r o n i s i n the bonding (even p a r i t y ) molecular o r b i t a l JI+ and the i n t e r n u c l e a r spacing i s a d i a b a t i c a l l y increased t o a very I n t h i s l i m i t $+ reduces t o a l i n e a r combination o f 1s atomic l a r g e value. o r b i t a l s JI centered on t h e two protons 2 and r, 1 1 *+=-$g+-$r

J Z J Z

*

I f Qop = n2 i s the occupation number f o r $,,

then

has nonvanishing m a t r i x elements between $+ and t h e n e a r l y However, because Q degenerate antibonafng s t a t e $-

J.R. SCHRIEFFER

186

i t f o l l o w s t h a t JI+ i s n o t an eigenfunction o f Q and the mean square f l u c t u a t i o n i s given by OP

For t h i s simple example,

The usual quantum theory o f measurement i n t e r p r e t a t i o n o f t h i s f i c t i c i o u s f r a c t i o n a l charge i s t h a t w h i l e t h e average (expected) charge i s f r a c t i o n a l , o n l y integer charge w i l l be measured i n any s i n g l e experiment. How i s t h i s r e l a t e d t o the kinks? The e s s e n t i a l p o i n t i s t h a t the o f f diagonal elements o f the k i n k charge operator vanish e x p o n e n t i a l l y i n the l i m i t o f w i d e l y spaced kinks, r a t h e r than appoaching a constant [4, see Eq. (23)] f o r t h e H, example. Therefore t h e one s o l i t o n s t a t e approaches an eigenfunction o f Q f o r widely spaced kinks. This c r u c i a l d i f f p r e n c e a r i s e s from t h e f a c t t h a t thepe i s a unique e l e c t r o n i c ground s t a t e o f KK f o r any spacing, w i t h no low l y i n g exc i t e d states, a l l e x c i t e d s t a t e s having an energy o f order o r g r e a t e r than A . Furthermore, even these s t a t e s are n o t e x c i t e d by a very long wavelength potent i a l since a slowly varying p o t e n t i a l o n l y couples t o the t o t a l charge o f the system i n the v i c i n i t y o f t h e kink. Therefore, t h e fundamental f r a c t i o n a l charge o f the k i n k i s sharp and a r i s e s t o t a l l y from t h e d e p l e t i o n o f s t a t e s from the f i l l e d valence band when a s o l i t o n i s formed.

I f Qop measures the up s p i n charge on the s o l i t o n , then f o r KO,

A t o t a l l y separate b u t somewhat confusing question i s the spacial d i s t r i b u t i o n o f charge located i n t h e gap states. The discussion i n t h e previous paragraph concerns c o n f i g u r a t i o n s i n which there are no e l e c t r o n s i n t h e gap center s t a t e s o r when these s t a t e s are completely f i l l e d . I n t h i s case t h e r e i s no degeneracy, and a slowly vgrying f i e l d cannot admix e x c i t e d states. Suppose, however, t h a t we consider KIK so t h a t t h e r e i s one s p i n up d e c t r o n i n t h e gap center s t a t e o f K. This con9iguration i s degenerate w i t h K K1; however, t o r f i n i t e spacing there i s mixing which produces the analog o f t h 8 $ + s t a t e o f H

Jz

K

ii l o

1

-

+ - KoK1

Q

.

I n t h i s case, t h e gap center e l e c t r o n f l u c t u a t e s between K and f l u c t u a t i n g K charge whose p r o b a b i l i t y d i s t r i b u t i o n i s

k

leading t o a

187

KINKS OF FRACTIONAL CHARGE IN QUASI-ONE DIMENSIONAL SYSTEMS

which i s t o be compared w i t h (24) showing a sharp d i s t r i b u t i o n a t Q = +. Clearly, the f r a c t i o n a l charge associated w i t h t h e vacuum d e f i c i t ( o r vacuum p o l a r i z a t i o n ) i s a sharp quantum number, and i t i s t h i s we r e f e r t o when d i s A f i c t i c i o u s f r a c t i o n a l charge can e n t e r from a cussing f r a c t i o n a l charge. t o t a l l y d i f f e r e n t and u n i n t e r e s t i n g e f f e c t , namely t h a t analogous t o charge f l u c t u a t i o n s i n H,. CONCLUSIONS While the noninteger charge e f f e c t s discussed above w i l l be screened by the d i e l e c t r i c constant o f the medium, as i n any s o l i d the n e t charge macroscopically c a r r i e d i s the unscreened charge. The shot noise experiment should measure t h i s charge. F r a c t i o n a l i z e d quantum numbers may be observable i n other systems, such as domain w a l l s i n magnets and 3He texture^.^ Recently, Goldstone and WilczeklO have shown how these ideas can be generalized t o r e l a t i v i s t i c f i e l d t h e o r i e s i n higher dimensions. ACKNOWLEDGMENTS The author i s g r a t e f u l t o Drs. S. Kivelson and W. P. Su f o r h e l p f u l discussions. This work was supported i n p a r t by the National Science Foundation under Grant No. DMR80-07432, and Grant No. PHY77-27084. REFERENCES

1.

M. J. Rice, A. R. Bishop, J . A. Krumhansl, and S. E. T r u l l i n g e r , Phys. Rev. L e t t . 2 , 432 (1976).

2.

R. Jackiw and C. Rebbi, Phys. Rev. D.

3.

J. Hubbard, Phys. Rev. B.

4.

W. P. Su, J. R. S c h r i e f f e r , and A. J. Heeger, Phys. Rev. L e t t . (1979); Phys. Rev. B 11, 2099 (1980).

5.

M. J. Rice, Phys. L e t t .

6.

For a review o f the r e l a t i o n between f r a c t i o n a l l y charged kinds i n condensed matter physics and r e l a t i v i s t i c f i e l d theory, see R. Jackiw and J. R. S c h r i e f f e r , Nucl. Phys. @ [FS 31, 253 (1981).

7.

W. P. Su and J. R. S c h r i e f f e r , Phys. Rev. L e t t . 4 6 , 738 (1981).

8.

The author i s indebted t o D r . A. K. Kerman and V. Weisskopf f o r r a i s i n g t h i s question. The discussion below r e l i e s on work o f D r . S. A. Kivelson and the author, t o be published.

9.

Jason Ho, p r i v a t e communication.

10.

F. Wilczek and J. Goldstone, t o be published.

2,3398

(1976).

lJ, 494 (1978).

42,

1698

E,152 (1979).

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (4s.) @ North-Holland PublishingCompany, 1982

189

THEORETICALLY PREDICTED DRUDE ABSORPTION BY A CONDUCTING CHARGED SOLITON I N DOPED-POLYACETYLENE M. J . Rice Xerox Webster Research Center Webster , NY 14580, USA With the a i d o f a novel quantum theory o f t h e charged s o l i t o n we conclude t h a t the observation o f a " d i s c r e t e Drude absorption'' would c o n s t i t u t e an experimental v e r i f i c a t i o n o f charged-sol i t o n t r a n s p o r t i n doped-polyacetylene. The suggestion [1,2] t h a t charged s o l i t o n s may be generated i n trans-polyacet y l e n e [ ( C H I ] by l i g h t acceptor o r donor doping has stimulated much subsequent experimental 'and t h e o r e t i c a l i n t e r e s t i n t h i s subject. [3] A recent development o f considerable i n t e r e s t has been the r e p o r t by the Penn group [4] o f a range o f dopant concentration f o r which the homogeneously doped polymer e x h i b i t s metal 1i c c o n d u c t i v i t y w i t h an e s s e n t i a l l y zero P a u l i s p i n s u s c e p t i b i l i t y . Although elect r o n t r a n s p o r t a r i s i n g from a small b u t f i n i t e Fermi-level d e n s i t y o f l o c a l i z e d one-electron states cannot be r u l e d out, [ 5 ] t h e conclusion t h a t the m e t a l l i c l i k e d.c. c o n d u c t i v i t y r e s u l t s from t h e t r a n s p o r t o f charged-solitons i s s e r i ously suggested f o r the f i r s t time. I n order t o conceive o f an experiment t h a t could d i s t i n g u i s h between s o l i t o n and e l e c t r o n t r a n s p o r t we have t h e o r e t i c a l l y investigated the Drude absorption o f a conducting charged-soliton i n (CH) . Remarkably, we f i n d t h a t i n a d d i t i o n t o the usual Drude absorption expected fir a d i f f u s i n g f r e e - c a r r i e r , the charged s o l i t o n ' s Drude absorption possesses a v i b r a t i o n a l component which leads t o a d i s c r e t e absorption band a t t h e frequency w. o f the s o l i t o n ' s i n t e r n a l breathing mode.[6,7] This band i s a consequence of tAe coupling o f t h e i n t e r n a l and t r a n s l a t i o n a l motions o f t h e s o l i t o n which endows the mobile charged s o l i t o n with an o s c i l l a t i n g e l e c t r i c d i p o l e moment proportional t o i t s mean v e l o c i t y o f t r a n s l a t i o n . The observation o f t h i s " d i s c r e t e Drude absorption'' would c o n s t i t u t e an experimental v e r i f i c a t i o n of charged-sol i t o n t r a n s p o r t i n doped-(CH),. I n order t o c a l c u l a t e t h e charged s o l i t o n ' s Drude absorption we employ t h e r e s u l t s o f a quantum theory which we have developed f o r the charged s o l i t o n . The l a t t e r theory, which w i l l be published i n an independent a r t i c l e , [ 8 ] i s a s t r a i g h t f o r w a r d extension t o quantum mechanics o f a c l a s s i c a l Hami l t o n i a n theory o f the s o l i t o n r e c e n t l y introduced by Rice and Mele.[6] I n t h e quantum theory the s o l i t o n i s described by the wavefunction +(x,a) where ( $ ( ~ , ! 2 ) ( dxda ~ determines the p r o b a b i l i t y t h a t the center o f the s o l i t o n be found between t h e spac i a l p o i n t s x and x + dx w i t h a l e n g t h between 2 and Q + &.. The frequencydependent c o n d u c t i v i t y o f the free-charged s o l i t o n , u(w), i s

where IE,) denote the energy eigenvalues o f the f r e e s o l i t o n , R denotes t h e volume o f t h e system, P(Ea) = exp(-pE,)/Z,exp(-pE,) a b i l i t y o f f i n d i n g the s o l i t o n i n t h e s t a t e with energy E, T = 3/kBp, and v (4) i s t h e m a t r i x element YO! c a m

v

Y,

(9) = i

J dZ! J dx V* (x,!L)v(q,!Z)V,(x,ll) 0

-00

Y

= Ey - ,E, YP i s t h e proba t temperature w

190

M.J. RICE

I n (3) M (Q)denotes the c l a s s i c a l length-dependent t r a n s l a t i o n a l mass o f t h e s o l i t o n f6] and p, = iMVx i s the s o l i t o n ' s t r a n s l a t i o n a l momentum operator. The energy eigenvalues o f t h e f r e e s o l i t o n are determined by t h e Schrodinger equation Wa(x,2)

= EaVa(x,Q)

(4)

where t h e Hamiltonian operator H i s H = (1/4) (Mi(!2)-lp;

+

pa Mi(2) -1)

Here, Mi(a) denotes the c l a s s i c a l length-dependent i n t e r n a l i n e r t i a l mass-of the s o l i t o n , [ 6 ] p = i Va i t s i n t e r n a l momentum operator, and Vi(!2) = A!2 + Be i t s i n t e r n a l po?ential oenergy. A and B are p o s i t i v e constants which determine and e q u i l i b r i u m dength 2! o f the c l a s s i c a l s t a t i c s o l the formation energy E i t o n according t o the f a k i l i a r r e l a t i o n s E = 2dAB and 2 = JA/B. I f we denote by MS and M. the t r a n s l a t i o n a l and interna? i n e r t i a l magses o f t h e l a t t e r s o l i t o n , we haJe according t o the c l a s s i c a l Hamiltonian theory, [ 6 ] M.(11) = M.!2 /Q and M ( a ) = M Q /a. An extension o f (4) and (5) t o provide a q u a h a 1 des&r?pt i o n %f a chdrled s o l i t o n bound t o an i o n i c dopant molecule w i l l be discussed elsewhere. [8] The eigenspectrum defined by Eqs. (4) and (5) and t h e l a t t e r s t a t e d Q-dependences o f the c l a s s i c a l masses, may be solved f o r e x a c t l y t o y i e l d t h e f r e e s o l i t o n eigenvalues

2 4 2 Zk2)1/2 = [l + ( Z / K ) (n + 1/21] (M sc + co

.

where n = 0,1,2,. . , i s an i n t e r n a l v i b r a t i o n quantum number and k = (A/L)s, w i t h s = f l,&Z,..., s p e c i f i e s t h e q u a n t i z a t i o n o f t& & o l i t o n ' s t r a n s l a t i o n a l and ~ / 2denotes t h e momentum along a chain o f l e n g t h L (L +$). M c20 dimensionless parameter ~ / 2= (2M.Q A/ 2 ) . Thd l a t t e r ' $ magnitude s p e c i f i e s the e x t e n t t o which t h e nature of'tt?e i n t e r n a l motion o f t h e s o l i t o n i s quantum mechanical. C l e a r l y , i n the l i m i t K + 01, (6) y i e l d s t h e c l a s s i c a l energy spectrum o f the s o l i t o n . For (CH)x we estimate ~ / 2 3. The corresponding eigenfunctions are

where

191

THEORETICALLY PREDICTED DRUDE ABSORPTION

i n which, f o r j # 0, the c o e f f i c i e n t s a . are determined by t h e recurrence r e l a t i o n aj+l = - a . ( n - j ) / [ ( j + l ) ( j + l + ~ ) ] andJao by the normalization o f y k(x,!2). I n (7a) Ak denotes J the c h a r a c t e r i s t i c l e n g t h Ak = Q , ~ - l [ l + (

~ / M , C ~ ) ~4 ] . With

the use o f t h e r e l a t i o n K = 2Eo / w., where IN. = 2(B/2M.2 )JI i s t h e c l a s s i c a l b r e a t h i n d mode o f t'h8 s t a t i c s o l i t o n , (6) harmonic frequency[6] o f the $rnaf may be expanded f o r 2k2/MS<<E t o read

where M = M K/(K+Zn+l) may be regarded as the e f f e c t i v e s o l i t o n t r a n s l a t i o n a l mass in%he J i b r a t i o n a l sub-band n. We note t h a t i t f o l l o w s from (7) t h a t , f o r small k, the expectation value o f the s o l i t o n length 2 i s j u s t ce> = 2 (K+2n+l)/K f o r the band n. This increase i n w i t h v i b r a t i o n a l e x c i t a t i o n i s rgsponsible f o r the corresponding decrease i n M The remarkable c o n t r a s t between t h e simp l e harmonic nature o f t h e v i b r a t f o n a l eigenspectrum o f (8) and t h e decidedly anharmonic nature o f the associated wavefunctions (7b) w i l l be commented on i n an independent publication. [a] With the use o f (81, (7) and the assumption t h a t u(w) may be evaluated t o y i e l d ~ ( w )= uo ( l - i w / r ) - '

EoS

and wi>>kBT,

eq. (1) for

+ (kgT/ wi)

where w lnnl

=

I

dwn(e)e

*,I

(2)

0

= -(l+~)-"*. which, from eq. (7), gives 2 / 2 I n evaluating (1) we have i n troduced a phenomenological Ed)ns#nt Drude l i f e t i m e T = Iand a n a t u r a l w i d t h r. f o r - t h e i n t e r n a l breathing mode o f t h e s o l i t o n . u i s t h e d.c. c o n d u c t i v i t y u' = R le2r/M and the f i r s t term o f (9) c o r r e s p o n d t o t h e Drude absorption o p d i n a r i l y exbgcted f o r a d i f f u s i n g f r e e - c a r r i e r . The second term i s t h e d i s c r e t e Drude absorption a t wi a r i s i n g from the coupling o f the s o l i t o n ' s i n t e r n a l and t r a n s l a t i o n a l motions. I t s o r i g i n i s c l e a r l y apparent from a c l a s s i c a l i n t e r p r e t a t i o n o f the s o l i t o n ' s v e l o c i t y operator (3). Since, c l a s s i c a l l y , $ i s o s c i l l a t i n g about i t s e q u i l i b r i u m value 2 , the v e l o c i t y v = p /MS(2) o f t h e s o l i t o n i n a s t a t e o f constant t r a n s l a t i 8 n a l momentum p has %n o s c i l l a t o r y component. As the s o l i t o n i s charged i t consequently i s etdowed w i t h an o s c i l l a t i n g e l e c t r i c d i p o l e moment which i s p r o p o r t i o n a l t o i t s mean v e l o c i t y o f translation. The i n t e g r a t e d o s c i l l a t o r strength o f the d i s c r e t e Drude absorpt i o n , S. = S ( k T/ w.) ( l + ~ ) - l , i s t h q e f o r e , understandably, p r o p o r t i o n a l t o the absblute' t e d e r a t l r e T S = ( a e2/2MSo) i s t h e i n t e g r a t e d o s c i l l a t o r strength o f t h e ordinary Drude abkorption.

M.J. RICE

192

I

0

0

I

I

1

2

3

w/w,

Figure 1: The r e a l and imaginary p a r t s o f o(w)/ao vs w/wi f o r the parameter values s t a t e d i n t h e t e x t .

computed from eq. (9)

I n Fig. 1 we have p l o t t e d t h e r e a l and imaginary p a r t s o f a(w)/a as a f u n c t i o n o f w/w. f o r t h e reprgsentative choice o f parameter values r =%., r . = w./5, k T = u] /5 and K = 2E /w. = 6. To date, an accurate microscopit c a l c u l a t i o n o! wi ?or the chargedSso\iton i n (CH) has n o t been undertaken. Following a phenomen-qlogical approach, Rice and A l e [ 6 ] have established t h e estimate 1280 cm , whereas Su e t a l . , [7] have a r r i v e d a t t h e r e s u l t wi 780cm on the basis o f a s i m p l i f i e d microscopic model o f (CH),. An experimental search f o r t h e d i s c r e t e Drude absorption would be made d i f f i c u l t i f the value o f w . l i e s close t o e i t h e r o f t h e two dopant-dependent i n t e n s e l y I R activ_e modes albeady observed [9] f o r inhomogeneously doped-(CH) a t 900 and 1370 cm l , r e s p e c t i v e l y . The l a t t e r modes have been i d e n t i f i e d flO] as spec i f i c i n t r i n s i c s t r u c t u r a l v i b r a t i o n a l e x c i t a t i o n s o f t h e charged s o l i t o n s . It i s hoped t h a t t h i s paper w i l l s t i m u l a t e an experimental search f o r t h i s poss i b l e absorption band.

F i n a l l y , i t i s i n t e r e s t i n g t o note t h a t the r e l a t i v e l y small value o f ~ / 2f o r (CH) i m p l i e s t h a t t h e bond a l t e r n a t i o n amplitude o f n a t u r e ' s simplest polymer has \he character o f a quantum f i e l d . The relevance o f t h e quantum f i e l d theory l i t e r a t u r e [ll] t o t h e present problem i s t h e r e f o r e c a l l e d t o a t t e n t i o n . We thank G i l l Stengle, John Bardeen and Eugene Mele f o r several u s e f u l d i s cussi ons. REFERENCES

[l] Rice, M. J., Phys. L e t t . 7 l A , 152 (1979).

[2] Su, W. P., S c h r i e f f e r , J. R . , and Heeger, A. J., Phys. Rev. L e t t . (1979). (31 See, e.g.,

2, 1698

Proceedings o f I n t e r n a t i o n a l Conference on Low-Dimensional Synt h e t i c Metals (Helsingor, Denmark, August 10-15, 19801, t o be published i n Chemica S c r i p t a (1981).

193

THEORETICALLY PREDICTED DRUDEABSORPTION

[4] Ikekata, S., Kaufer, J., Woener, T.,

Pron, A., Druy, M. A., Sirak, A., Heeger, A. J., and MacDiarmid, A. G., Phys. Rev. L e t t . 45, 1123 (1980): see, also, Peo, M., Roth, S. and Hocker, J., Proc. I n t e r . Conf. on LowDimensional Synthetic Metals (Helsingor, Denmark, Aug. 10-15, 1980) t o be published i n Chemica S c r i p t a (1981); Epstein, A. J., Rommelmann, H., Druy, M. A., Heeger, A. J., .and MacDiarmid, A. G., p r e p r i n t 1980.

[5] Mele, E. J., and Rice, M. J., Phys. Rev. B (submitted). [6]

Rice, M. J., and Mele, E. J., S o l i d State Commun.

[7] Su, W. P., (1980).

S c h r i e f f e r , J. R.,

and Heeger, A.

2, 487

(1980).

J., Phys. Rev. B

22,

2099

[8] Rice, M. J., and Mele, E. J., t o be published. [9] Fincher, C. R., Jr., Ozaki, M., Heeger, A. J., and MacDiarmid, A. G., Phys. Rev. B 3, 4140 (1979). [lo] Mele, E. J., and Rice, M. J., Phys. Rev. L e t t 45, 926 (1980). [ll] Jackiw, Rev. Mod. Phys.

49,

681 (1977).

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Nonlinear Problems: Present and Future 195

A.R. Bishop, D.K. Campbell, B. Nicolaenko (4s.)

0 North-Holland Publishing Company, 1982

POLARONS I N POLYACETYLENE

Alan R. Bishop and David K. Campbell Theoretical D i v i s i o n and Center f o r Nonlinear Studies Los Alamos National Laboratory Los Alamos, NM 87545 USA Recent t h e o r e t i c a l studies o f polyacetylene, (CH) have focussed on k i n k - l i k e s o l i t o n s i n the trans isomer. I t i s stown t h a t t h e same t h e o r e t i c a l models a l s o p r e m o l a r o n - l i k e s o l i t o n s i n both c i s - and W - ( C H ) . The e x p l i c i t f o r m h e s e polarons, t h e i r r e l a t i o n t o kinks, p o & i b l e i m p l i c a t i o n s f o r experiment, and open questions are discussed. INTRODUCTION During the l a s t few years the extensive t h e o r e t i c a l and experimental i n t e r e s t i n nonlinear e x c i t a t i o n s i n polyacetylene ((CH) ) has focused almost e x c l u s i v e l y on k i n k - l i k e s o l i t o n states. [l-101 This i s x h a r d l y s u r p r i s i n g , f o r a p a r t from t h e i r possible d i r e c t experimental i m p l i c a t i o n s [2,11] f o r t r a n s p o r t properties, doping mechanisms, and t h e observed semi-conductor metal t r a n s i t i o n i n (CHI , these k i n k s o l i t o n s , w i t h t h e i r b i z a r r e spin/charge assignments, [3] have s t i m b l a t e d t h e o r e t i c a l work [12,13,14] on t h e existence and r o l e o f " f r a c t i o n a l charge'' i n both s o l i d s t a t e systems and f i e l d theory models. Recently, however, i t has been discovered [15,16,17] t h a t t h e k i n k s o l i t o n s are not the only nonlinear e x c i t a t i o n s p r e d i c t e d by t h e o r e t i c a l models o f (CH)x. Indeed, i n both (numerical studies o f ) l a t t i c e models [15] and ( a n a l y t i c studies o f ) continuum theories, [16,17] a nonlinear "polaron" e x c i t a t i o n has been found. Although more conventional i n i t s p r o p e r t i e s than the k i n k s o l i t o n , the polaron, as we s h a l l see, i s the lowest energy e x c i t a t i o n a v a i l a b l e t o a s i n g l e electron. Experimentally, the polaron may thus p l a y an important r o l e i n t h e doping process o r i n e l e c t r o n i n j e c t i o n and, t h e o r e t i c a l l y , i t s presence f u r t h e r embelOne p a r t i c u l a r l y s i g n i l i s h e s the already r i c h s t r u c t u r e p r e d i c t e d f o r (CH) f i c a n t feature o f the polaron i s t h a t it, u n l i k e t h e k i n k s o l i t o n which can e x i s t only i n the trans isomer o f ( C H I , i s p r e d i c t e d t o occur i n both c i s and trans isomers. The possible i m p l i c a t i h s o f t h i s f o r comparative e x p e r x n t a l s t u d i e s o f cis and trans a r e o f great c u r r e n t i n t e r e s t . Thus i n t h i s a r t i c l e we s h a l l focus on t h i s o r e conventional" b u t nonetheless i n t e r e s t i n g nonlinear e x c i t a t i o n , describing i t s predicted nature i n d e t a i l and discussing p o t e n t i a l experimental imp1 i c a t i o n s b r i e f l y .

.

POLARONS I N m - ( C H ) , From the l a t t i c e model [3] f o r trans-(CH) , a standard sequence o f approximat i o n s [6] leads i n the c o n t i n u u m m i t 61an e f f e c t i v e a d i a b a t i c mean-field Hamiltonian i n terms o f t h e ( r e a l ) gap parameter A and e l e c t r o n f i e l d Y. The result i s

A.R. BISHOP, D.K. CAMPBELL

196

where w2/g2 i s t h e n e t e f f e c t i v e electron-phonon coupling constant, [3,6] ai i s

Q

the i t h Pauli matrix, V(y) =

(!$;),

and vF i s t h e Fermi v e l o c i t y .

[18]

For

l a t e r purposes we note t h a t i n e r i v i n g t h i s mean f i e l d Hamiltonian, t h e l a t t i c e which would add a term p r o p o r t i o n a l t o A2(y) i n (1) has been k i n e t i c energy e x p l i c i t l y ignored. Obviously, t h i s w i l l have no e f f e c t on the s t a t i c e x c i t a t i o n s we discuss below, b u t i t . w i l l prove s i g n i f i c a n t f o r on-going studies i n v o l v i n g the dynamics o f s o l i t o n s i n (CH),.

-

-

V a r i a t i o n o f H leads t o the s i n g l e p a r t i c l e e l e c t r o n wave f u n c t i o n equations

and the s e l f - c o n s i s t e n t gap equation

The summation i n (3) i s over occupied e l e c t r o n s t a t e s and s i s a s p i n l a b e l (suppressed i n (1) and (2)). Equations (2) and (3) are t h e continuum e l e c t r o n phonon equations f o r t r a n s (CH) , [5,6] and remarkably one can f i n d a n a l y t i c , closed form e x p r e s s i o f i r seveval n o n t r i v i a l solutions. [5,6,16,17] Unsurprisingly, t h i s i s r e l a t e d t o the ( p a r t i a l ) s o l i t o n features o f these equations. [16,17] Although we wish t o focus on the polaron s o l u t i o n s t o (2) and (3), f o r c l a r i t y and completeness we s h a l l b r i e f l y review t h e other solutions. F i r s t , the ground s t a t e o f t h e i n f i n i t e chain o f --(CH) e r a t e (see Figure l), and i s described by A(y) = + A.

A,

= W exp (-A -1)

i s t w o - f o l d degeno r Afy) = A -,, w i t h [5,6]

(4. a)

where

The parameters i n (4) include W, t h e f u l l one-electron bandwidth (e 10 eV i s trans-(CHI 1; v , t h e Fermi v e l o c i t y (vF = aW/2, [ l a ] where a i s t h e underi n =-(CH),); and, as i n d i c a t e d p r e v i o u s l y , l a t d c e sFacing (= 1.22

POLARONS IN POL YACETYLENE

197

u2/g2, the n e t e f f e c t i v e electron-phonon coupling constant (whose value i s i n 0.7 eV, eqn. (4.a) eqfect determined by (4)). Since experimentally A. 0.4. implies the dimensionless coupling A

.. H

H

H I

H I

...c A+ /;\ /\ /\ c.. .

Figure 1: (a) two (c) gap A =

I

CI

CI

CI

I

H

H

H

H

H

-

and (b) The two schematic bond s t r u c t u r e s corresponding t o t h e degenerate ground states f o r =-(CH) A p l o t f o r =-(CH) o f the energy ber u n i t l e n g t h versus band parameter A f o r spatialTy constant A. Note the degeneracy between f A and t h e l o c a l maximum ( i n d i c a t i n g i n s t a b i l i t y ) a t A = 0.

-

.

0

the s i n g l e e l e c t r o n energy spectrum i s given by c(k) = Fig. 2) and the corresponding e l e c t r o n wave f u n c t i o n s a r e simply plan8 waves, the e x p l i c i t forms o f which are given i n r e f s . [5,6, and 171. Physically, A # 0 implies a gap (= 2A , see Fig. 2) around the fermi surface i n t h e s i n g f e e l e c t r o n spectrum; t h a t q h i s gap should e x i s t i n t h e continuum model o f ( C H I i s a d i r e c t consequence o f t h e well-known P e i e r l s i n s t a I n chemical b i l i t y [19] o f one-'bimensional coupled electron-phonon systems. terms, A, # 0 indicates a "bond a l t e r n a t i o n " between s i n g l e and double bonds (see Fig. l), and i t s value i s p r o p o r t i o n a l t o t h e lengthening (shortening) o f a s i n g l e (double) bond w i t h respect t o the i d e a l i z e d uniform bond l e n g t h which would occur were A, = 0. For

-4

f

constant A

(s8;

198

A. R. BISHOP, D. K. CAMPBELL

a) -A0 0

-2

2

Figure 2: The s p a t i a l s t r u c t u r e o f t h e gap parameter (A(y)) and t h e associated e l e c t r o n i c l e v e l s f o r --(CH) f o r (a) the ground s t a t e ; (b) the k i n k s o l i t o n ; (c) t h e polaron. foblid l i n e s i n t h e l e f t - h a n d drawings i n d i c a t e A(y); dashed l i n e s show e l e c t r o n d e n s i t i e s f o r l o c a l i z e d states. The right-hand drawings show the e l e c t r o n i c l e v e l s i n the conduction band ( s i n g l e shading), valence band (crossed shading), and gap states. Q = charge and S = spin. The two-fold degeneracy o f t h e ground s t a t e o f trans-(CH) has the very import a n t consequence t h a t a l o c a l i z e d , f i n i t e e n e r f i n k sditon corresponding t o a l o c a l i z e d l a t t i c e "defect" can e x i s t ; t h i s k i n k (K) i n t e r p o l a t e s between the degenerate ground s t a t e s (see Fig. 2) and has t h e e x p l i c i t form

-

AK(y) = Aotanh VFy/Ao

.

-

(5)

There i s a l s o an a n t i - k i n k (k) s o l i t o n with Ak(y) = -A (y). The associated s i n g l e e l e c t r o n energy l e v e l s c o n s i s t o f extended c o n d u c t i h (E(k) = + 4 and valence (E(k) = J v q band s t a t e s [5,6,17] which a r e modffied' ( e s s e n t i a l l y "phase s h i f t e d ) fr8m t h e plane waves o f t h e ground s t a t e and an a d d i t i o n a l l o c a l i z e d "mid-gap" s t a t e ( a t E = 0, see Fig. 2). The e x p l i c i t form o f the wave f u n c t i o n f o r t h i s l o c a l i z e d h a t e i s

-

POLARONS IN POL YACETYLENE

Uo(y) = No sech A0y/vF

and Vo(y) = - i N o sech Aoy/vF

w i t h No =

For e i t h e r K o r i,t h i s "mid-gap" leading t o l o c a l i z e d e x c i t a t i o n s 6,121 i n d i c a t e d i n Fig. 2. It i n t e r e s t i n " f r a c t i o n a l charge."

s t a t e can be occupied by 0, 1, o r 2 electrons, w i t h t h e b i z a r r e spin/charge assignments [3,5, i s these states t h a t have e x c i t e d the recent [12-141

One v i t a l consequence o f t h e k i n k s o l i t o n ' s i n t e r p o l a t i n g between t h e degenerate ground states i s t h a t t h e r e i s a r e s t r i c t i o n u s u a l l y c a l l e d a " t o p o l o g i c a l c o n s t r a i n t " [12,17] on t h e production o f kinks. S p e c i f i c a l l y , i n the i n f i n i t e chain polymer, kinks can be excited from the ground s t a t e o n l y i n p a i r s o f k i n k and a n t i - k i n k . The i n t e r e s t e d reader should convince himself o f t h i s by showing t h a t the production o f a s i n g l e k i n k from t h e ground s t a t e t h a t i s , going from requires overcoming an the A(y) c o n f i g u r a t i o n i n Fig. 2a t o t h a t i n Fig. 2b i n f i n i t e ( f o r an i n f i n i t e l e n g t h polymer) energy b a r r i e r .

-

-

-

-

F i n a l l y , we come t o the polaron s o l u t i o n t o equations (2) and ( 3 ) . The r e c e n t l y discovered [16,17] a n a l y t i c form f o r the gap parameter o f t h e polaron can be w r i t t e n i n t h e revealing form (c.f., eq. (5))

where tanh 2Koyo = KovF/Ao

.

(7. b)

This c o n f i g u r a t i o n f o r A(y), which corresponds t o a s p a t i a l l y l o c a l i z e d deviat i o n from t h e ground s t a t e value (here chosen t o be + A ), i s sketched i n Fig. 2c. Again t h e s i n l e e l e c t r o n states include extended s l a t e s i n t h e conduction (&(k) = + and valence (E(k) = q-4 bands, which s t a t e s are phase-shifted by th% polaron "defect"; t h e expl i c f t ?orms o f t h e corresponding wave functions are given i n r e f . 17. I n addition, two l o c a l i z e d e l e c t r o n i c states, w i t h energies symmetrically placed, form i n t h e gap a t &+ 5 f uo where

-

-

ug

-J

=

The e l e c t r o n wave functions f o r these l o c a l i z e d s t a t e s are, f o r Uo(Y) = No

and

I ( l - i ) s e c h Ko(y+yo)

+

(l+i)sech Ko(y-yo)]

E

= +wo

A. R. BISHOP, 0.K. CAMPBEL L

200

V,(Y)

= No

where No = 4&, 1

{ ( l + i ) s e c h Ko(y+yo) + (1-i)sech Ko(y-yo)]

and f o r &=-wO

u-o

5

= +iVo and V-,

UIE=-,,,

(9b) :VIE=-u

0

= -iUo. 0

-

-

I t i s important t o note t h a t the polaron c o n f i g u r a t i o n f o r A(y) eq. (6) and the associated e l e c t r o n wave functions s a t i s f y the e l e c t r o n p a r t eq. (2) of the continuum equations f o r 9 KovF i n t h e allowed range 0 5 KovF 5 A., It i s

-

-

-

-

the s e l f - c o n s i s t e n t gap c o n d i t i o n eq. ( 3 ) which determines t h e s p e c i f i c value o f K v f o r an actual s o l u t i o n t o t h e coupled equations. Using some aspects o f %.o!iton theory, [17] one can, i n e f f e c t , convert t h e gap equation t o a s t r a i g h t f o r w a r d minimization problem. Apart from s i m p l i f y i n g t h e problem t e c h n i c a l l y , t h i s i s very appealing i n t u i t i v e l y , f o r the q u a n t i t y being m i n i mized i s e s s e n t i a l l y t h e ener o f t h e f u l l i n t e r a c t i n g electron-phonon system. I n p a r t i c u l a r , one f i n d s d e polaron-type c o n f i g u r a t i o n i n =-(CH), that t Ep(n++' n-, KO) = (n+

-

n- + 2)w0 +

4

KovF

-

-n4 wo t a n

-

where n are r e s p e c t i v e l y t h e occupation numbers which can be o n l y 0, 1, o r 2 o f $he l o c a l i z e d s t a t e s E+ = +UJ and e- = -w kntroducing 8 such t h a t KovF = Aosin8 and (see (8)) wo = A0co& and minimizpng E w i t h respect t o 8 g i v e P

-

.

From eq. (11) i t f o l l o w s d i r e c t l y t h a t f o r a stable, l o c a l i z e d polaron s o l u t i o n the electrons must be d i s t r i b u t e d i n one o f two c o n f i g u r a t i o n s : (1) n+ = 1, n- = 2, the " e l e c t r o n polaron" s t a t e ; or (2)

n+ = 0, n- = 1, the "hole polaron" s t a t e .

Before describing these s t a t e s i n d e t a i l , we note t h a t studying o t h e r p o s s i b l e configurations reveals some i n t e r e s t i n g phenomena. When n = n- --- f o r n+ = 0, 1, o r 2 (11) i m p l i e s 6 = n/2, so Kov = A , w = 0 fcorresponding t o t h e "mid-gap" state), and by (7.b) y + a. henceothe& p u t a t i v e trans-(CH), "bipolarons" separate i n t o an i n f i R i t e l y separated k i n k / a n t i - k w p a i r s , w i t h charges determined by the value o f nt = n-. When n+ = 0, n- = 2, 6 = 0, so w = A , K v = 0, and the "unoccupied polaron collapses t o the ground s t a t e , A'. O!herockoices o f n+ and n- lead simply t o combinations o f the above e x c i t a t?ons ( i n f i n i t e l y separated i n space); f o r example, n+ = 1, n- = 0 leads t o a c o n f i g u r a t i o n o f kink, a n t i - k i n k , and polaron.

---

Returning t o t h e t r u e , s t a b l e polaron states, we see t h a t f o r n+ = 1, n- = 2 ( e l e c t r o n polaron) o r n+ = 0, n- = 1 (hole polaron) eq. (11) y i e l d s 6 = n/4 o r

POLARONS IN POL YACETYLENE

KovF =

201

A0/G= wo

(12)

From eq. (10) the energy o f t h i s e x c i t a t i o n i s seen t o be

E =@A P n o

0.90Ao

(13)

From e ns. (61, (7.b), and (12) one sees t h a t the s p a t i a l e x t e n t o f t h e polaron ( = 11 f o r trans-(CH) ) i s s l i g h t l y l a r g e r than t h a t o f the k i n k (= 9 A f o r W - ( C H ) );this compirison i s made v i s u a l i n Fig. 2. Although general arguments sug&sting the existence o f polarons i n systems l i k e trans-(CH) that i s , (quasi)-one-dimensional P e i e r l s dimerized chains can b e v a n c e # , [20] , i t i s encouraging t o f i n d t h a t the continuum model, w i t h o u t any ad hoc addit i o n s , e x p l i c i t l y contains these e x c i t a t i o n s . I t i s a l s o heartening t h a t numeri c a l simulations o f the l a t t i c e model [15] have revealed a polaron e x c i t a t i o n whose s t r u c t u r e i s i n good agreement w i t h t h a t p r e d i c t e d by eq. (6), (7), and (12).

1

-

-

Another aspect o f the polaron f i e l d c o n f i g u r a t i o n (eq. (7)) and corresponding energy expression (eq. (10)) i s worth mentioning: namely, since t h e e l e c t r o n equations eqns. (2) are s a t i s f i e d f o r a l l K i n t h e range 0 < K 5 A /v , eq. (10) can be viewed as a s i n g l e parameter (K expression f o r tfie Rink/&!Thus, f o r example, k i n k i n t e r a c t i o n energy as a f u n c t i o n o f separ%tion, 5,. w i t h n+ = n- one f i n d s , f o r yo >> E l

-

-

9

with I K i l l where K changes w i t h y according t o eq. (7.5). T h i s expression represents a s l i g h t fhprovement on ppevious continuum estimates o f k i n k / a n t i k i n k forces and i s i n good agreement w i t h d i s c r e t e simulations. [21] Thus f a r we have t r e a t e d the polaron as a s t a t i c c o n f i g u r a t i o n , a s o l u t i o n t o the a d i a b a t i c mean f i e l d Hamiltonian, H i n eq. (1). For many experimental applications, i t i s v i t a l t o understand t h e dynamics o f k i n k / a n t i - k i n k and kinkA f i r s t step i n t h i s d i r e c t i o n i s t o r e i n s t a t e t h e polaron i n t e r a c t i o n s . l a t t i c e k i n e t i c energy term, AHw e x p l i c i t l y dropped i n d e r i v i n g H. I n t h e i t i o n a l term o f t h e form continuum l i m i t t h i s leads t o an

where M i s the mass o f a s i n g l e CH u n i t (" 13 m , w i t h m t h e p r o t o n ' s mass), a is i s the lattice,, spacing, and CL i s a constant &hose v a h e , i n trans-(CH) roughly 4.1 eV/A. [3] Unfortunately, t h e (time-dependent) e q u a t i o n s a t fd low from H z H + AH have not been solved f o r n o n - t r i v i a l l y time-dependent A, and thus dhe can asL&t say nothing d e f i n i t i v e about k i n k and polaron dynamics i n the continuum e l e c t r o n phonon models. However, some i n t r i g u i n g p a r t i a l r e s u l t s i n t h i s area4can be obtained i n two ways. F i r s t , one can r e f e r t o the phenomenological @ models [8,9] o f trans-(CH)x t o discuss dynamics. Here one f i n d s

202

A.R. BISHOP, D.K. CAMPBELL

t h a t c o l l i s i o n s between g4 kinks (and polarons [17,22]) have a f a s c i n a t i n g and complex s t r u c t u r e which suggests p o s s i b l y i n t e r e s t i n g e f f e c t s i n t r a n s p o r t and recombination processes i n ( C H I Much o f t h i s s t r u c t u r e i n o4 appears due t o a k i n k shape o s c i l l a t i o n mode. [yi3] Since arguments f o r a s i m i l a r mode i n t h e theory have r e c e n t l y been advanced, [24] a s i m i l a r complex continuum (CH) dynamics might'be expected. Second, f o r small v e l o c i t y kinks and polarons, one can crudely estimate kinematic e f f e c t s by simply making a G a l i l e a n boost o f t h e s t a t i c s o l u t i o n and r e t a i n i n g o n l y lowest order, nonvanishing terms i n the v e l o c i t y ; i n e f f e c t , t h i s gives an approximation t o the s o l i t o n ' s i n e r t i a l mass and thus gives some idea o f i t s r o l e i n t r a n s p o r t phenomena. For k i n k s o l i t o n s , For t h e polaron, r e p l a c i n g y by the r e s u l t s are given i n r e f s . 3 and 6. = y-v t i n (6) and e v a l u a t i n g the expression i n (14) gives P

y

m

J

4 { sech4 KO(;

-m

a

4 + yo) + sech KO(;

2 2 -2 sech KO(? + yo) sech Ko(y

-

-

yo)

yo))

.

Evaluating the i n t e g r a l s (using (7.b) c r u c i a l l y ) and d e f i n i n g M as the t o t a l P c o e f f i c i e n t o f v2 i n (15) leads t o

P

2

(Kl)~ ( OK Fv )

M =- ( K v ) 16a M2 o F

where

f(KoVF)

=3 8

-

2

tn

moAo3 (KoVF 1

' t VF}

($+KovF)

(16.b)

'O-~O'F

.9 m , which shows t h a t t h e polaron has an Using eq. (12), we f i n d t h a t m even smaller e f f e c t i v e mass t h a n k h e k i n t , f o r which mK 5me. [3,6] POLARONS I N =-(CH),

From the perspective o f the nonlinear e x c i t a t i o n s we are discussing, t h e most important d i f f e r e n c e between t h e and Cis isomers o f (CH) i s that for cis-(CHIx, the two chemical s t r u c t u r e s corresponding t o t h e p h a t i v e ground s t a t e do not have the same energy. Thus t h e r e i s a unique, nondegenerate ground s t a t e for-=-(CH) This s i t u a t i o n i s shown i n Figure 3. An immediate consequence i s t h a t si#ce t h e r e are n o t two degenerate ground s t a t e s f o r k i n k s o l i tons t o connect, there are no k i n k s o l i t o n s o l u t i o n s i n G - ( C H ) ! A p h y s i c a l l y i n t u i t i v e understanding o f t h i s s i t u a t i o n f o l l o w s by consideving what would happen i f one t r i e d t o create a k i n k / a n t i - k i n k p a i r from t h e ground s t a t e . Cons i d e r t h e case r e l e v a n t t o &-(CH)x where 6E/2 i s small (see Fig. 3) so t h e

trans

.

-

-

POLARONS IN POLYACETYLENE

203

Figure 3: (a) and (b) - The two schematic band s t r u c t u r e s corresponding t o t h e actual cis-(CH) ground s t a t e ((a), the c i s - t r a n s o i d c o n f i g u r a t i o n ) and t h e m e t a s A b l e s t a t e ((b), the &-'configuration.) (c) A p l o t f o r *-(CH) o f t h e energy per u n i t l e n g t h versus band gap parameter_ A f o r s p a d a l l y constant A. Note t h e unique ground s t a t e a t A = A. and the metastable s t a t e o f A = Ams.

-

metastable and ground s t a t e have almost the same energy per u n i t length, One could then imagine c o n s t r u c t i n g a c o n f i g u r a t i o n t h a t i n t e r p o l a t e d between A and A f o r a s p a t i a l distance d, andothen (an a n t i - k i n k , say), remalned i n A i l t e r p o l a t e d between A and A (a kin@. This c o n f i g u r a t i o n would thus l o o k l i k e eq. (6), w i t h d =mfy Fo% f i n i t e d, the c o n f i g u r a t i o n would have f i n i t e f o r d + 0)) i t s energy would become i n f i n i t e l i k e energy, b u t since bE/E (6E/E).d. I n t h i s sense, one can say t h a t the p u t a t i v e l i n k s i n c&-(CH) are "confined." O f course, r e f e r r i n g t o eg. (6) we see t h a t permanently confingd KK only polaronp a i r s have t h e same s t r u c t u r e as polarons: Thus, i n &-(CH),, nonlinear e x c i t a t i o n s w i l l e x i s t .

. >8,

204

A. R. BISHOP, D.K. CAMPBELL

To make t h i s precise, i t i s very useful t o study an e x p l i c i t model o f e - ( C H ) (due t o Brazovskii and Kirova [16]) which i s both a n a l y t i c a l l y solvable an8 r e l a t e d , i n a c e r t a i n l i m i t , t o the model we have examined-for trans-(CH) . One assumes [16] t h a t the gap parameter can be w r i t t e n as A(y) + Ax, w h e r e m i s s e n s i t i v e t o e l e c t r o n feedback (as f o r t r a n s ) b u t A 'is a co8s t a n t , J x t r i n s i c component.[25] This ansatz can be motivated i n t%rms o f the e f f e c t s a r i s i n g from molecular o r b i t a l s o t h e r than t h e one included e x p l i c i t l y i n eq. (11, (21, and (3). [16]

=m)

The mean f i e l d Hamiltonian f o r @-(CHIx

then becomes [16]

Although i n t h e i n t e r e s t o f s i m p l i c i t y we. have n o t i n d i c a t e d t h i s e x p l i c i t l y , i t i s very important t o note t h a t a l l the physical parameters i n cis-(CH) l a t t i c e spacing, e f f e c t i v e e l e c t r o n phonon coupling, band width, fer?iirvelocfty, band gap - can have d i f f e r e n t values from those_ i n trans. Comparing (1) and (17) we see t h a t w i t h the replacement A(y) + A(y) m y ) + A t h e e l e c t r o n equations f o r t r a n s are the same as f o r c i s . Thus, m u t a t j s mutandis, t h e s t r u c t u r e o f the e m o n spectrum and the e i g e n f u n c t i o n s b e m s a m e . The gap equation, however, i s modified t o read

-

which leads t o d i f f e r e n t c o n s t r a i n t s on s o l u i i o n s afid on bound s t a t e eigenvalues. S p e c i f i c a l l y , the ground s t a t e has the form A(y) = A. = Ai + Ae where

A:

5

-1

(19. a)

Wc exp(-Ac )

and

(19. b)

Here f o r c l a r i t y we have i n d i c a t e d e x p l i c i t l y t h a t f o r c & - ( C H I t h e dimensionsee eq. (4.b) and f u l l band width, W , a#e n o t necessarl e s s coupling, A, i l y the same as f o r Z - ( C H ) It can be shown d i r e c t b t h a t t h e r e a r e no k i n k s o l u t i o n s t o the gap equatfon (la), i n c o n f i r m a t i o n o f our e a r l i e r i n t u i t i v e arguments. For polaron c o n f i g u r a t i o n s , however, we expect s o l u t i o n s , and indeed we f i n d t h a t e x a c t l y t h e same f u n c t i o n a l form given by equations (7) can s a t i s f y t h e continuum equations f o r c&-(CH) , provided t h a t K ( o r u ) i s chosen appropriately. To determine K we f o l l c h our previous pr8cedur8 and again convert t h e gap equation t o a miRimization problem for the energy o f the f u l l i n t e r a c t i n g e l e c t r o n phonon system. For fi-(CH), t h e energy expression becomes

-

.

-

-

-

POLARONS IN POL YACETYLENE

~oy[tanh-l(Ko~F~o)-KovFfio]

+

.

where y = A /A Again i n t r o d u c i n g we can w r i t 8 tfie'equation minimizing E

o

=

205

sine

(n+

-

n- + 2)

-e

N

P

-

such t h a t K v = A s i n 0 and wo = iocosO, w i t h respect €0 8 9s

3.

y tane

, o 5 0 5 n/2

.

I n the l i m i t o f zero e x t r i n s i c gap, A = 0, y = 0, and as expected eq. (21) reduces t o the trans-(CH) r e s u l t o f 8s. (11). For y > 0, eq. (21) unlike equation (11) - h a s s o l u t f o n s f o r a l l combinations o f n+ and n-. These are summarized i n Table I f o r the s p e c i f i c c a s e o f y = 1.

-

Table I

-

-

n-

1

4

!

0

0

2

+2

0.71

1.43

BIPOLARON (h,h)

0

1

1

+1

0.38

0.98

POLARON (h)

0

2

0

0

0

0

GROUND STATE

1

0

3

+1

0.95

1.96

TRIPOLARON (h, h,e)

1

1

2

0

0.71

1.43

BIPOLARON (e-h)

1

2

1

-1

0.38

0.98

POLARON (e)

2

0

4

0

1.11

2.36

QUADRIPOLARON

2

1

3

-1

0.95

1.96

TRIPOLARON (e,e,h)

2

2

2

-2

0.71

1.43

BIPOLARON (e,e)

n+

g

o

INTERPRETATION

(e,e,h,h)

-

The c&-(CH) polaron states f o r various values o f N = n+ n- + 2 f o r t h e case o f y = 1. f = 2 nn+ i s the e l e c t r i c charge o f t h e state, 0 i s the angle defined i n t h e t e x t (given here i n radians), and E fi gives the f u l l energy o f the e x c i t a t i o n . The i n t e r p r e t a t i o n column gives t h g g h e r a l nature o f t h e e x c i t a t i o n (polaron, bipolaron, ...) and t h e type ( e l e c t r o n 5 e, hole 5 h).

-

-

trans-

O f p a r t i c u l a r i n t e r e s t are the "bipolaron" s o l u t i o n s i n which n+ = n-. I n ( C H I the analogous s o l u t i o n s were i n f i n i t e l y separated KK p a i r s . Indeed, one can %ee from eq. (20) t h a t y e s s e n t i a l l y plays the r o l e o f a confinement parame t e r , leading t o a term i n the energy which increases l i n e a r l y ( r e c a l l eq. (7.b)) w i t h the k i n k / a n t i - k i n k "separation," 2y0. This i s the e x p l i c i t r e a l i z a t i o n o f our e a r l i e r h e u r i s t i c argument about k i n k confinement.

206

A.R. BISHOP, D.K. CAMPBELL

From Table I we see t h a t both the polaron w i d t h (2yo) and depth increase w i t h increasing occupation number, N = n+ + 2 n-.

-

As i n t h e case o f trans, the problem o f polaron dynamics i n &-(CHI i s largely open. One can, o f u r s e , estimate t h e i n e r t i a l mass as before, ht the more i n t e r e s t i n g possible e f f e c t s o f c o l l i s i o n dynamics - recombination, long-1 i v e d oscillatory states have thus f a r eluded q u a n t i t a t i v e d e s c r i p t i o n .

-

IMPLICATIONS AND DISCUSSION Since polarons have o n l y r e c e n t l y been discovered [13,16,17] i n the t h e o r e t i c a l models f o r (CH),, the p o s s i b l e i m p l i c a t i o n s i n p a r t i c u l a r , f o r experiment - o f t h e i r existence are n o t y e t completely understood. Obviously the existence o f polarons and kinks i n trans-(CH) , contrasted w i t h t h e existence o f ( m u l t i - ) provrdes a n a t u r a l d i s t i n c t i o n between these two polarons only i n isomers. To understand t i e q u a n t i t a t i v e i m p l i c a t i o n s o f t h i s d i s t i n c t i o n f o r experiment, however, f u r t h e r t h e o r e t i c a l studies s t r e s s i n g t h e dynamics o f k i n k and polaron i n t e r a c t i o n s are needed. Two aspects o f dynamics seem p a r t i c u l a r l y crucial. F i r s t , w i t h i n the time-de endent mean f i e l d approximation (see eqns. (1) and (14)), the nature o h n d kink-polaron i n t e r a c t i o n s must be understood. Second, one must go beyond the mean f i e l d approximation t o study the f u l l quantum dynamics o f (CH) . Although t h e r e i s , as y e t , l i t t l e quantit a t i v e progress i n e i t h e r o f the& areas, t h e r e c e n t l y noted connection between model r e l a t i v i s t i c f i e l d t h e o r i e s and the continuum theory o f (CH) o f f e r some q u a l i t a t i v e suggestions. [17] F i r s t , t h i s connection suggests that! l o n g - l i v e d o s c i l l a t o r y s t a t e s may e x i s t i n (CH) and t h a t these may be important i n (CH) dynamics. Second, known r e s u l t s on 4uantum dynamics i n f i e l d theory r a i s e t h g p o s s i b i l i t y t h a t quantum f l u c t u a t i o n s can d e s t a b i l i z e t h e polaron i n (CHI , causing i t t o disappear from the theory. Although p r e l i m i n a r y quantum Monte Carlo studies [26] do show l a r g e quantum f l u c t u a t i o n s i n (CHI , a n a l y t i c e s t i mates [24] o f the f i r s t order quantum c o r r e c t i o n s i n d i c a t e t h a t ( t o t h i s order, a t l e a s t ) the polaron i s n o t d e s t a b i l i z e d . C l e a r l y , f u r t h e r q u a n t i t a t i v e work on these problem i s required.

-

&-(m

Despite the absence o f c r i t i c a l q u a n t i t a t i v e d e t a i l , i t i s p o s s i b l e t o i n d i cate a number o f i n t e r e s t i n g , p o t e n t i a l experimental i m p l i c a t i o n s o f polarons. F i r s t , on energy grounds [16,17] one knows t h a t s i n g l e e l e c t r o n s added by doping o r by i n j e c t i o n - t o c i s - o r trans-(CH) should form polarons. When many electrons are added, i t i e n e r g e t m l y fa6orable f o r them t o form k i n k / anti-kink pairs ( i n o r multi-polarons ( i n Thus f o r --(CH) , the very l i g h t l y doped m a t e r i a l an average o f 5 one e l e c t r o n ( o r hole) pgr (CH) chain might r e f l e c t polaron t r a n s p o r t p r o p e r t i e s , whereas the more h e a v f l y q e d material would e x h i b i t (charged) k i n k t r a n s p o r t behavior. In Unfortunately t h i s simple cis-(CH),, one would n o t expect t h i s d i f f e r e n c e . p i c t u r e w i l l be complicated by, among o t h e r features, t h e presence o f n e u t r a l samples and the u n c e r t a i n t i e s over kinks "quenched" i n t o undoped --(CHI isomerization from c i s t o t r a n s upon dopifig. Nonetheless, i f polarons are pre, then t h e i r o p t i c a l absorption [27] sent i n l i g h t l y dop=e-and=-(CH) i s s u f f i c i e n t l y d i f f e r e n t from t h a t o f d n k s [28] t h a t experiments comparing "mid-gap" and i n f r a r e d absorption should be able t o d e t e c t t h e d i f f e r e n c e . [27]

-

trans)

-

-

cis).

7

The d i s t i n c t i o n between allowed nonlinear e x c i t a t i o n s i n c i s and trans-(CH) suggests [16] a p l a u s i b l e scenario f o r t h e experimental l y o b s e r v e d m conX t r a s t s between these isomers upon p h o t o i n j e c t i o n ( e x c i t a t i o n o f an electron-hole p a i r by intense r a d i a t i o n ) . I n trans-(CH) , p h o t o i n j e c t i o n leads t o a photoc u r r e n t which has been i n t e r p r e t e n 6 , 2 9 l X a s r e s u l t i n g from a charged kink/ anti-kink pair. I n cis-(CH) , one observes instead a photoluminescence, suggesting the r e c o m b i n m o n o? t h e e-h p a i r which c o u l d f o l l o w because o f t h e e f f e c t i v e confinement o f the p u t a t i v e kinks ( t h e l o c a l i z a t i o n o f m u l t i - p o l a r o n

POLARONS IN POL YACETYLENNE

207

states i n c i s aids recombination). P r e c i s e l y the p r e v i o u s l y discussed theoreti c a l dynamical studies are what i s r e q u i r e d t o make t h i s q u a l i t a t i v e p i c t u r e more precise. I n conclusion, we r e i t e r a t e t h a t recent t h e o r e t i c a l developments [13,16,17] have shown t h a t t h e nature o f nonlinear e x c i t a t i o n s i n polyacetylene may be even more i n t r i g u i n g and complex than previously believed. The c e n t r a l remaining challenge i s t o e s t a b l i s h - o r disprove d e f i n i t i v e l y t h e relevance o f t h i s elegant t h e o r e t i c a l s t r u c t u r e t o t h e r e a l material.

-

ACKNOWLEDGEMENTS We are g r a t e f u l t o many colleagues f o r t h e i r advice and encouragement. Special thanks are due t o J. L. Bredas, J. R. S c h r i e f f e r , and W. P. Su f o r enlightening discussions o f t h e i r numerical simulations and t o S. Etemad f o r h i s shared i n s i g h t s i n t o the experimental s i t u a t i o n . REFERENCES See f o r example, D. Bloor, Proc. Amer. Chem. S O C . , Houston, A p r i l 1980; t o appear i n J. Chem. Phys. For recent reviews see a r t i c l e s i n Proc. I n t . Conf. on Low-Dimensional Synt h e t i c Metals (Helsingor, Denmark, August 1980) i n Chemica S c r i p t a 2 (1981); Physics i n One Dimension, eds. J. Bernasconi and T. Schneider (Springer Verlag, 1981); and A. J. Heeger and A. G. MacDiarmid, p. 353-391 i n The Physics and Chemistry o f Low Dimensional Solids, ed. L. Alchcer (Reidel , 1980). Su, W. P., S c h r i e f f e r , J. R., and Heeger, A. J., Phys. Rev. L e t t . (1979); Phys. Rev. B 2 , 2099 (1980).

42,

1698

42, 408 and 416 (1977). Brazovskii, S. A., JETP L e t t e r s 28, 606 (1978) (trans. of Pisma Zh. Eksp. Teor. F i z . 28, 656 (1978)); and Soviet Phys. JETP 51, 342 (1980) (trans. o f Zh. Eksp. Teor. Fiz. 'a, 677 (1980)). Kotani, A.,

J. Phys. SOC. Japan

Takayama, H., Lin-Liu, Y. R. and Maki, K., Phys. Rev. B 3,2388 (1980); Krumhansl, J. A., Horovitz, B. , and Heeger, A. J., S o l i d State Commun. 2 , 945 (1980); Horovitz, B., S o l i d State Commun. 2 , 61 (1980). Horovitz, B.

,

Phys. Rev. L e t t .

Rice, M. J., Phys. L e t t . 7lJ,

5 , 742

(1981).

152 (1979).

Rice, M. J., and Timonen, J., Phys. L e t t .

El 368

(1979).

[lo] Mele, E. J., and Rice, M. J., i n Chemica S c r i p t a IJ, 21 (1981). [ll] See a r t i c l e s by Etemad S., and Rice, M. J., these proceedings. [12] Jackiw, R. and Rebbi, C.,

Phys. Rev. D 13, 3398 (1976); Jackiw, R., S c h r i e f f e r , J . R., Nuc. Phys. B 190, 253 (n81).

[13] Su, W. P., and S c h r i e f f e r , J. R., Phys. Rev. L e t t .

5 , 738

and

(1981).

[14] Rice, M. J., and Mele, E. J., " P o s s i b i l i t y o f S o l i t o n s w i t h charge * e / 2 i n Highly Correlated 1:2 Salts o f TCNQ," Xerox Webster p r e p r i n t , 1981.

A.R. BISHOP, D.K. CAMPBELL

208

[15] Su, W. P., and S c h r i e f f e r , J. R., Proc. Nat. Acad. Sci. 77, 5526 (Physics) (1980); Bredas, J. L. , Chance, R. R., and Silbey, R., Proceedings o f I n t e r national Conference on Low-Dimensional Conductors, Molecular C r y s t a l s and L i q u i d Crystals (Gordon Breach), t o be published.

[16] Brazovskii, S. , and Kirova, N.,

Pisma Zh. Eksp. Teor. Fiz.

[17] Campbell, D. K. and Bishop, A. R . , Phys. Rev. B

24,

2,

6 (1981).

4859 (1981); Nuc. Phys.

B ( i n press).

[18] Following the standard conventions i n t h e continuum model, we use u n i t s w i t h h = 1. [19] P e i e r l s , R. E., Quantum Theory o f S o l i d s (Clarendon Press, Oxford, 1955) p. 108; Allender, D. Bray, J. W., and Bardeen, J . , Phys. Rev. B. 9, 119 (1974). [20] Bishop, A. R.,

S o l i d State Commun.

Y. R., unpublished.

[21] Lin-Liu,

and Maki, K.,

2 , 955

(1980).

Phys. Rev.

822

5754 (1980); Kivelson, S.

[22] The phenomenological

$4 theories, i f coupled t o a phenomenological e l e c t r o n f i e l d describing o n l y the l o c a l i z e d , "gap" s t a t e s , do c o n t a i n b o t h the k i n k s o l i t o n s ( o f a l l charges) and t h e polaron.

[23] Campbell, 0. K., and Wingate, C., [24] Nakahara, N.

, and

Maki, K.,

i n preparation.

p r e p r i n t (1981).

[25] I n another context t h i s ansatz has been discussed by Rice; see Rice, M. S., Phys. Rev. L e t t . 37, 36 (1976). [26] Su, W. P.,

unpublished.

[27] Fesser, K., Bishop, A. R . , and Campbell, 0. K.

,

i n preparation.

1281 Gammel, 3. T. and Krumhansl, J. A. Phys. Rev. B 24, 1035 (1981). Maki, K. and Nakahara, N., Phys. Rev. B 23, 5005 (1981E Horovitz, B., p r e p r i n t (1981); Kirelson, S. , Lee, T.-Y. , T i n - L i u , Y. R., Peschel, I. , and Yu, L . , p r e p r i n t (1981). [29] Etemad, S. , these Proceedings.

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North-Holland Publishing Company, 1082

209

LIGHT SCATTERING AND ABSORPTION IN POLYACETYLENE Shahab Etemad and Alan J. Heeger Department of Physics University of Pennsylvania Philadelphia, PA 19104, U.S.A.

Recent experimental results on the lattice dynamics, band structure and photoexcitations of polyacetylene are reviewed, and interpreted in terms of the soliton model. The data indicate that injection of an electron-electron (hole-hole) pair by chemical doping or photoinjection of an electronhole pair invariably distort the lattice leading to formation of a soliton-antisoliton pair. It is concluded that unlike traditional semiconductors, the polyacetylene lattice is inherently unstable in the presence of electron and/or hole excitations. INTRODUCTION It consists of weakly Polyacetylene is the simplest conjugated polymer (1). coupled chains of CH units forming a pseud -one-dimensional lattice. Three of the four carbon valence electrons are in sp8 hybridized orbitals; two of the 6-type bonds construct the 1-d lattice while the third forms a bond with the hydrogen side group. The 120° bond angle between these three electrons can be and c&-(CH& satisfied by two possible arrangements of the carbons; trans-(CH with two and four CH monomer per unit cell respecively -Fig.$). -

.

-

-

L

C

C

C

/P\ -/a\-/p\ -/p\ -?)\-/p\ -/p\ -/p R The lattice structure of trans-(CH), showing - - - - - - allthepossible arrangeL a/(?\a/m\a/m\/m\ L ments of the distortion E F T ? pattern for a polymer chain. .- ! - - The lattice structure t of a cis-(CH), showing L /( ? / \I /p\ /p\ /p\ R the distortion pattern m m m for the nondegenerate & i - - cis-transoid ground Figure 1

R

( ? m ( ? ( ? m ( ? ( ? ( ?

/(?\ /(?\ /(?\

4

-

e

/(?\

state.

-

-

e

-

/(?\

-

-

I

O I Q )

TRANS-(CH),

(?

0

(?

210

S. ETEMAD, A.J. HEEGER

I n e i t h e r isomer the remaining valence e l e c t r o n has t h e symmetry o f a 24 o r b i t a l w i t h i t s charge densitv lobes perpendicular t o t h e plane defined by the other three. I n terms o f an energy-band description, the c-bonds form low-lying completely f i l l e d bands, w h i l e the T - b o n d corresponds t o a h a l f f i l l e d conduction band provided a l l t h e bond lengths are equivalent. As a r e s u l t of P e i e r l s i n s t a b i l i t y of the 1-d metal (21, t h e bondlengths are not equivalent and a l t e r n a t e i n s i z e r e s u l t i n g i n opening o f an associated P e i e r l s gap a t t h e Fermi level. Simple estimates lead t o a p i c t u r e o f polyacetylene as a broad band pseudo-onedimensional semiconductor. I n order t o describe t h e P e i e r l s d i s t o r t e d s t a t e we consider, as an i n i t i a l approximation, a model i n which theTT-electrons o f trans-(CH), are t r e a t e d i n a t i g h t b i n d i n g approximation and t h e C - e l e c t r o n s areassumed t o move a d i a b a t i c a l l y w i t h the nuclei. L e t u, be a c o n f i g u r a t i o n coordinate f o r displacement o f the nth CH group along the molecular s,ynmetry a x i s (x), where u= ,O f o r t h e undimerized chain. The Hamiltonian i s

where t o f i r s t order i n the u's, t

n+l,n = t-(3(un+1-un) o

[*I

M i s t h e mass o f the CH u n i t , K i s the s p r i n g constant f o r t h e CF-energy when expanded t o second order about the e q u i l i b r i u m undimerized systems and c k (ens) creates ( a n n i h i l a t e s ) a 7 - e l e c t r o n o f s p i n s on t h e nri, CH group. The band s t r u c t u r e o f the p e r f e c t i n f i n i t e dimerized trans s t r u c t u r e i s shown i n

Figure 2 The band s t r u c t u r e o f =-(CH),.

LIGHT SCATTERING AND ABSORPTION IN POL YACETYLENE

Fig.

2. u

21 1

I n t h i s p e r f e c t s t r u c t u r e , t h e displacements a r e o f t h e f o r m =

-r

131

( - l ) n ug

where f corresponds t o t h e two p o s s i b l e degenerate s t r u c t u r e s ( d o u b l e bonds p o i n t i n g " l e f t " and double bonds p o i n t i n g " r i g h t " ) . The t r a n s f e r i n t e g r a l s f o r the p e r f e c t chain are

(

tn+l,n=

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to-tl to+tl

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"double" bond

The o v e r a l l bandwidth i s 4 b - 8-10 eV; whereas t h e energy gap 2 A = 4 t , depends on t h e magnitude o f t h e d i s t o r t i o n . F o r example, i n trans-(CH), f o r 2A = 4 t , - 1.5 eV and B E 6 . 9 e V / l , as i n f e r r e d f r o m t h e a n a m o f t h e o p t i c a l a b s o r p t i o n c o e f f i c i e n t and t h e l a t t i c e dynamic r e s u l t s ( s e e below), uo which m i n i m i z e s t h e ground s t a t e energy can be o b t a i n e d t h e value o f from

a

= 4pu,

t o be u, .-.025 4 compared w i t h an i n t e r s i t e s e p a r a t i o n o f a = 1.28 a l o n g t h e c h a i n d i r e c t i o n . As a r e s u l t o f t h e l a r g e bandwidth and u n s a t u r a t e d r - s y s t e m , (CH), i s analoguous t o t h e t r a d i t i o n a l i n o r g a n i c semiconductors. The t r a n s v e r s e The l a r g e n e a r e s t t r a n s f e r i n t e g r a l t, i s , however, much l e s s t h a n t,. neighbor i n t e r c h a i n s e p a r a t i o n o f c.4A ( 4 ) i m p l i e s N tl . l e v compared t o t,- 2.5 eV; t h u s j u s t i f y i n g t h e pseudo-one-dimensionality o f t h e system. There has been a growing i n t e r e s t i n t h e s t u d y o f t h e p h y s i c a l p r o p e r t i e s o f (CH), due t o t h e f a c t t h a t t h e s t r u c t u r e o f trans-(CHI, e x h i b i t s a broken symmetry and has a t w o - f o l d degenerate g r o u n m t e . ( s e e F i g . 1 ) As a r e s u l t , one expects s o l i t o n - l i k e e x c i t a t i o n s , i n t h e f o r m o f bond a l t e r n a t i o n domain w a l l s , c o n n e c t i n g t h e two degenerate phases. The p r o p e r t i e s o f s o l i t o n s i n I CH), have been e x p l o r e d t h e o r e t i c a l l y i n s e v e r a l r e c e n t papers (5-8), which show t h a t t h e c o u p l i n g o f t h e s e c o n f o r m a t i o n a l e x c i t a t i o n s t o t h e T - e l e c t r o n s l e a d s t o unusual e l e c t r i c a l and magnetic p r o p e r t i e s . The p o s s i b i l i t y o f e x p e r i m e n t a l s t u d i e s o f such s o l i t o n s i n p o l y a c e t y l e n e , t h e r e f o r e , r e p r e s e n t s a unique o p p o r t u n i t y t o e x p l o r e n o n l i n e a r phenomena i n condensed m a t t e r physics. I n an e f f o r t t o c h a r a c t e r i z e t h e s o l i t o n e x c i t a t i o n s o f p o l y a c e t y l e n e , Su, S c h e i e f f e r , and Heeger (SSH) c o n s i d e r e d t h e two domains w i t h L domain t o t h e l e f t and R domain t o t h e r i g h t o f a domain w a l l o r s o l i t o n as shown i n F i g (1CI. They determined t h e p r o p e r t i e s o f t h e s o l i t o n ( c r e a t i o n energy, w i d t h , mass, spin, e t c . ) i n terms o f t h e m i c r o s c o p i c parameters o f e q u a t i o n ( 1 ) b y m i n i m i z i n g t h e g r o u n d - s t a t e energy o f t h e system f o r a displacement p a t t e r n which reduces t o t h e L and R phases as one moves t o t h e f a r l e f t o r r i g h t . Using a staggered o r d e r parameter t r i a l f u n c t i o n W , ,

they with in f i.e., they bond

f i n d t h a t due t o t h e c o m p e t i t i o n between t h e e l a s t i c t e r m ( i n c r e a s e s a decrease i n {)and t h e condensation energy ( i n c r e a s e s w i t h an i n c r e a s e ) t h e s o l i t o n i n trans-(CHI, i s spread o v e r a p p r o x i m a t e l y 15 l a t t i c e s i t e s , 2 f = 15 U s i n g x a d i a b a t i c a p p r o x i m a t i o n f o r t h e e l e c t r o n i c motion, e s t i m a t e t h e t r a n s l a t i o n a l mass o f s o l i t o n ; i.e., t h e i n e r t i a o f t h e a l t e r n a t i o n p a t t e r n o f e q u a t i o n 6 t o m o t i o n a l o n g t h e chain, t o be

.

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4

2

c71

m 9 - M( .%) s 3 a<

The enormous reduction i n t h e s o l i t o n mass from t h e mass o f a CH u n i t i s due t o the small amplitude o f the bond a l t e r n a t i o n . Using t h e experimentally determined values o f Q and A (see below), we estimate M ~ - J V where M~ w\c i s t h e e l e c t r o n mass. Since t h e r e i s complete degeneracy ( t h e s o l i t o n can be anywhere), and since the mass i s small t h e s o l i t o n i s expected t o be h i g h l y mobile, b u t by topological c o n s t r a i n t s confined t o move on a given chain 19,lO). Associated w i t h t h e s o l i t o n i s a l o c a l i z e d e l e c t r o n i c s t a t e s i t u a t e d a t midgap; i.e., half-way between t h e bonding valence band and t h e antibonding conduction band. SSH have p o i n t e d o u t t h a t t h e s o l i t o n s i n trans-(CH), are not the phase s o l i t o n s which correspond t o a 180" phase s h i f t m h e e l e c t r o n i c charge-density wave, r a t h e r they are amplitude s o l i t o n s w i t h reversed chargeelectron spin r e l a t i o n s (5). As a r e s u l t , i f t h e l o c a l i z e d s t a t e contthe s o l i t o n i s neutral, w i t h s p i n 4, and t h e r e f o r e i s paramagnetic. On t h e other hand, i f the l o c a l i z e d s t a t e i s empty (doubly occupied) t h e s o l i t o n i s p o s i t i v e l y (negatively) charged, w i t h spin 0, and non-magnetic. The s o l i t o n c r e a t i o n energy i n trans-(CH),; i.e., energy t o generate a mid-gap state, i s estimated by SSH t o m r~ .43 eV which i s l e s s than h a l f t h e gap, .7eV. The a n a l y t i c a l equation f o r E, has been found t o be

2E s =n 4- A < 2 A

181

using t h e continuum l i m i t approximation t o t h e SSH model (7). This r e s u l t i s o f fundamental importance t o t h e nature of t h e photoexcitations and t h e process o f chemical doping i n --(CH), Equation ( 8 ) c l e a r l y s t a t e s t h a t ( w i t h i n the model described by equation 1 , from energetic considerations trans-(CH)x i s unstable t o the presence o f an electron-hole (e-h) p a i r , o r an e a - h ) p a i r i n o r near the conduction (valence) band, and d i s t o r t s t o form two midgap states o f appropriate charge.

.

In t h i s paper we review the r e s u l t s o f t h e recent l i g h t s c a t t e r i n g and absorption We show t h a t t h e strong coupling o f experiments i n cis- and =-(CHI,. these nonlinear e x c i t a t i o n s t o t h e e l e c t r o n i c s t r u c t u r e o f t h e polymer leads d i r e c t l y t o a number o f unusual r e s u l t s which cannot be explained i n terms o f " l i n e a r i z e d " concepts. The organization o f t h i s paper i s as follows: We f i r s t sumnarize t h e r e s u l t s o f a recent study o f t h e l a t t i c e dynamics o f trans-(CH), , i n c l u d i n g both t h e unperturbed polymer and t h e doped polymer c o n t a i n i n g charged s o l i t o n s a t d i l u t e concentrations. The generation o f s o l i t o n s upon doping rearranges t h e bondlengths g i v i n g r i s e t o antisymmetric displacement p a t t e r n about t h e s o l i t o n center. The l a t t i c e dynamics i s l o c a l l y perturbed, inducing i n f r a r e d a c t i v e modes which correspond t o t h e o s c i l l a t i o n o f charge across s o l i t o n center. The d e t a i l e d q u a n t i t a t i v e agreement between c a l c u l a t e d and experimentally observed i n f r a r e d a c t i v e v i b r a t i o n a l modes (IAVM) provides s p e c i f i c experimental evidence f o r doping through s o l i t o n generation i n polyacetylene. The appearance o f t h e I A V M i s concurrent w i t h generation o f mid-gap s t a t e r e a d i l y detectable i n t h e absorption spectra o f l i g h t l y doped polyacetylene. The experimental r e s u l t s can n a t u r a l l y be understood i n terms o f the s o l i t o n model; t h e presence o f a s o l i t o n breaks the t r a n s l a t i o n symmetry o f t h e l a t t i c e , thereby removing o s c i l l a t o r s t r e n g t h from t h e interband o p t i c a l t r a n s i t i o n and c r e a t i n g a mid-gap absorption band due t o t h e o p t i c a l t r a n s i t i o n between a band s t a t e and t h e s o l i t o n l e v e l . The observation o f doping induced IAVM concurrent w i t h t h e appearance o f mid-gap s t a t e s confirms t h e t h e o r e t i c a l p r e d i c t i o n t h a t trans-(CHX, l a t t i c e i s unstable t o t h e presence o f e-e (h-h) p a i r i n t h e c o n d a n (valence) band.

LIGHTSCATTERING AND ABSORPTION IN POL YACETYLENE

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To avoid p o s s i b l e c o m p l e x i t i e s associated w i t h s o l i t o n generation b y chemical doping we have a l s o considered experimental c o n f i g u r a t i o n s t h a t have p e r m i t t e d us t o study t h e response o f t h e l a t t i c e t o d i r e c t p h o t o - i n j e c t i o n o f e-h p a i r s . The existence o f t h e two isomers o f p o l y a c e t y l e n e has p r o v i d e d t h e conceptual and experimental b a s i s f o r t h e study o f t h e p h o t o e x c i t a t i o n s i n t h e undoped polymer i n terms o f photogeneration o f s o l i t o n s . The r e s u l t s o f two complementary experiments; i.e. ,. photoluminescence and p h o t o c o n d u c t i v i t y i n c i s (CH), and trans-(CHX, are presented. For &-(CHI , we f i n d band edgel u m i n e s c e n f i t h e s c a t t e r e d l i g h t spectrum near %he i n t e r b a n d a b s o r p t i o n edge, b u t we a r e unable t o d e t e c t any photoconductive response. I s o m e r i z a t i o n o f t h e same sample quenches t h e luminescence s i g n a l , and gives r i s e t o t h e appearance o f a l a r g e photoconductive response. The s p e c t r a l dependence o f t h e p h o t o c o n d u c t i v i t y i n trans-(CH) can be i n t e r p r e t e d i n terms o f charged s o l i t o n s , photogenerated either-ctly ? t h r e s h o l d h =4 A h ) o r i n d i r e c t l y t h r o u g h c o u p l i n g o f t h e l a t t i c e t o e l e c t r o n - h o l e p a i r e x c i t a t i o n s ( f i c ~ ) 2 b ) . The o b s e r v a t i o n o f luminescence i n cis-(CH), , b u t n o t i n trans-(CHI ; and t h e o b s e r v a t i o n o f photoconductivit-n trans-(CH , b u t m n c&-(CH), provide confirmation o f t h e proposal t h a t solare he photogenerated c a r r i e r s . I n trans-(CH) t h e degenerate ground s t a t e leads t o f r e e s o l i t o n e x c i t a t i o n s , absence o f ‘bind edge luminescence, and p h o t o c o n d u c t i v i t y . I n c&-(CH), t h e non-degenerate ground s t a t e leads t o confinement o f t h e photogenerated c a r r i e r s , absence o f photoconductivity, and t o t h e band edge luminescence. I A V M OF CHARGED SOLITONS (12) S p e c i f i c evidence f o r t h e f o r m a t i o n o f charged s o l i t o n s upon doping has emerged from i n f r a r e d s t u d i e s o f t h e donor and acceptor s t a t e s i n l i g h t l y doped polyacetylene. I n a s e r i e s o f experiments w i t h v a r i o u s dopants i n b o t h (CH), (15) and (CD), (12) i t was found t h a t upon d i l u t e doping new a b s o r p t i o n modes appeared i n t h e I R region, w i t h remarkable i n t e n s i t y . The doping induced modes are p r i m a r i l y p o l a r i z e d p a r a l l e l t o t h e chain d i r e c t i o n and t h e i r i n t e n s i t i e s grow i n p r o p o r t i o n t o t h e dopant l e v e l , becoming comparable t o any o f t h e i n t r i n s i c I R l i n e s o f t h e undoped polymer a t about 0.1%. Thus t h e dopin induced I R modes have oscil1,ator s t r e n g t h s enhanced by approxi m a t e l y l # ; such a l a r g e enhancement must a r i s e from c o u p l i n g o f t h e new v i b r a t i o n a l modes (induced by doping) t o t h e e l e c t r o n i c o s c i l l a t o r s t r e n g t h o f t h e polyene chain. These r e s u l t s are q u i t e general; t h e same modes can be observed f o r i o d i n e and AsF, (p-type), and f o r Na doping (n-type) ( 1 5 ) . Thus these i n t e n s e absorptions a r e n o t due t o s p e c i f i c v i b r a t i o n s o f t h e dopant molecules, nor t o d e t a i l e d i n t e r a c t i o n s between t h e dopants and t h e polymer chain. The observed g e n e r a l i t y suggests t h a t t h e i n t e n s e I R a b s o r p t i o n modes a r e i n t r i n s i c f e a t u r e s o f t h e doped (CHI* and ICD), chains. Mele and R i c e have been a b l e t o s u c c e s s f u l l y i n t e r p r e t these r e s u l t s i n terms o f a o r trans-(CD (16-17). t h e o r y o f t h e l a t t i c e dynamics o f s o l i t o n s i n --(CH), I n t h e frequency range o f t h e v i b r a t i o n a l modes t h e y have feu-Vera i n t e r n a l modes p e c u l i a r t o t h e s o l i t o n s t r u c t u r e I n p a r t i c u l a r , t h r e e o f these modes were found t o be s t r o n g l y i n f r a r e d active, d e r i v i n g t h e i r o s c i l l a t o r s t r e n g t h from i n t e r a c t i o n s w i t h t h e r - e l e c t r o n s . Mele and R i c e have shown t h a t t h e dominant motions associated w i t h t h e i n f r a r e d a c t i v e v i b r a t i o n a l modes (IAVM) o f a s o l i t o n i n v o l v e an antisymmetric c o n t r a c t i o n o f t h e s i n g l e ( o r double) bonds on one s i d e o f t h e s o l i t o n c e n t e r and an expansion on t h e other, thus d r i v i n g charge back and f o r t h across t h e s o l i t o n center. Due t o t h e s p a t i a l spread o f t h e s o l i t o n over about 15 s i t e s , t h e y f i n d t h a t t h e asymmetric o s c i l l a t i o n o f t h e e l e c t r o n i c charge generates a l a r g e o s c i l l a t i n g d i p o l e moment c o n s i s t e n t w i t h t h e e x p e r i m e n t a l l y observed enhanced o s c i l l a t o r s t r e n g t h o f t h e dopant induced IAVM.

f

S. ETEMAD, A.J. HEEGER

214

The f o r c e f i e l d model constructed by Mele and Rice r e l i e d on a s i n g l e adjustable parameter (3 which included t h e e f f e c t o f nonlocal coupling o f t h e atomic d i s placements through t h e i r i n t e r a c t i o n w i t h t h e extended T - e l e c t r o n i c s t a t e s (16). The value o f was determined by a f i t o f the c a l c u l a t e d zone center frequencies o f gerade modes t o t h e observed Raman l i n e s i n unperturbed trans-(CH), and (CD), (121. Despite considerable change i n t h e Raman spectpolyacetylene upon deuteration, t h e f i t t o t h e Raman l i n e s i n both (CH), and (CD), i s obtained w i t h the same value o f 8=6.9 eV/A. These major changes which occur i n t h e phonon spectrum on going from (CH), t o (CD1, a r i s e since t h e increase i n mass i s suff i c i e n t l y l a r g e compared t o t h e mass o f the carbon atom t h a t n o n - t r i v i a l changes occur (i.e., beyond a simple s c a l i n g by M-”’ ) i n t h e normal modes. A comparison o f the I A V M o f s o l t t o n s i n =-(CH)

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Figure 3 The comparison o f t h e experimentally derived and t h e t h e o r e t i c a l l y calculated additional absorption due t o t h e infrared active v i b r a t i o n a l modes (IAVM) o f s o l i t o n s i n trans-(CH), and L X D ) x

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i n F i g 3. The dashed curves are t h e experimental r e s u l t s f o r a concentration o f approximately 0.1% and t h e s o l i d curves are t h e r e s u l t o f the c a l c u l a t i o n s

215

LIGHT SCATTERING AND ABSORPTION I N POL YACETYLENE

w i t h no a d j u s t a b l e parameter ( t h e v a l u e o f F 6 . 9 eV/A was a l r e a d y f i x e d ) . Bes i d e s t h e l a r g e f r e q u e n c y s h i f t s upon d e u t e r a t i o n , t h e d e t a i l e d s t r u c t u r e o f the s o l i t o n IAVM t h a t i s driven by t h e higher frequency s t r e t c h i n g o s c i l l a t i o n s i s a l s o changed. F i r s t , t h e a b s o r p t i o n i s c o n s i d e r a b l y b r o a d e r and s l i g h t l y asymmetric, w i t h a s h o u l d e r on i t s h i g h energy side. The w i d t h fmeasured as f u l l w i d t h a t h a l f t h e peak a m p l i t u d e ) o f t h i s mode i s changed by a f a c t o r o f f i v e , f r o m about 50 cm-' i n (CH), t o about 250 cm-l Secondly, t h e r e l a t i v e i n t e g r a t e d o s c i l l a t o r s t r e n g t h s (A)o f t h e i n (CD),. two a b s o r p t i o n modes i s changed. The e x p e r i m e n t a l l y determined r a t i o i n (CH),,R(900)/R(1370), i s about 2/3, whereas t h e analogous r a t i o i n ( C D L , R(780)/Y2(1120) i s about 3/2. As shown i n F i g . 3, t h e c a l c u l a t e d i n f r a r e d a b s o r p t i o n a s s o c i a t e d w i t h t h e I A V M o f s o l i t o n s i s i n r e m a r k a b l y good agreement w i t h t h e e x p e r i m e n t a l r e s u l t s . F i r s t , t h e frequencies a r e i n agreement w i t h t h e e x p e r i m e n t a l r e s u l t s t o an accuracy o f a few p e r c e n t f o r a l l t h e I A V M f o r b o t h (CH), and (CD), Secondly, t h e i n c r e a s e d w i d t h and asymmetry o f t h e h i g h e r f r e q u e n c y mode i n (CD), appears t o be accounted f o r b y t h e b l u e s h i f t i n g o f t h e two degenerate modes i n t h e low f r e q u e n c y l i n e o f --(CH), F i n a l l y as i n t h e case f o r (CH), , t h e c a l c u l a t e d a b s o l u t e i n t e g r a t e d o s c i 1 l a t o r s t r e n g t h s correspond t o -. 20% o f t h e t o t a l a s s o c i a t e d w i t h t h e added charge, and t h e y agree w i t h t h e experimental values t o w i t h i n a f a c t o r o f 2-3. The r e l a t i v e o s c i l l a t o r s t r e n g t h s o f t h e d i f f e r e n t modes a r e even more accurate. I n (CHI, , t h e frequency mode i s more i n t e n s e b y about 3:2, whereas i n (CD), we f i n d t h a t t h e h i h e r frequency a b s o r p t i o n i s more i n t e n s e b y -3:2. I n b o t h cases t h e exper mental and t h e o r e t i c a l i n t e n s i t y r a t i o s a r e i n c l o s e agreement.

.

.

lower

+

The u n u s u a l l y p r e c i s e account o f t h e l a t t i c e dynamics o f t h e u n p e r t u r b e d polymer and c h a r a c t e r i s t i c v i b r a t i o n s o f t h e l o c a l d e f o r m a t i o n a s s o c i a t e d w i t h t h e presence o f s o l i t o n s i n b o t h (CHI, and ICD), i s achieved b y f i x i n g t h e s i n g l e a d j u s t a b l e parameter B = 6.9 eV/A. As i n d i c a t e d above, t h i s v a l u e o f (3 i m p l i e s a h o r i z o n t a l displacement o f @0.025A f o r t h e CH u n i t , o r -0.044A f o r t h e bond l e n g t h change, w i t h an energy gap o f 2 A = 1.5eV. U s i n g eq. 7 t h e i n f e r r e d v a l u e o f t h e s o l i t o n mass i s m, -me; i.e., quantum e f f e c t s a s s o c i a t e d w i t h t h e s o l i t o n s t a t e s may dominate i t s p h y s i c a l p r o p e r t i e s . SOLITON MID-GAP STATE ( 1 1 ) I n t h i s s e c t i o n we examine, f r o m a band s t r u c t u r e p o i n t o f view, t h e response o f --(CH), l a t t i c e t o t h e presence o f an e l e c t r o n - e l e c t r o n ( h o l e - h o l e ) p a i r i n o r near t h e conduction(va1ence) band. I n c o n t r a s t t o t h e case o f c o n v e n t i o n a l semiconductors, we f i n d t h a t donor ( a c c e p t o r ) doping does n o t r e s u l t i n g e n e r a t i o n of l o c a l i z e d s t a t e near t h e c o n d u c t i o n ( v a l e n c e ) band. Instead, independent o f t h e n a t u r e o f t h e dopant, a d d i t i o n o r removal o f e l e c t r o n s t o o r f r o m t h e polymer c h a i n i s i n v a r i a b l y accommodated b y f o r m a t i o n s o l i t o n s t a t e s near mid-gap. C a l c u l a t i o n s o f t h e a b s o r p t i o n c o e f f i c i e n t (a)show t h a t a s o l i t o n k i n k on a c h a i n suppresses t h e i n t e r b a n d t r a n s i t i o n , whereas t r a n s i t i o n s i n v o l v i n g t h e s o l i t o n l e v e l a r e f o u n d t o have a s i g n i f i c a n t l y enhanced a b s o r p t i o n c r o s s - s e c t i o n . The r e s u l t s a r e i n agreement with t h e experimental a b s o r p t i o n s p e c t r a o b t a i n e d f r o m W - ( C H ) , l i g h t l y doped w i t h AsF,

.

Takayama, L i n - L i u and Maki (TLM) ( 7 ) c o n s i d e r e d t h e continuum l i m i t o f t h e l i n e a r c h a i n model i n t r o d u c e d b y SSH t o d e s c r i b e (CHI,. Their analytical r e s u l t s a r e i n agreement w i t h t h e n u m e r i c a l r e s u l t s o f SSH and a l l o w e x p l i c i t c a l c u l a t i o n o f wave f u n c t i o n s and m a t r i x elements. The e f f e c t i v e H a m i l t o n i a n f o r t h e continuum model i s

S.ETEMAD,

216

H = - i v F3a2%4

A.J. HEEGER

+A(x)q

PI

w i t h t h e Fermi v e l o c i t y vF = 2 taa. T h e C ' s are P a u l i matrices, a and to are the l a t t i c e constant and nearest neighbor t r a n s f e r i n t e g r a l , r e s p e c t i v e l y , o f t h e uniform (undimerized) chain, and A ( x ) i s the order parameter d e s c r i b i n g the dimerization pattern. To calculateoc(o), Suzuki e t a1 (11) have f i r s t obtained the m a t r i x elements o f the momentum operator, which can be expressed i n the continuum model i n t h e form px = 2M ir where MA= - i j + ( x , y, z)& +(x-a, y, z ) dxdydz and # i s t h e atomic.rr)jp o r b i t a l o f t h e carbon atoms. They f i n d t h a t f o r a p e r f e c t dimerized c h a i t (A(x) Ao), t h e interband absorption c o e f f i c i e n t (per carbon atom), f o r t r a n s i t i o n s from valence band (VB) t o conduction band (CB), i s

7

2 where A = 116frhe2 IM,I 2 ) /m,ncWEg, E=, 2 h a , W 4t,, and meand e are t h e e l e c t r o n mass and charge, c i s the v e l o c i t y o f l i g h t , and n i s t h e index o f r e f r a c t i o n . Although oci(*)) diverges as ( h d - Eg) , t h i s square r o o t s i n g u l a r i t y w i l l be smeared out by disorder, i n t e r c h a i n coupling. o r f l u c t u a t i o n s .

For a trans-(CH), chain w i t h a s t a t i c k i n k ( ~ ( x )= A tanh(x/S)), t h e s o l i t o n f o r m a t G n e r g y takes the minimum value $ A e w i t h = A +/A,, and one bound s t a t e appears a t mid-gap. Using t h e r e s u l t s o f TLM Suzuki e t a1 c a l c u l a t e d f o r t r a n s f t i o n s between a s o l i t o n l e v e l ( S ) and t h e band s t a t e s t o be

I={,

4,

2

The f a c t o r (7Tt0/a) i n eq. 12 i n d i c a t e s an enhancement o f the s o l i t o n t r a n s i t i o n r e s u l t i n g d i r e c t l y from t h e d e l o c a l i z a t i o n o f the s o l i t o n wave f u n c t i o n 70. 7a and K'{./a) (%/a>>l). For W = 10 eV and Eg = 1.5 eV, one f i n d s 5, As a r e s u l t , t h e t r a n s i t i o n s i n v o l v i n g the s o l i t o n l e v e l should be observable even a t extremely small concentrations. The interband w; (a)(per carbon atom) i n the presence o f a k i n k i s correspondingly decreased; i.e.,

-

@ * ( f a )= @(Ld)(l-

i

i

R2 s. -2 -) 2N

a

I t i s i n t e r e s t i n g t o note t h a t the presence o f a s o l i t o n on a chain breaks the t r a n s i t i o n a l symmetry o f t h e l a t t i c e . As a r e s u l t , t h e momemtum conserving ( v e r t i c a l ) interband t r a n s i t i o n i s quenched and c$ i s due t o n o n - v e r t i c a l terms I11,18).

The o p t i c a l absorption spectra o f =-(CH)

X

are shown i n Fig. 4.

LIGHTSCATTERING AND ABSORPTION IN POLYACETYLENE

Figure 4 Absorption spectra o f trans-(CH),: c u r v e 1, undoped; c u r v e 2 , 0.01% AsF,; c u r v e 3, 0.1% AsF, ; c u r v e 4, compensated b y NH3; and c u r v e 5, 0.5% AsFS. The i n s e t shows t h e temperature dependence o f &(a) f o r undoped cn trans-(CH)x. z

b 0

Curves 1 t h r o u g h 5 on F i g . 4 show t h e e f f e c t s o f doping; I-undoped, 2 v e r y l i g h t l y doped w i t h AsF, ( a b o u t 0.01%), 3 - l i g h t l y doped with AsF, ( a b o u t 0.1%), and 4-subsequently compensated w i t h NH3; a l l d a t a a r e f r o m t h e same (CH), f i l m . The doping was c a r r i e d o u t i n situ u s i n g extreme c a r e so t h a t t h e r e s u l t s c o u l d be d i r e c t l y and q u a n t i t a t i v e 5 compared. Curve 5 on F i g . 4 was o b t a i n e d w i t h a s e p a r a t e f i l m doped t o a somewhat h i g h e r l e v e l ( 4 . 5 % ) ; q u a n t i t a t i v e comparison was p o s s i b l e t h r o u g h n o r m a l i z a t i o n o f t h e a b s o r p t i o n c u r v e p r i o r t o doping. The e f f e c t o f temperature on undoped (CH), i s shown i n t h e i n s e t ; d a t a o b t a i n e d a t 77 K and 4.2 K a r e i n d i s t i n g u i s h a b l e . The s t r o n g a b s o r p t i o n band w i t h edge a t 1.4 eV and peak a t 1.95 eV has been a t t r i b u t e d t o t h e d i r e c t i n t e r b a n d t r a n s i t i o n i n a one-dimensional ( 1 - d ) band s t r u c t u r e , and can be viewed as a r i s i n g f r o m a t r a n s i t i o n from t h e 1-d peak i n t h e d e n s i t y o f s t a t e s i n t h e valence band t o t h a t in t h e c o n d u c t i o n band. The r o u n d i n g appears t o s h i f t t h e p o s i t i o n o f t h e peaks i n t h e VB and CB d e n s i t i e s o f s t a t e s b y about 0.2 eV. I n a d d i t i o n t o t h e main peak, a weak a b s o r p t i o n (see t h e i n s e t t o F i g . 4 ) i s observed c e n t e r e d near 0.9 eV, corresponding t o a t r a n s i t i o n between t h e peak i n VB d e n s i t y o f s t a t e s and a

217

218

S.ETEMAD, A.J. HEEGER

l e v e l i n s i d e t h e gap. R e l a t i v e t o t h e VB edge, t h e gap s t a t e occurs a t about 0.7 eV; i.e., near mid-gap. On doping w i t h AsF5 t h e low energy a b s o r p t i o n grows w i t h i n c r e a s i n q c o n c e n t r a t i o n and s h i f t s t o w a r d lower energy; compensation decreases b( almost t o t h e same l e v e l as b e f o r e doping. The s t r e n g t h o f t h e main a b s o r p t i o n decreases on doping w i t h Ask , b u t i t does n o t r e c o v e r a f t e r subsequent compensation w i t h NH3. The c h a r a c t e r i s t i c f e a t u r e s o f t h e l o w energy and main a b s o r p t i o n bands can b e e x p l a i n e d i f we assume t h a t t h e doping proceeds t h r o u g h f o r m a t i o n o f p o s i t i v e charged s o l i t o n s (S') and t h a t t h e l o w energy a h s o r p t i o n band i s a s s o c i a t e d w i t h t h e t r a n s i t i o n f r o m t h e v a l e n c e band i n t o t h e Sc l e v e l t o f o r m a n e u t r a l s o l i t o n ( S o ) . The low energy a b s o r p t i o n , seen i n t h e l o w t e m p e r a t u r e d a t a o f i n s e t t o F i g . (41, i n d i c a t e s t h a t s o l i t o n s t a t e s a r e a l s o p r e s e n t i n t h e undoped m a t e r i a l . As t h e number o f S+ i n c r e a s e s w i t h AsF5 doping, t h e s t r e n g t h o f t h e low energy t r a n s i t i o n (S' t, e+ S o ) grows p r o p o r t i o n a l l y . F u r t h e r , t h e c a l c u l a t i o n s p r e s e n t e d above p r e d i c t t h a t t h e i n t e r b a n d t r a n s i t i o n w i l l be suppressed w i t h t h e i n t r o d u c t i o n o f s o l i t o n k i n k s , a g a i n i n agreement w i t h t h e e x p e r i m e n t a l r e s u l t s . On r e a c t i n g w i t h NH, t h e S+ l e v e l i s compensated, and t h e low energy band decreases a c c o r d i n g l y . On t h e o t h e r hand, t h e i n t e r b a n d t r a n s i t i o n does n o t r e c o v e r t o i t s i n i t i a l s t r e n g t h on compensation, s i n c e t h e ?T-electron k i n k s remain on t h e chain; o n l y t h e charged c e n t e r i s compensated. The i n t e g r a t e d i n t e n s i t y o f t h e low e n e r g y a b s o r p t i o n band a f t e r ( c u r v e 3 ) i s about one t e n t h o f t h a t o f t h e main a b s o r p t i o n band sample. We i n f e r f r o m eq. 12 a v a l u e f o r 71-*( <,/a) -lo1 i n good t h e t h e o r e t i c a l value. The u n i f o r m decrease i n i n t e n s i t y o f t h e t r a n s i t i o n i s a l s o e v i d e n t i n c u r v e 3 and corresponds t o about a r e l a t i v e t o t h e undoped sample.

0.1% d o p i n g o f t h e undoped agreement w i t h interband 10% r e d u c t i o n

PHOTOEXCITATIONS I N POLYACETYLENE (13-14) I n t e r e s t i n p h o t o e x c i t a t i o n s t u d i e s o f (CHI, has been s t i m u l a t e d b y t h e r e c e n t c a l c u l a t i o n s o f Su and S c h r i e f f e r (191, who c o n s i d e r e d d i r e c t i n j e c t i o n o f an e-h p a i r and s t u d i e d t h e t i m e e v o l u t i o n o f t h e system. T h e i r p r i n c i p a l r e s u l t was t h e c o n c l u s i o n t h a t i n trans-(CH), a p h o t o - i n j e c t e d e-h p a i r e v o l v e s t o a s o l i t o n - a n t i s o l i t o n psi-a t i m e o f o r d e r o f t h e r e c i p r o c a l o f an o p t i c a l phonon frequency. T h i s i s t h e c e n t r a l e x p e r i m e n t a l q u e s t i o n addressed i n t h i s s e c t i o n : Are s o l i t o n s p h o t o g e n e r a t e d i n p o l y a c e t y l e n e ? The proposed p h o t o g e n e r a t i o n o f charged s o l i t o n s (20) has c l e a r i m p l i c a t i o n s which can be checked t h r o u g h s t u d i e s o f photoluminescence and p h o t o c o n d u c t i v i t y . We t h e r e f o r e p r e s e n t and d i s c u s s i n t h i s s e c t i o n t h e e x p e r i m e n t a l r e s u l t s o b t a i n e d w i t h t h e s e two complementary t e c h n i q u e s i n c i s - and trans-(CH), In t h e s c a t t e r e d l i g h t spectrum f r o m cis-(CH), , we f i n E r e l a t i w b r o a d luminescence s t r u c t u r e peaking a t 1 .9 eV, n e a r t h e i n t e r b a n d a b s o r p t i o n edge, t o g e t h e r w i t h a s e r i e s o f m u l t i p l e o r d e r Raman l i n e s (14,21).(see 'Fig561

.

LIGHT SCATTERING AND ABSORPTION IN POL YACETYLENE

219

E (eV) 1.60

1.80

2.20

2.00

2.40

h

+ .-

c

3

-e 0

v

H

Figure 5A Photoluminescence and m u l t i p l e overtone Raman s t r u c t u r e from cis-(CH), obtained a t 7 K. The scattered l i g h t i n t e n s i t y i s p l o t t e d as a f u n c t i o n o f frequency s h i f t (AS) away from t h e 2.54 eV laser excitation.

v)

c ,-

-

c

-

->c

--

I

I

I

I

I

I

I

1

-

-

-

t v ) -

2

w

-

--

I - -

z

'.

F i g u r e 56

Scattered l i g h t spectrum from trans-(CH), obtained a t 7 K using 2.54 eV laser e x c i t a t i o n . T h e m n s i t y o f any luminescence near t h e onset o f interband absorption ( 1 . 5 eV) i s l e s s than t h a t of the c&-(CH), peak (Fig. 5A) by a t l e a s t a f a c t o r o f f i f t y . Through a measurement o f t h e e x c i t a t i o n spectrum, we have found t h a t t h e luminescence turns on sharply f o r e x c i t a t i o n energies greater than 2.05 eV, implying a Stokes s h i f t o f 0.15 eV. The main peak i n t h e luminescence s t r u c t u r e a t 1.9 eV has a h a l f - w i d t h o f about 0.1 eV and a Lorentzian energy p r o f i l e . The luminescence i n t e n s i t y i s e s s e n t i a l l y independent o f temperature from 7K t o room temperature. Isomerization o f t h e same sample t o --(CH), quenches t h e luminescence; we f i n d no i n d i c a t i o n o f luminescence near t h e interband absorption edge o f =-(CH), even a t temperatures as low as 7 K. (Fig. 5 6 ) (14).

S.ETEMAD, A.J. HEEGER

220

The photocurrent (I

Ph

)

spectrum for W - ( C H &

Figure 6 Comparison of the photocurrent (log I ),t and optical absorpei on coefficient (00 as a function o f photon energy in =-(CH) . Note the threshold of ?ph near 1 eV, well below the onset of interband absorption. Iph and o( are compared on a linear scale in the inset.

is plotted in Figure 6

E

0

and compared with the energy dependence of the absorption coefficient for trans-(CH),, o< (W) (13). Iph has a threshold at 1.0 eV, well below the interband absorption edge at 1.5 eV, implying the presence of states deep inside the gap. This is clearly shown in the inset to Fig. 6 where the photoresponse and absorption are compared on a linear scale. The threshold for the generation of free carriers is better seen in the logarithmic plot of Iph versus photon energy (Fig. 6). The absence of structure in o( ( w ) at the onset of Iph implies a low quantum efficiency at threshold. The free carrier generation efficiency rises exponentially above the threshold and changes to a slow increase above the onset of the interband transition. Measurements of Iphin &-(CH) were also attempted. Under similar conditions any photoresponse in samples o'i 80% --rich (CH) was below the noise level of our experiment. The upper limit on Iph in cis-?CH), is more than three orders of magnitude smaller than I* in transTH), In situ isomerization of the same film resulted in the s i z e a b l q shown in7ig. 6.

.

The observation of luminescence in c&-(CH), but not in trans-(CHI,, and the observation of photoconductivity in trans-(CH), but not r&(CH) provide confirmation of the proposal that solitons are the photogenerated csrriers. In =-(CH), , the degenerate ground state leads to free soliton excitations, absence of the band edge luminescence and photoconductivity. In &-(CH),, the non-degenerate ground state leads to confinement of the photogenerated carriers, absence of photoconductivity, and to the observed recombination luminescence.

22 1

LIGHTSCATTERING AND ABSORPTION IN POL YACETYLENE

I n t r a d i t i o n a l semiconductors , p h o t o c o n d u c t i v i t y and r e c o m b i n a t i o n luminescence a r e i n t i m a t e l y r e l a t e d , and b o t h a r e observed a f t e r p h o t o e x c i t a t i o n . Photoc o n d u c t i v i t y i n d i c a t e s t h e presence o f f r e e c a r r i e r s generated b y t h e absorbed photons. A l t h o u g h t h e subsequent r e c o m b i n a t i o n o f t h e s e photogenerated c a r r i e r s can t a k e p l a c e e i t h e r r a d i a t i v e l y o r n o n - r a d i a t i v e l y , r e c o m b i n a t i o n luminescence i s commonly observed, a t l e a s t a t low temperatures. The fundamental d i f f e r e n c e s between such t r a d i t i o n a l d a t a and t h o s e o b t a i n e d f r o m p o l y a c e t y l e n e can be seen by comparison o f t h e (CH), r e s u l t s w i t h r e s u l t s o b t a i n e d f r o m cadmium s u l f i d e (CdS). The luminescence and m u l t i p l e o r d e r Raman s c a t t e r i n g d a t a However, i n from CdS (22) a r e s i m i l a r t o t h e r e s u l t s o b t a i n e d f r o m &-(CH), CdS, a s t r o n g p h o t o c o n d u c t i v e response i s observed f o r photon e n e r g i e s j u s t above t h e band edge, ( 2 3 ) whereas i n c&-(CH), s i g n i f i c a n t p h o t o g e n e r a t i o n of f r e e c a r r i e r s i s n o t observed even f o r photon e n e r g i e s 1 eV above t h e band edge. I s o m e r i z a t i o n t o --(CH), quenches t h e luminescence a t a l l temperatures, b u t t u r n s on t h e p h o t o c o n d u c t i v i t y . I n n e i t h e r isomer i s t h e t r a d i t i o n a l combination o f e f f e c t s observed. These unique e x p e r i m e n t a l r e s u l t s , t h e r e f o r e , l e a d us t o c o n s i d e r t h e proposed p h o t o g e n e r a t i o n o f s o l i t o n s i n more d e t a i l .

.

A schematic diagram o f t h e p h o t o g e n e r a t i o n o f a charged s o l i t o n - a n t i s o l i t o n p a i r i n --(CH), i s shown i n F i g . 7.

M

M

M

M

M

-/-

-A+B

I

I

M

A-

I

Figure 7 a. b.

.

Band diagram and chemical s t r u c t u r e diagram f o r trans-(CH), The band diagram shows s h e m a t i c a l l y t h e absorptia photon and t h e c r e a t i o n o f an e-h p a i r . A trans-(CH)x c h a i n c o n t a i n i n g a charged s o l i t o n - a n t i s o l i t o n p a r s i n c e r e g i o n s A and B a r e degenerate, t h e s o l i t o n s a r e f r e e and can move a p a r t w i t h no c o s t i n energy. The c o r r e s p o n d i n g band diagram shows t h e mid-gap s t a t e s a s s o c i a t e d w i t h t h e two s o l i t o n s ; one empty (+) and t h e o t h e r doubly occupied ( - ) . The s o l i t o n d e f o r m a t i o n s a r e shown as l o c a l i z e d on a s i n g l e s i t e , whereas t h e def o r m a t i o n should be spread o v e r about 15 l a t t i c e c o n s t a n t s .

S. ETEMAD,

222

A.J. HEEGER

The incident photon (for h0),2b) generates an e-h pair within the rigid lattice (Fig. 7a). The system rapidly evolves to a charged soliton pair (Fig. 7b) as sec, their results After a time of order shown by Su and Schrieffer.(l9) imply the presence of two kinks separating degenerate regions. Because of the precise degeneracy of the A and B phases, the two charged solitons are free to move in an applied electric field and contribute to the photoconductivity. The simultaneous absence of band edge luminescence in trans-(CH), even at the lowest temperatures is consistent with the proposed photogeneration of charged soliton-antisoliton pairs. In this case, band edge luminescence cannot occur since there are no electrons and holes. The charged carrier pair consists of two kinks with associated mid-gap states which cannot give rise to band edge luminescence. is not present in %-(CHI,, The topological degeneracy in %-(CH) so that soliton photogeneration would no? lead to photoconductivity in the cis-isomer. Since the cis-transoid configuration (Fig. 8a has a lower

Figure 8 a. b.

c.

cis-(CH), ; cis-transoid configuration (lower energy). cis-(CH), ; trans-cisoid configuration (higher energy). A $-(CH), chain containing 'a charged soliton-antisoliton pair. Since the regions A and B are not degenerate, the solitons are confined; the farther apart the greater the energy. T h e t o n deformations are shown as localized on a single site, whereas the deformation should be spread over many lattice sites.

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223

energy t h a n t h e t r a n s - c i s o i d c o n f i g u r a t i o n ( F i g . 8b), domain w a l l s would s e p a r a t e non-degenerate r e g i o n s ( F i g . 8 c ) . A l t h o u g h n e i t h e r o f t h e l i m i t i n g forms o f F i g . 8a and 8b i s l i t e r a l l y c o r r e c t , we may t h i n k o f t h e ground s t a t e as e s s e n t i a l l y e q u i v a l e n t t o F i g . 8a, w i t h t h e s t r u c t u r e o f F i g . 8b b e i n g s l i g h t l y h i g h e r i n energy, Consider t h e n t h e p h o t o e x c i t a t i o n o f a charged s o l i t o n - a n t i s o l i t o n p a i r as shown s c h e m a t i c a l l y i n F i g . 8c. The energy required i s Etot

= 2Es + n AEo

where E, i s t h e energy f o r c r e a t i o n o f a s i n g l e s o l i t o n (analogous t o t h e s o l i t o n c r e a t i o n energy i n trans-(CHI , n i s t h e number o f CH monomers s e p a r a t i n g t h e two k i n k s , a m b E ,i s t h e energy d i f f e r e n c e between t h e two c o n f i g u r a t i o n s ( F i g . 8a and 8b), expected t o be a s m a l l f r a c t i o n o f t h e f u l l gap 2 A - r 2.0 eV i n cis-(CH), Thus, as t h e y b e g i n t o form, t h e two s o l i t o n s would be " c o n f i n e d " 3 r bound i n t o a p o l a r o n - l i k e e n t i t y ; t h e f a r t h e r apart, t h e g r e a t e r t h e energy. As a r e s u l t , one expects t h e photogenerated p a i r t o q u i c k l y recombine, thereby, quenching t h e p h o t o c o n d u c t i v i t y i n G - ( C H ) ,

.

.

The luminescence r e s u l t s i n --(CH), as d e s c r i b e d e a r l i e r p r o v i d e some i n s i g h t i n t o t h e dynamics o f system. The energy g i v e n o f f t o phonons d u r i n g t h e f o r m a t i o n o f t h e l a t t i c e d i s t o r t i o n would l e a d t o a Stokes s h i f t o f t h e luminescence r e l a t i v e t o t h e minimum energy needed t o make a f r e e e-h p a i r . As i n d i c a t e d above, a Stokes s h i f t o f AEs = 0.15 eV i s observed r e l a t i v e t o t h e onset of p h o t o e x c i t a t i o n a t 2.05 eV. The magnitude o f DE i s , however, much l e s s t h a n t h a t expected f o r t h e f o r m a t i o n o f e i t h e r two ' f r e e s o l i t o n s o r two f r e e polarons. I n t h e case o f two f r e e s o l i t o n s , t h e c o r r e s p o n d i n g e l e c t r o n i c l e v e l s would be near mid-gap w i t h t h e luminescence e n e r g y g o i n g t o z e r o i n t h e l i m i t o f (AEo/2A)+ 0. I n t h i s l i m i t f o r a w e l l - s e p a r a t e d e l e c t r o n p o l a r o n and hole-polaron, (24,25) t h e c o r r e s p o n d i n g luminescence e n e r g y would be h q = * A / J i =1.4A w i t h an i m p l i e d Stokes s h i f t o f A€s= ,6d. The much s m a l l e r e x p e r i m e n t a l v a l u e of DE, ,+ .07A, t h e r e f o r e . i n d i c a t e s t h a t t h e absence of a degenerate ground s t a t e i n cis-(CH), has made q u a l i t a t i v e changes i n t h e dynamics of p h o t o e x c i t a t i o n s - 5 t h e two isomers. The s u g g e s t i o n t h a t confinement o f photogenerated c a r r i e r s p l a y s an i m p o r t a n t r o l e i n G - ( C H ) , may p r o v i d e an e x p l a n a t i o n f o r t h e d r a m a t i c d i f f e r e n c e i n t h e l i g h t s c a t t e r i n g s p e c t r a o f t h e two isomers as shown i n F i g . 5. B r a z o v s k i i and K i r o v a have p o i n t e d o u t t h a t i n g e n e r a l t h e t o t a l gap i n a comnensurate P e i e r l s d i s t o r t e d system a r i s e s f r o m a c o m b i n a t i o n o f two terms ( 2 4 ) ; i.e., A(x) =Ae

+

A; ( X I

rl61

where A J x ) i s t h e i n t r i n s i c P e i e r l s gap s t a b i l i z e d b y T - e l e c t r o n s , and Ae i s an e x t r i n s i c c o n t r i b u t i o n which i s i n c o m p r e s s i b l e and a r i s e s f r o m T - e l e c t r o n A - 0 and t h e y suggest a v a l u e o f A = . 3 eV f o r d i m e r i z a t i o n . I n trans-(CHI, c i s - ( C H k . T h i s i-reasonabfe agreement w i t h t h e l a r g e r o p t i c a f gap i n cis-(CH),; i.e., 2Acis 2.0eV compared t o ?A+, 1.5 eV. A s i z e a b l e A , t o g e t h e r w i t h a r e l a t i v e l y s m a l l Stokes s h i f t i g i c a t e t h a t because o f t h e confinement t h e d i s t o r t i o n o f G - ( C H ) , l a t t i c e i n t h e presence o f an e l e c t r o n h o l e p a i r may be i n t h e s h a l l o w p o l a r o n l i m i t d e s c r i b e d b y B r a z o v s k i i and K i r o v a l a t t i c e where l a r g e ( 2 6 ) . T h i s i s t h e o p p o s i t e l i m i t t o t h e case o f trans-(CH), a m p l i t u d e d i s t o r t i o n s i n A ( x ) a r e p r e c u r s o r t o th-nation o f a solitona n t i s o l i t o n p a i r upon p h o t o i n j e c t i o n o f an e l e c t r o n - h o l e p a i r (191. Since t h e s c a t t e r e d l i g h t s p e c t r a i n p r i n c i p l e r e f l e c t t h e response o f t h e l a t t i c e t o t h e p h o t o e x c i t e d c a r r i e r s , t h e d r a m a t i c d i f f e r e n c e between t h e e x p e r i m e n t a l r e s u l t s from and =:(CH), may indeed be due t o t h e confinement.

-

G-

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The s p e c t r a l dependence o f t h e p h o t o c o n d u c t i v i t y i n trans-(CH)xprovides s u p p o r t i v e evidence f o r t h e proposed photogeneration o f s o l i t o n s . The minimum energy r e q u i r e d f o r photogeneration o f a p o s i t i v e and n e g a t i v e s o l i t o n p a i r i s 2(2A/~r) (20 (7,24) Since t h e d i r e c t photogeneration o f a s o l i t o n p a i r r e q u i r e s a s i g n i f i c a n t l a t t i c e d i s t o r t i o n simultaneous w i t h t h e e l e c t r o n i c t r a n s i t i o n , t h e quantum e f f i c i e n c y near t h r e s h o l d i s expected t o be s m a l l and t o increase e x p o n e n t i a l l y as hwapproaches 28, i n a manner s i m i l a r t o an Urbach t a i l ( 2 7 ) . For hw)ZA, t h e i n t e r b a n d t r a n s i t i o n has a quantum e f f i c i e n c y o f o r d e r u n i t y f o r d i r e c t e-h p a i r p r o d u c t i o n so t h a t t h e o v e r a l l s o l i t o n - a n t i s o l i t o n photogeneration process should be o n l y a weak f u n c t i o n o f energy. The experimental r e s u l t s are i n agreement; Iph r i s e s r a p i d l y and e x p o n e n t i a l l y above 1 eV 2(?4/fi) and more s l o w l y above 1.5 eV= 2 A . Thus t h e s p e c t r a l dependence o f Jph f o r trans-(CH), i s c o n s i s t e n t w i t h photogeneration o f charged s o l i t o n - a n t i s o l i t o n p a i r n h e knee i n Iph a t h w c 2 A , t h e onset o f i n t e r b a n d absorption, must be c o n t r a s t e d w i t h t h e sharp r i s e t y p i c a l l y seen a t t h a t p o i n t i n t r a d i t i o n a l semiconductors where e l e c t r o n s and/or holes a r e t h e p h o t o i n j e c t e d carriers.

.

The broadening o f t h e a b s o r p t i o n edge can be expected from s t a t i c d i s o r d e r , o r from a combination o f quantum and thermal f l u c t u a t i o n s i n A f x ) , t h e l o c a l energy gap parameter (8,24). Such dynamical f l u c t u a t i o n s generate random dynamical d i s t o r t i o n s i n t h e c r y s t a l . Since t h e e l e c t r o n i c s t a t e s t h a t c o n t r i b u t e t o t h e absorption edge are determined i n accord w i t h t h e Frank-Condon p r i n c i p l e by t h e instantaneous sec) c o n f i g u r a t i o n o f t h e l a t t i c e , these f l u c t u a t i o n s l e a d t o a d i s t r i b u t i o n o f s h o r t l i v e d s t a t e s (-lo-'' sec) below t h e band edge. As a r e s u l t , t h e absorption edge i s smeared and an Urbach t a i l i s generated. For temperatures l e s s than a t y p i c a l phonon energy (nw, r ~ 1500 l cm-' ) B r a z o v s k i i has shown t h a t t h e energy dependence o f such an Urbach a b s o r p t i o n t a i l v a r i e s as (6,241 A hw 3 1 2 ) m(w) d exp(- - ( 2 - -) 17 no, D f o r iiw<2A. I n t h i s case, t h e associated m i c r o f i e l d ( 2 7 ) a r i s e s from t h e quantum f l u c t u a t i o n s i n s t e a d o f d i s o r d e r o r o p t i c a l phonons as i s t h e case f o r i o n i c o r covalent semiconductors. A f i t o f t h e exponential r i s e o f t h e photoc u r r e n t below t h e band edge (see Fig. 61 t o equation 16 r e s u l t s i n lic;bd0.2eV, i n agreement w i t h t h e l a t t i c e dynamic r e s u l t s f o r Z - ( C H ) (12,161.

x

CONCLUSION The experimental r e s u l t s from p o l y a c e t y l e n e reviewed i n t h i s paper show t h a t a number o f unusual e f f e c t s a r e observed i n t h e l a t t i c e dynamics (monitored b y I R and Raman spectroscopy), t h e band s t r u c t u r e (seen through o p t i c a l absorption), and t h e p h o t o e x c i t a t i o n dynamics (monitored b y photoluminescence and p h o t o c o n d u c t i v i t y ) . A l l o f these r e s u l t s have been c o h e r e n t l y i n t e r p r e t e d i n terms of t h e s o l i t o n model. The t h e o r e t i c a l i n t e r p r e t a t i o n o f t h e experimental r e s u l t s has been based on a simple 1-d Hamiltonian, which describes t h e P e i e r l s d i s t o r t i o n i n a system o f n o n i n t e r a c t i n g e l e c t r o n s w i t h s t r o n g c o u p l i n g t o t h e l a t t i c e . S o l i t o n s are t h e n a t u r a l n o n l i n e a r e x c i t a t i o n s o f such a system. I t i s shown t h a t , u n l i k e t h e t r a d i t i o n a l semiconductors, where e l e c t r o n s and holes can be s t a b l e e x c i t a t i o n s , i n p o l y a c e t y l e n e t h e l a t t i c e i s i n h e r e n t 1 u n s t a b l e t o t h e i r presence. I n j e c t i o n o f an e l e c t r o n - e l e c t r o n ( h o d - a i r b y chemical doping o r p h o t o i n j e c t i o n o f an e l e c t r o n - h o l e p a i r d i s t o r t the l a t t i c e leading t o formation o f a s o l i t o n - a n t i s o l i t o n p a i r i n a t i m e s c a l e o f JO-" seconds; thus p r e v e n t i n g t h e system from r e l a x i n g b y any o t h e r means EFFECTS I N LATTICE DYNAMICS: The l o c a l d i s t o r t i o n generated b y t h e presence o f a s o l i t o n on a c h a i n g i v e s r i s e t o a s e t o f i n f r a r e d a c t i v e

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v i b r a t i o n a l modes which a r e due t o t h e l a r g e a m p l i t u d e o s c i l l a t i o n s o f charge across t h e s o l i t o n c e n t e r . The agreement between t h e model c a l c u l a t i o n s and t h e e x p e r i m e n t a l l y observed I A V M r e f l e c t s t h e s p e c i f i c o d d - f u n c t i o n c h a r a c t e r o f t h e s o l i t o n wave f u n c t i o n . We have been a b l e t o achieve a q u a n t i t a t i v e understanding o f t h e Raman f r e q u e n c i e s and t h e s o l i t o n induced i n f r a r e d a b s o r p t i o n i n trans-(CH), and trans-(CD)A. The success o f t h e t h e o r y i m p l i e s t h e b a s i c Val i d i t y o f t h e one-=ron approach, which Coulomb i n t e r a c t i o n s p l a y i n g o n l y a r e l a t i v e l y m i n o r r o l e . The d e t a i l e d agreement between t h e c a l c u l a t e d and e x p e r i m e n t a l l y observed i n f r a r e d a c t i v e v i b r a t i o n a l modes p r o v i d e s s p e c i f i c e x p e r i m e n t a l evidence o f s o l i t o n doping i n p o l y a c e t y l e n e . EFFECTS SEEN I N THE BANDSTRUCTURE: E x p l i c i t evidence f o r g e n e r a t i o n o f s o l i t o n mid-gap s t a t e s , from a j o i n t t h e o r e t i c a l and e x p e r i m e n t a l s t u d y o f t h e e f f e c t s of d i l u t e doping on t h e o p t i c a l a b s o r p t i o n s p e c t r a o f trans-(CHI,, have been reviewed. C a l c u l a t i o n s w i t h i n t h e s o l i t o n model has l e a d o t h r e e s p e c i f i c p r e d i t i o n s : ( 1 ) A s o l i t o n a b s o r p t i o n band, independent o f t h e n a t u r e o f t h e dopant, s h o u l d appear near t h e m i d d l e o f t h e a b s o r p t i o n gap. ( 2 ) The i n t e n s i t y o f t r a n s i t i o n s i n v o l v i n g t h e s o l i t o n l e v e l s h o u l d be enhanced b y about two o r d e r s o f magnitude. ( 3 ) The e x i s t e n c e o f a s o l i t o n on a c h a i n s h o u l d suppress u n i f o r m l y t h e e n t i r e i n t e r b a n d t r a n s i t i o n , A l l o f t h e s e e f f e c t s a r e observed e x p e r i m e n t a l l y w i t h magnitudes i n agreement w i t h t h e t h e o r e t i c a l r e s u l t s . EFFECTS SEEN I N PHOTOEXCITATIONS: The combined r e s u l t s o f two complementary and experiments; i.e. , photoluminescence and p h o t o c o n d u c t i v i t y i n c&-(CH)x trans-(CH), i n d i c a t e t h a t t h e p h o t o i n j e c t e d c a r r i e r s a r e i n d e e d t h e s o l i t o n a n t i s o l i t o n p a i r s . The p r i n c i p l e r e s u l t s a r e summarized i n t h e f o l l o w i n g t a b l e :

C i s- (CH Ix

Trans-(CH),

Ground S t a t e

Non-degenerate

Degenerate b y symmetry

Nonlinear excitation

Confinement

Stable soliton-antisol i t o n pair

Band-edge 1umi nescence

Yes

No

Photoconductivity

No

Yes

The e x p e r i m e n t a l r e s u l t s p r e s e n t e d t h u s c o n f i r m t h e t h e o r e t i c a l e x p e c t a t i o n t h a t t h e trans-(CH), l a t t i c e i s u n s t a b l e t o t h e presence o f any p a i r of e l e c t r o n o r h o l e e x c i t a t i o n s . The s t r o n g c o u p l i n g o f t h e e l e c t r o n i c s t r u c t u r e t o t h e l a t t i c e p r o v i d e s a n a t u r a l l y f a s t r o u t e f o r t h e l a t t i c e t o d i s t o r t and t r a n s f o r m t h e s i m p l e charge e x c i t a t i o n p a i r s i n t o s o l i t o n - a n t i s o l i t o n p a i r s . As a r e s u l t o f t h e s e unusual f e a t u r e s , p o l y a c e t y l e n e i s f u n d a m e n t a l l y d i f f e r e n t f r o m c o n v e n t i o n a l semiconductors. The s i m p l e one-dimensional s t r u c t u r e o f t r a n s - p o l y a c e t y l e n e p r o v i d e s t h e means f o r e x p e r i m e n t a l r e a l i z a t i o n o f a variet-mathematical problems. F o r example, t h e concept o f s o l i t o n as a n o n l i n e a r s o l u t i o n o f t h e D i r a c e q u a t i o n has been of c o n s i d e r a b l e mathematical i n t e r e s t . The r e c e n t mapping (25,28) of SSH H a m i l t o n i a n o n t o t h e two dimensional D i r a c e q u a t i o n has shown a u n i f y i n g base t o be shared b y condensed m a t t e r and p a r t i c l e p h y s i c i s t s The p o s s i b i l i t y o f experimental s t u d i e s o f s o l i t o n s i n p o l y a c e t y l e n e i s , t h e r e f o r e , o f broad i n t e r e s t t o many f i e l d s o f physics.

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226

A.J. HEEGER

Acknowledgement: This work i s a r e v i e w o f t h e r e s u l t o f f r u i t f u l c o l l a b o r a t i o n s w i t h T. -C. Chung, L Lauchlan, A. G. MacDiarmid, M. Ozaki, A. Pron, and N. Suzuki a t t h e U n i v e r s i t y o f Pennsylvania, and E. G. Mele and M. J. R i c e o f Xerox. The review was prepared under support from t h e Army Research O f f i c e (DAAG29-81-K-0058). References: 1.

For a review o f t h e i n i t i a l works on p o l y a c e t y l e n e see: Heeger, A. J. and MacDiarmid, A. G., The Physics and Chemistry o f Low Dimensional S o l i d s Ed. by L o u i s Alcacer (D. Reidel P u b l i s h i n g Co., 1980) p. 353.

2.

H i g h l y Conducting One-Dimensional Solids, Ed. b y J. T. Devreese, R. P. Evrard, and V. E. van Doren (Plenum Press, N. Y. 1978) c o n t a i n s s e v e r a l reviews on t h i s t o p i c .

3.

Rice, M. J., and S t r a s s l e r , S.,

4.

Baughman, R. H., Hsu, S. L., Phys., 68, 5405 ( 1978).

5.

Su, W. P., S c h r i e f f e r , J. R. and Heeger, A. J., Phys. Rev. L e t t . (1979); i b i d . Phys. Rev. B 12, 2099 (1980).

6.

Rice. M. J.,

7.

Takayama, H.,

8.

B r a z o v s k i i , S . , JETP L e t t .

9.

Nechtschein, M., Devreux, F., Greene, R. L., 6. B., Phys. Rev. L e t t . 44, 356 (1980).

10.

Weinberger, B. R., Ehrenfreund, E., Pron, A., Heeger, A. J. and MacDiarmid, A. G . , J. Chem. Phys. 72, 4749 (1980).

11.

Suzuki, N., Ozaki, M., Etemad, S., Heeger, A. J. and MacDiarmid, A. G., Phys. Rev. L e t t . 3, 1209 (1980); Erratum Phys. Rev. L e t t . 45, 1483, (1980).

12.

Etemad, S., Pron, A., Heeger, A. J., MacDiarmid, A. G., Rice. M. J., Phys. Rev. 823, 5137 (1981).

13.

Etemad, S., M i t a n i , M., Ozaki, M., Chung, T. -C., Heeger, A. J., MacDiarmid, A. G. S o l i d S t a t e Communications ( i n press).

14.

Lauchlan, L., Etemad, S., Chung. T. -C., A. G., Phys. Rev. B ( i n press).

15.

Fincher, C. R., Jr., Ozaki, M., Phys. Rev. !3H, 4140 (1979).

16.

Mele, E. J. and Rice, M. J.,

17.

Mele, E. J.,

S o l i d S t a t e Commun. l3, 125 (1973).

Pez, G. P.,

Phys. L e t t . 7lJ,

and S i g n o r e l l i ,

A. J.,

J. Chem.

2, 1698

152 (1979).

21, 2388 (1980). JETP 78, 677 (1980).

L i n - L i u . Y. R. and Maki, K . , Phys. Rev. B

28,

and Rice, M. J.,

656 (1978): i b i d .

Clarke, T. C. and S t r e e t ,

Heeger, A. J.,

Mele, E. G., and

and MacDiarmid,

Heeger, A. J., and MacDiarmid, A. G.,

S o l i d S t a t e Commun Phys. Rev. L e t t .

3, 339

45,

and

(1980).

926 (1980).

LIGHTSCATTERING AND ABSORPTION IN POL YACETYLENE

18.

Gamal , J. T. and Krumhansel

19.

Su, W. P. and S c h r i e f f e r , J. R . , Proc. Nat. Acad. o f Sci.

20.

Etemad, S., Ozaki, M., Heeger, A. J . and MacDiarmid, A. G., Proc. o f t h e " I n t e r n a t i o n a l Conf. on Low Dimensional S y n t h e t i c Metal ," H e l s i n g o r , Denmark, August 1980, Chemica S c r i p t a ( i n press).

21.

Lichtmann, L. S., 869 (1980).

36,

, J.

227

A. ( p r e p r i n t ) , (1981).

Sarhangi, A. and Fitchen, D. C.,

77,5626

(1980).

S o l i d S t a t e Commun.

22.

L e i t e , R. C. C., Scott, J. F., and Damen, T. C., Phys. Rev. L e t t . 22, 780 (1969); K l e i n , M. V. and Parto, S. P. S., Phys. Rev. L e t t . 11, 782 (1969).

23.

See, f o r example, Bube, R. H., P h o t o c o n d u c t i v i t y o f S o l i d s , Wiley and Sons, N . Y. 1960, p. 230 and 391.

24.

B r a z o v s k i i , S. and Kirova, N . , Pisma Zh. Eksp. Teor. F i z .

25.

Campbell, D. and Bishop, A.,

26.

Using t h e r e s u l t s o f Ref. 24, t h e experimental value o f t h e Stokes s h i f t i m p l i e s a shallow d i s t o r t i o n , w i t h a maximum v a l u e o f .15A and f u l l w i d t h o f -.40a, i n t h e gap parameter A ( x ) .

27.

See DOW, J. D. and R e d f i e l d , D., therein.

28.

Jackiw, R. and S c h r i e f f e r , J. R.,

Phys. Rev. B

3, 48-59

3,6

(1981).

(1981).

-

Phys. Rev.

B 5, 594 (1972) and references

Phys. B 190 [FS3],

253 (1981).

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North-Holland Publishing Company, 1982

229

QUASI SOLITONS:

A CASE STUDY O F THE DOUBLE SINE-GORDON EQUATIONS

P . Kumar and R.R.

Holland

Physics Department U n i v e r s i t y of F l o r i d a G a i n e s v i l l e , F l o r i d a 32611

We r e p o r t h e r e r e s u l t s from t h e n u m e r i c a l s i m u l a t i o n of t h e s c a t t e r i n g between a s o l i t o n a n t i s o l i t o n p a i r f o r t h e generalized double sineGordon e q u a t i o n .

INTRODUCTION The d o u b l e sine-Gordon e q u a t i o n (DSGE) is a c u r i o u s n o n - l i n e a r p a r t i a l d i f f e r e n t i a l equation ' h a t it a d m i t s two d i s t i n c t classes of s o l i t o n s o l u t i o n s . F u r t h e r m o r e , u n l i k e an i d e a l s o l i t o n e q u a t i o n , where t h e s o l u t i o n s u n d e r g o e l a s t i c s c a t 5 e r i n g , t h e s c a t t e r i n g between DSGE s o l i t o n s is w e a k l y i n e l a s t i c . D u r i n g a c o l l i s i o n , small a m p l i t u d e waves are e m i t t e d t h a t c a r r y away t h e e n e r g y . The c o n s e q u e n c e s of t h e weak i n e l a s t i c i t y are two-fold. On t h e one hand, t h e k i n e m a t i c s of t h e c o l l i s i o n p r o c e s s is changed. On t h e o t h e r h a n d , p z r t u r b a t i o n t h e o r y m e t h o d s ha e been d e v e l o p e d by Kaup and N e w e l l and by S c o t t and Mclaughlin' a n d a p p l i e d to cases where sine-Gordon e q u a t i o n is w e a k l y p e r t u r b e d . I f t h e DSGE c a n be e x p r e s s e d as a p e r t u r b a t i o n on t h e s i n e Gordon e q u a t i o n , t h e p r e s e n t numerical s i m u l a t i o n can p r o v i d e a test of t h e a n a l y t i c approaches. The DSGE is a c l a s s o f n o n - l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s d e s c r i b e d by 2 6J 0 2 sin$(cos$ + a ) (1) a t t - c $xx - 1-a Here t h e s u b s c r i p t s d e n o t e a d e r i v a t i v e and t h e p a r a m e t e r a d e s c r i b e s d i f f e r e n t members of t h e c l a s s . F o r a=O, w e r e c o v e r t h e sine-Gordon e q u a t i o n . Th eq t ' n a p p e a r s i n s e v e r a l p h y s i c a l contexts. 1 case o f I n s u p e r f l u i d %e,YB-19 a s w e l l a s i n a spe a takes self-induced transparency (SIT) with l e v e l degeneracy, t h e v a l u e 1 / 4 and h a s been a n a l y z e d p r e v i o u s l y . A d i f f e r e n t v e r s i o n of E q . ( l ) ,w i t h RHS c a r r y i n g a minus s i g n also o c c u r s f r e q y g y t l y i n d i f f e r e n t p h y s i c a l c o n t e s t s e.g. p o l i n g p r o c e s s i n e more common e q u a t i o n f o r S I T w i t h l e v e l PVF 2 d e g e n e r a and('f cy. However t h i s e q u a t i o n d o e s n o t c a r r y t h e two d i s t i n c t classes of s o l i t o n s , it is more l i k e a p e r t u r b a t i o n 1 SG e q u a t i o n and h a s been s t u d i e d a n a l y t i c a l l y by B u l l o y g h e t . a l .

747

99

W

Eq.(l)

0 2 (cos$+a) , I 2(1-a ) ( 1 ) . I t c o n s i s t s o f v a l l e y s o f minima

may be d e r i v e d from a p o t e n t i a l

drawn s c h e m a t i c a l l y i n F i g .

V($) =

230

P. KUMAR.

R.R. HOLLAND

at 4 = 4 = cos-l(-a) and at 0 = 2n-4 02separated by maxima The prefytor l/(l-a ) ensures that the at nn; n20 ,I ,2.. potential has curvature w at the minima. The two solitons correspond to the system b?ing at adjacent minima asymptotically (x+-- and +-) and connected via the potential maxima. The type I1 soliton ( - I $ ~ , $ ~ )is connected via the

.

Fig. 1. The po_tfntial V(g) as a function of 4 and are at 0, = cos ( - a )

X.

The minima

higher peak at 4=0 and is more massive than the type I soliton ( 4 ,2n-$ ) which is connected via the lower peak at + = n . There areotopo18gical implications in that a type I soliton must be followed either by a type I1 soliton or a type I antisoliton ( 4
e=

I

cotanh[(x-ut)/cJl-v -2-2-

= tanh[(x-ut)/cdl-v

I

type I

(2

type 11

(3

where 5 = c/w is a characteristic length in the problem and u=vc is the velociey of the soliton. The energy of these solutions is given by 2Acwo EI =

-

f1-a22Acwo

En--J1-a2-

+ a(40-n))

(sin+o + a+o)

(4 (5

QUASI-SOLITONS: THE DOUBLE SINE-GORDON EOUATION

231

where A is an overall energy scale factor, clearly E12< EII. The limit a=l is singular. If ye drop the prefactor (1-a ) in the potential, it becomes a cos $/2 potential. The solitons then are ( - n , n ) solutions and are given by tan$/2 =

x-ut G2-

5J

E = 2nAw 2 5 0

Linear stabilitv analvs s shows that these sol tons are stable. The normal mode*spect;um in each case includes a bound state at w=O, corresponding to the translational mode and a continuum of wave like sxcita3ions2w30se frequency is given by the dispersion relation w = w + c k The phonons are not reflectionless anymore. In a 8ollision between a soliton-antisoliton (SS) pair, many phonons are emitted that take the energy away and make the collision inelastic.

.

NUMERICAL SIMULATION RESULTS Computer simulation of soliton dynamics has historically been an important and essential source of information. Beginning with the works of Fermi Pasta and Ulam, numerical simulation has been an integral part of analyses of sol n dynamics. ve been These($? The extensively reviewed by Bullought'? and by Makhankov. numerical study DSG (for a=1/4) ha een previously reported by Maki a?j)Kumar,(Pf Kitchenside et.al.72P and by Shiefman and Kumar. We study here the a dependence of the properties of DSG solitons. As before, ( 3 ) the numerical calculations were done using discrete space-time steps. At the initial time, the soliton-antisoliton pair was described by Eqs.(2) or ( 3 ) . The solitons were launched towards each other from positions sufficiently far apart ( 5 ~ )s o that the effect of interaction was negligible. At any time, the data set consisted of two vectors $(x.,t) and @(x.,t-At) which was used to generate $(x.,t+At) using simble trapezoiAa1 rule integrations. The profile bectors were shifted in time and g(xi,t-At) was dropped. In case of SS bound states, $(x=O) was stored separately at all times. Such a procedure applied directly, caused wild fluctuations in @(x.,t+At). These were errors due to the discrete lattice, amplified by the non-linear terms in the partial differential equation. This instability could be controlled by averaging nearest neighbor values on the space lattice at each time. The remaining discrete lattice errors were particularly bothersome for small a where damping effects were large. For a=l, we chose Ax = .0255 and At was 5% smaller. For larger a, Ax = .ly (and At, 5% smaller) was found sufficient. For a type I S s pair, the results of this simulation are described in Figs. (2-5). In Fig. (2), we show the classical version of the scattering matrix for a=.l. For small incoming velocities of the SS pair, the final state is a bound state where the pair oscillates with a characteristic frequency. In a simple description of soliton as a particle moving in a potential wall, this frequency should monotonically decrease with the soliton energy. There is considerable structure in the energy dependence of the oscillation frequency. It does go to zero at a

232

P. KUMAR. R.R. HOLLAND

c h a r a c t e r i s t i c v e l o c i t y V $ ( e n e r g y ) t h a t is a mea u r e of r h e e n e r g y B t h e SS p a i r loss on c o l l i s i o n . F o r v e l o c i t i e s l a r g e r t h a n Vc, u n d e r g o e s a h a r d core r e p u l s i v e f p l l i s i o n L F o r v e l o c i t y h i g h e r t h a n a n o t h e r c r i t i c a l v e l o c i t y Vc , t h e SS p a i r is c o n v e r t e d i n t o a t y p e 11 p a i r . T h e s e f e a t u r e s are s i m i l a r to o n e s r e p o r t e d e a r l i e r f o r a = . 2 5 , w i t h t h e e x c e p t i o n of t h e e n e r g y d e p e n d e n c e of t h e o s c i l l a t i o n f r e q u e n c y . The e x t r a s t r u c t u r e i n t h e f r e q u e n c y was m i s s e d e a r l i e r i n t h e w i d e l y s e p a r a t e d v e l o c i t y v a l u e s s t u d i e d . I n F i g . ( 5 1 , wf show e t w o c r i t i c a l v e l o c i t i e s as a f u n c t i o n of a . F o r a=O, V and Vg' a r e b o t h e q u a l t o z e r o . They i n c r e a s e r a p i d 9 w i t h a b u t f o r small t h e y r e m a i n e q u a l . An i d e a l s i n e - G o r d o n SS p a i r t r a n s m i t s t h r o u g h e a c h o t h e r a n d , it c o r r e s p o n d s t o p a r t i c l e c o n v e r s i o n w 0.9

-

-

- 0.7

0.7

0.5

-

-

-

- 0.5 -

' *

*. 0.3 -

- 0.3

0

0

-

c

0.I

hand s c a l e r e f e r s t o t h e frequency w h i l e t h e r i g h t hand s c a l e r e f e r s to t h e v e l o c i t y of outgoing SS p a i r s . The v e r t i c a l arrows indicate the thresholds f o r incoming v e l o c i t i e s .

0.

-

* . I

0.I

I

- 0.1

**

** I

0.3

I

I

0.5

I

1

0 .I

I

1

0.9

i n t h e afO case. The h d - c o r e r e p u l s i o n a p p f a r s o n l y f o r a>.07 a t which t i m e t h e V: and 'V! c u r v e s s e p a r a t e . Vc r e a c h e s a maximum f o r a=.2 and t h e n d e c r e a s e s f o r h i g h e r a . Energy c o n s e r v a t i o n r e q u i r e s f o r t h e a d e p e n d e n c e of V,':

The o b s e r v e d v a l u e s of V ': to t h e i n e l a s t i c effects.

are s l i q t t l y d i f f e r e n t from Eq.(8) d u e T h e s e Vc r a p i d l y a p p r o a c h o n e so t h a t

QUASI-SOLITONS: THE DOUBLE SINE-GORDON EQUATION

Q

W

~0.3

pout

-

-

-

- 0.5

0.5

*

-

*

m

-

e

*-•

0.3

0.1

-

-

-

- 0.1

. '

W

I

I

vout

0.7

/ /

c

-

0.1

-

0.I

0.5

/

:*

-

0.3

0.3

F i g . 3. The scattering matrix f o r a = . 3 . The l e f t hand s c a l e r e f e r s to t h e frequency while t h e r i g h t hand s c a l e r e f e r s to t h e v e l o c i t y of outgoing SS p a i r s . The v e r t i c a l arrows indicate the thresholds f o r incoming v e l o c i t i e s .

I

a = 0.6

1.

233

0.3

0.I

F i g . 4 . The scattering matrix The l e f t f o r a=.6. hand scale refers t o the frequency while t h e r i g h t hand scale r e f e r s to t h e v e l o c i t y of outgoing SS p a i r s . The v e r t i c a l arrows indicate the thresholds for incoming v e l o c i t i e s .

P. KUMAR, R.R. HOLLAND

234

Fig. 5. The critical velocities as a function of a . Th solid line shows VL' if the collisions were energy conserving. are the calculated while (01 are calculated Vc.

"C

0.9

0.7

0.5

0 0

0

0

0

0

1

1

0.3

0.I 1

0.1

1

1

0.3

1

1

0.5

1

0.7

1

0.9 U

for a2.3, the range of velocities for particle conversion becomes negligibly small. The final velocity after particle conversion, for an energy conserving collision is given by

which describes the data surprisingly well provided that we use the observed value of V, (instead of Eq. (8)). The effects of damping are more important at small a , as described earlier. Yet another measure of inelasticity is the rate of decay of the bound state amplitude. It decays in time with a power law as t-". The exponent n varies from n=1.5 for a=.l to n=.2 for a = . 9 . The characteristic time scale also increases with a . In all cases however, the frequency of the bound state remains well defined. The damping remains a weak effect and the solitons can be described as quasi-solitons.

OUASI-SOLITONS: THE DOUBLE SINE-GORDON EQUATION

235

We thank A.C. Scott for encouraging discussions. This work was supported by NSF grant no. DMR 8006311 and by the Research Corporation. REFERENCES Maki, K. and Kumar, P., Phys. Rev. B14 (1976) 3920. Kitchenside, P.W. Bullough, R.K. and Candrey, P.J., Creation of Spin Waves in h e B , in: Bishop, A . and Schneider, T. (eds), Solitons and Condensed Matter Physics (Springer-Verlag, Berlin 1978). Shiefman, J. and Kumar, P., Physica Scripta 20 (1979) 435. Kaup, D.J. and Newell, A.C., Phys. Rev. B18 (1978) 5162. McLaughlin, D.W. and Scott, A.C., Phys. Rev. A 18 (1978) 1652. Hopfinger, A.J. Lewanski, A . J . , Sluckin, T.J. and Taylor, P.L., Solitary Wave Propagation as a Model for poling in PvF2, in: Bishop and Schneider (red.) Solitons and Condensed Matter Physics (Springer-Verlag, Berlin 1978). Bullough, R.K., Solitons in: Interaction of Radiation with Condensed Matter Vol. I, IAEA, Vienna 1977. Makhankov, V., Comp. Phys. Comm. 21 (1980) 1.

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 8. Nicolaenko (%is.) @ North-Holland Publishing Company. 1982

237

CLASSICAL FIELD THEORY WITH Z ( 3 ) SYMMETRY H e r b e r t M. Ruck Nuclear Science D i v i s i o n Lawrence B e r k e l e y L a b o r a t o r y University o f California Berkeley, CA 94720

ABSTRACT We p r e s e n t s o l u t i o n s and some o f t h e i r p r o p e r t i e s o f a c l a s s i c a l v e c t o r f i e l d model i n two-dimensional Minkowski space w i t h i n t e r n a l symmetry Z(3)--the c y c l i c group o f o r d e r t h r e e .

1.

INTRODUCTION

C y c l i c symmetric f i e l d t h e o r i e s a r e o f i n t e r e s t i n s o l i d s t a t e p h y s i c s where t h e symmetry f o l l o w s f r o m t h e l a t t i c e s t r u c t u r e o f t h e c r y s t a l s 1 9 2 and i n elementary p a r t i c l e p h y s i c s where t h e c o l o r t h e o r y o f quarks and gluons imposes c y c l i c t r a n s f o r m a t i o n s o f t h e quantum numbers3. I n t h i s s h o r t communication we a r e d e a l i n g w i t h t h e mathematical s t r u c t u r e o f a system o f n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s t h a t i s i n t e r e s t i n g i n i t s e l f . Some p h y s i c a l a p p l i c a t i o n s a r e g i v e n elsewhere4.

2.

THE Z ( 3 ) MODEL

A. The Lagrangian d e n s i t y o f t h e model5 i s a f u n c t i o n a l o f a two-component v e c t o r f i e l d 3 = [ b l ( t , x ) , b 2 ( t , x ) ] E Rp i n one time, one space dimensions xu = ( t , x ) E Rp, p , =~ 0 , l :

V(b1,62) i s t h e p o t e n t i a l p a r t o f t h e Lagrangian t h a t c o n t a i n s t h e polynomial s e l f - i n t e r a c t i o n o f the s c a l a r f i e l d s :

The m e t r i c i s gpV = d i a g ( + l , - 1 ) , p , v = 0 , l ; ap = alaxp, a$ = a! - a t . The parameters x, II a r e p o s i t i v e x > 0, u > 0. The c o n s t a n t y s d e ermined such t h a t t h e p o t e n t i a l ( 2 ) i s p o s i t i v e d e f i n i t e V 2 0. The parameter v i s chosen t o be p o s i t i v e v > 0. The group t r a n s f o r m a t i o n s o f t h e f i e l d s t h a t l e a v e t h e Lagrangian d e n s i t y ( 1 ) i n v a r i a n t a r e ( i ) a r e f l e c t i o n o f 62: 61

+

61

, 62 + -62

and ( i i ) a Z ( 3 ) d i s c r e t e r o t a t i o n by 120' o f t h e f i e l d s :

(3)

238

H. M. RUCK

6, + b1cos(2nn/3) - b 2 s i n ( 2 n n / 3 )

,

(4a)

b2 +

,

(4b)

b1sin(2nn/3) + b2cos(2nn/3)

w i t h n = 0,1,2. The d i s c r e t e symmetry Z ( 3 ) Eq. ( 4 ) i s imposed by t h e c u b i c t e r m i n t h e p o t e n t i a l ( 2 ) . When v i s s e t z e r o t h e Lagrangian o b v i o u s l y has f u l l SO(2) r o t a t i o n a l symmetry. There e x i s t t h r e e more r e p r e s e n t a t i o n s o f t h e L a g r a n g i a n (1): i n terms o f complex f i e l d s , i n p o l a r c o o r d i n a t e s , and i n terms o f a m a t r i x f i e l d . 6. With t h e complex f i e l d s 6 = 41 + ib2, becomes:

6*

=

61 - i d 2 t h e L a g r a n g i a n

The r e f l e c t i o n symmetry ( 3 ) becomes a complex c o n j u g a t i o n :

6 + 6* and b* + b ,

(61

and t h e d i s c r e t e r o t a t i o n ( 4 ) becomes a phase t r a n s f o r m a t i o n :

b + bexp(i2nn/3)

, 6* +

6*exp(-i2~n/3)

.

(7)

C.

I n p o l a r c o o r d i n a t e s t h e Z ( 3 ) i n v a r i a n c e i s e a s i l y seen. The f i e l d s a r e = p s i n e . The p o t e n t i a l p a r t ( 2 ) o f . t h e Cagrangian becomes:

41 = Pcose, and 62

V(p,e)

= xp4

- vp3

cos(3e) -

pp

2

-

.

y

(8)

The r e f l e c t i o n t r a n s f o r m a t i o n ( 3 ) becomes a r e f l e c t i o n o f t h e a n g l e 8,-e

8:

,

(9)

and t h e r o t a t i o n a l t r a n s f o r m a t i o n ( 4 ) a t r a n s l a t i o n a l o p e r a t i o n :

e

+e

+ 2rn/3

,n

=

0,1,2

.

(10)

The o n l y e-dependent f a c t o r i n (8) c o s ( 3 e ) i s e v i d e n t l y i n v a r i a n t under t h e mappings ( 9 ) and (10).

D. The l a s t f o r m i s i n terms o f a m a t r i x f i e l d t h a t l i n k s t h e model t o a S U ( 2 ) representation. Define t h e m a t r i x f i e l d :

where 01, a2, 63 a r e t h e P a u l i m a t r i c e s . d e n s i t y (1) as a t r a c e ( T r ) e x p r e s s i o n :

Then we can w r i t e t h e L a g r a n g i a n

CLASSICAL FIELD THEORY WITH Z(3) SYMMETRY

239

I i s t h e 2 x 2 dimensional u n i t m a t r i x . The r e f l e c t i o n symmetry ( 3 ) becomes t h e mapping:

;3

03;03

Y

and t h e Z ( 3 ) t r a n s f o r m a t i o n t h e mapping:

i+o+;o

,

w i t h t h e operator:

o

, 03

= exp(-inno2/3)

=

I

,

n = 0,1,2

.

T h i s m a t r i x r e p r e s e n t a t i o n i s a s c a l a r v e r s i o n o f t h e SU(2) Yang-Mills f i e l d 6 : 3

A

= I.1

A:aa

,

a=I

w i t h t h e i m a g i n a r y component set t o z e r o A$ = 0. The r e p r e s e n t a t i o n (11) i s s u i t a b l e f o r g e n e r a l i z a t i o n t o more t h a n two f i e l d components b y an expansion i n terms o f t h e g e n e r a t o r s o f t h e SU(3) group.

3.

SOLITON SOLUTIONS The Euler-Lagrange f i e l d equations i n two dimensions are:

F i r s t we a r e l o o k i n g f o r t i m e i n d e endent s o l u t i o n s o f t h e f i e l d e q u a t i o n s 7 . One dimensional s o l u t i o n s of t h e 7 t h e o r y have found a p p l i c a t i o n s i n p o l y a c e t y l e n e 8 . For f u r t h e r d i s c u s s i o n o f s o l i t o n s i n (CH), see Ref. 9. I n a d d i t i o n , r e f e r e n c e s t o t h i s problem can be found i n t h i s volume o f t h e proceed ings.

9

Plane wave s o l u t i o n s i n h i g h e r d i m e n s i o n a l space can be o b t a i n e d f r o m t h e one dimensional s o l u t i o n by t h e mapping10 x 3 k,x,/k,k,. I n order t o solve t h e f i e l d equations, i t i s useful t o consider t h e t o p o l o g y o f t h e p o t e n t i a l V(61,62) as a s u r f a c e d e f i n e d on t h e 61,62 space. There a r e t h r e e minima-also c a l l e d vacua-denoted by QlyR2,R3, where t h e p o t e n t i a l i s zero, l o c a t e d on a c i r c l e o f r a d i u s 6~ a t an a n g l e o f 120' f r o m each o t h e r . The c o o r d i n a t e s o f t h e minima are:

bv i s t h e v a l u e o f t h e vacuum f i e l d determined as t h e p o s i t i v e r o o t o f t h e e q u a t i o n aV(p,O)/ap = 0. I n t h e c e n t e r o f t h e 61.62.plane t h e p o t e n t i a l i s e l e v a t e d V(0,O) = -y > 0. For l a r g e v a l u e s o f t h e f i e l d s 6f + 63 >> 66 t h e p o t e n t i a l has steep w a l l s due t o t h e p o s i t i v i t y o f A. here are t h r e e The d i s t a n c e f r o m t h e c e n t e r o f saddle p o i n t s located a t 60", 180° and 240'. t h e saddle p o i n t s i s g i v e n by t h e p o s i t i v e r o o t o f t h e e q u a t i o n a V ( p Y n / 3 ) / a p = 0.

H.M. RUCK

240

S o l u t i o n s t u n n e l i n g f r o m one vacuum t o t h e o t h e r a r e o b t a i n e d by t h e method o f t r a j e c t o r i e s i n f i e l d spacell. When t h e parameters A,

V,

u,

Y

a r e r e l a t e d t o each o t h e r i n t h e f o l l o w i n g

way:

t h e geometry o f t h e p o t e n t i a l arranges i t s e l f such t h a t t h e saddle p o i n t between two vacua l i e s on t h e s t r a i g h t l i n e c o n n e c t i n g t h e s e two vacua. The t u n n e l i n g f r o m one vacuum t o t h e o t h e r w i l l now o c c u r a l o n g t h i s s t r a i g h t l i n e t h r o u g h t h e sadd l e p o i n t

.

There a r e s i x p a i r s o f s o l u t i o n s t o t h e f i e l d e q u a t i o n s c o n n e c t i n g t h e minima p a i r w i s e . Three p a i r s o f s o l u t i o n s a r e generated by t h e mapping ( 4 ) ; t h e o t h e r t h r e e obtained by t h e r e f l e c t i o n ( 3 ) . We use t h e n o t a t i o n a = ( 3 ~ / 2 ) 1 / 2 6 v . The b a s i c set o f s o l u t i o n s i s : .

1.

The s o l u t i o n g o i n g f r o m 01 t o Q2, i.e.

B1(x) = 1 b v [ l

+

3tanh(ax)];

b2(x) =

+

b(+m)

7 6 bV[1 -

E

4,

tanh(ax)l

+

f ~ ( - ~E)

9 is

.

(19)

The t r a j e c t o r y i s t h e a l g e b r a i c r e l a t i o n independent o f x:

b1

+

J5 6,

=

B,

.

(20)

The next two p a i r s a r e o b t a i n e d f r o m ( 1 9 ) by r o t a t i o n o f 120" and 240'.

2.

$(+-I E

The s o l u t i o n t u n n e l i n g f r o m R2 t o R3 w i t h t h e a s y m p t o t i c v a l u e s ~ 2 , E a3 i s

d(-m)

The t r a j e c t o r y i s

1

61=-28, 3.

.

(22)

The s o l i t o n t u n n e l i n g f r o m Q3 t o

B1(x) =

1

bv[l - 3tanh(ax)l

: 62(x)

=

a1

&(+a) E

J5 b,[1 -7

+

sl3,

&(-m)

tanh(ax)]

E

R1 i s

,

(23)

with the trajectory

Q1- 43

b2

=

6,

.

A l l t r a j e c t o r i e s t o g e t h e r [Eqs. (20),(22), and ( 2 4 ) ] f o r m an e q u i l a t e r a l t r i a n g l e w i t h t h e c o r n e r s i n t h e minima. o f t h e p o t e n t i a l .

4.

PROPERTIES OF THE SOLITON SOLUTIONS

The energy-momentum t e n s o r i s a f u n c t i o n a l o f t h e f i e l d s and t h e i r d e r i v a t i v e s d e f i n e d as:

(24)

CLASSICAL FIELD THEORY WITHZ(3) SYMMETRY

241

The k i n e t i c and p o t e n t i a l energy o f t h e f i e l d c o n f i g u r a t i o n g i v e n b y any o f t h e Eqs. (19),(21), o r ( 2 3 ) a r e equal. T h e i r sum i s t h e energy d e n s i t y :

.

(x) = 3 9 IB,[COS~(~X)]-~ 4

T

00

(26)

The i n t e g r a t e d f i e l d energy g i v e s t h e mass o f t h e s o l i t o n :

J

M =

.

dx TOO(x) = aB, 3

-03

The o t h e r components o f t h e f i e l d t e n s o r - t h e p r e s s u r e T11-van i s h .

f i e l d momentum To1 and t h e f i e l d

The l i n e a r e x t e n s i o n ( a n a l o g o f t h e r a d i u s i n one d i m e n s i o n ) o f t h e energy d i s t r i b u t i o n i n c o n f i g u r a t i o n space i s d e f i n e d by:

R

dx x

= [3f

TOO(x)I

p

0

dx T O O ( x ) 1 l /2

,

(28)

0

[Too(x) = T m ( - x ) ] , such t h a t a b o x - l i k e energy d i s t r i b u t i o n o f a r b i t r a r y h e i g h t and l e n g t h 2s g i v e s t h e c o r r e c t answer R = s. The dependence on a and bv i s e x p l i c i t :

R = (0.80305 ...)aWith B = radius:

-y

1 / 2 -1

6,

.

> 0 t h e mass o f t h e s o l i t o n can be w r i t t e n i n terms o f t h e l i n e a r

M = (1.1438 ...)B(2R)

5.

(29)

.

(30)

RESONANCES

E x c i t e d s t a t e s o f t h e s c a l a r f i e l d s a r e g i v e n b y f l u c t u a t i o n s i n t i m e about t h e c l a s s i c a l f i e l d s 61 and 62. Again we c o n s i d e r t h e s o l u t i o n Eq. ( 1 9 ) . The p e r t u r b a t i v e expansion o f t h e f i e l d s 1 2 (Q2 and 93 a r e n e g l e c t e d i n t h e f i e l d equations) : 6;(t,x)

= 61(x)

+

Q(t,x)

; b;(t,x)

1

= 62(x) --9P(t,x)

s

4-

( 3 11

i s chosen such t h a t t h e new f i e l d s 6; and 6; s a t i s f y t h e same a l g e b r a i c r e l a t i o n ( 2 0 ) as t h e c l a s s i c a l f i e l d s . W i t h t h i s expansion t h e e n e r g y spectrum o f t h e resonance has two l e v e l s :

where t h e e n e r g i e s and t h e p e r t u r b a t i o n s are:

,

( t , x ) = Acos(Et) sech(ax) t a n h ( a x )

t,x)

= Acos(Et)[l

-

sech2(ax)]

.

(33)

(34)

H.M. RUCK

242

6.

TWO SOLITON INTERACTIONS

I n physical applications t h e behavior o f m u l t i c e n t e r s o l u t i o n s i s i m p o r t a n t . I have n o t found any e x a c t m u l t i c e n t e r s o l u t i o n i n t h i s model.

I w i l l s h o r t l y a n a l y z e an ansatz f o r a f i e l d c o n f i g u r a t i o n w i t h two c e n t e r s a t x and x?. There i s an o r d e r i n of t h e c e n t e r s i m p l i e d x2 Q 71. The t r i a l f u n c t i o n c o n t a i n s l i n e a r l 3 9 1 l and q u a d r a t i c forms o f t h e s i n g l e s o l i t o n s o l u t i o n s Eq. ( 1 9 ) l o c a t e d a t x i and t h e s o l u t i o n Eq. ( 2 3 ) w i t h t h e s i g n o f 62 changed [symmetry Eq. ( 3 ) ] l o c a t e d a t x2:

61( 2 ) (x,x1,x2) +

sin

2

= cos

~1 6 , [ 1

(E)

6, ( 2 ) (X,X 1 ,X 2 ) +

E

2

(E)

+

T1 4,fi

3[1

- [l -

tanh(a(x

+

3 t a n h ( a ( x - xl))][-l

= cos

s i n2 ( E )J38 6,[1

+

+

+

-

x,))

i s a v a r i a t i o n a l parameter w i t h v a l u e s

+

E

-

3tanh(a(x

t a n h ( a ( x - x,))

t a n h ( a ( x - xl))][l

-tanh

-

x ) 2

tanh

t a n h ( a ( x - x,))]

E [O,n/2].

I f one o f t h e c e n t e r s i s removed t o i n f i n i t y , t h e f i e l d s (35,36) reduce t o exact s i n g l e s o l i t o n s o l u t i o n s . T h i s l i m i t process i s independent o f t h e v a l u e o f E. I f x2 + -m, t h e f i e l d s (35,36) reduce t o Eq. ( 1 9 ) and f o r x 1 + + - t h e f i e l d s reduce t o Eq. (23) w i t h t h e s i g n o f 62 i n v e r t e d .

The s t a t i c p o t e n t i a l energy between two s o l i t o n s a t a d i s t a n c e d = x 1 i s t h e d i f f e r e n c e between t h e t o t a l f i e l d energy and t w i c e t h e mass o f an isolated soliton:

-

x2

rn

P(d,E) = /.dx

Thi)(x,d,c)

-

2M

.

(37)

-w

A n u m e r i c a l c a l c u l a t i o n shows t h a t t h e p o t e n t i a l i s n e g a t i v e , i.e. t h e two s o l i t o n s a t t r a c t each o t h e r when t h e y come c l o s e t o g e t h e r . F o r E = 0 ( p u r e l i n e a r c o m b i n a t i o n ) we o b t a i n a lower bound and f o r E = K/2 ( p u r e q u a d r a t i c f o r m ) an upper bound f o r t h e p o t e n t i a l energy. P(d,O)

Q P(d,E)

Q P(d,n/2)

;

Q0

E

.

E (O,n/2)

(38)

The p o t e n t i a l converges f a s t t o z e r o when t h e d i s t a n c e d becomes s l i g h t l y l a r g e r t h a n t h e l i n e a r e x t e n s i o n o f t h e p a r t i c l e s d > 2R. The q u a l i t y o f t h e ansatz Eqs. (35,36) can be checked by computing t h e a c t i o n f o r t h i s f i e l d c o n f i g u r a t i o n . The a c t i o n p e r u n i t t i m e i s : m

S(d,E)

= -

J

dx Thi)(x,d,E)

.

(39)

-W

F o r l a r g e d i s t a n c e s d >> R t h e a c t i o n i s -2M independent o f E . F o r f i x e d d i s t a n c e s i n t h e i n t e r a c t i o n r e g i o n , t h e a c t i o n has a minimum f o r E = K / 2 and a maximum f o r E = 0. -2M Q S ( d , n / 2 )

GS(d,e)

GS(d,O)

QO

;

E

E (O,n/2)

.

(40)

CLASSICAL FIELD THEORY WITHZ(3) SYMMETRY

I n t h e s t a t i c t r i a l f u n c t i o n s ( 3 5 , 3 6 ) t h e q u a d r a t i c f o r m o f two s i n g l e s o l i t o n s o l u t i o n s m i n i m i z e s t h e a c t i o n a t any d i s t a n c e and seems t h e r e f o r e t o be p r e f e r r e d t o t h e l i n e a r form. The net e f f e c t i s t h a t t h e s t a t i c s o l i t o n s a t t r a c t each o t h e r . T h i s i s c o n s i s t e n t w i t h t h e f i n d i n g s i n t h e v4 model a t low v e l o c i t i e s . H i g h v e l o c i t y s o l i t o n s c o l l i d i n q w i t h ea h. o t h e r may s c a t t e r backward showing a r e p u l s i v e p o t e n t i a l at high ~ p e e d l ~ . ~ ~ . CONCLUSIONS We analyzed a f i e l d t h e o r y model w i t h Z ( 3 ) i n t e r n a l symmetry i n 1 + 1 dimensions. Exact s o l i t o n s o l u t i o n s a r e found i n one space dimension. An ansatz f o r t h e two-center f i e l d shows t h a t t h e s t a t i c p o t e n t i a l between two solitons i s attractive. ACKNOWLEDGMENTS

I am g r a t e f u l t o David K. Campbell f o r many h e l p f u l and i n t e r e s t i n g d i s c u s s ions. T h i s work was supported by t h e D i r e c t o r , O f f i c e o f Energy Research, D i v i s i o n o f N u c l e a r P h v s i c s o f t h e O f f i c e o f H i g h Energy and N u c l e a r P h y s i c s o f t h e U.S. Department o f Energy under C o n t r a c t W-7405-ENG-48.

243

244

H.M. RUCK

REFERENCES R.8. Potts, Some generalized order-disorder t r a n s f o r m a t i o n s , Proc. Cambr. P h i l . SOC. 4s (1952) 106-109. 2. A.N. Berker, S. Ostlund and F.A. Putnam, Renormalization-group treatment o f a P o t t s l a t t i c e gas f o r k r y p t o n adsorbed o n t o graphite, Phys. Rev. 617 (1978) 3650-3665. 3. A. de Rujula, H. Georgi and S.L. Glashow, A t h e o r y o f f l a v o r mixing, Ann. o f Phys. 109 (1977) 258-313. H.M. R u c k T o l y n o m i a l chromodynamics i n 1+1dimensions I . Confinement o f 4. quarks and t h e s t r u c t u r e o f c m p o s i t e p a r t i c l e s , Lawrence Berkeley Laboratory p r e p r i n t LBL-12316 (February 1981). 5. F. Constantinescu and H.M. Ruck, Quantum f i e l d t h e o r y P o t t s model, J. Math. Phys. 19 (1978) 2359-2361; Phase t r a n s i t i o n s i n a continuous t h r e e s t a t e s model X t h d i s c r e t e gauge symmetry, Ann. o f Phys. 115 (1978) 474-495. 6. C.N. Yang and R. M i l l s , Conservation o f i s o t o p i c s f i and i s o t o p i c gauge invariance, Phys. Rev. 96 (1954) 191-195. 7. H.M. Ruck, S o l i t o n s i n c y c l i c symmetry f i e l d t h e o r i e s , Nucl. Phys. 8167 (1980) 320-326. 8. M.J. Rice, Charged n-phase k i n k s i n l i g h t l y doped polyacetylene, Phys. L e t t . 71A (1979) 152-154. W.P. SCJ.R. S c h r i e f f e r and A.J. Heeger, S o l i t o n s i n polyacetylene, Phys. 9. Rev. L e t t . 42 (1979) 1698-1701; S o l i t o n e x c i t a t i o n s i n polyacetylene, Phys. Rev. 822 (1x0) 2099-2111. 10. P.6. F t , S o l i t a r y waves i n n o n l i n e a r f i e l d t h e o r i e s , Phys. Rev. L e t t . 32 (1974) 1080-1081. 11. R. Rajaraman, S o l i t o n s o f coupled s c a l a r f i e l d t h e o r i e s i n two dimensions, Phys. Rev. L e t t . 42 (1979) 200-204. 12. H.M. Ruck, Quark confinement i n f i e l d t h e o r i e s w i t h d i s c r e t e gauge symmetry Z(3), i n : K.B. Wolf (ed.), Group t h e o r e t i c a l methods i n physics, L e c t u r e Notes i n Physics, 135 (Springer Verlag, B e r l i n , 1980). 13. A.E. Kudryavtsev, m i t o n l i k e s o l u t i o n s f o r a Higgs s c a l a r f i e l d , JETP L e t t . 22 (1975) 82-83. 14. V.G. M a a n k o v , Dynamics o f c l a s s i c a l s o l i t o n s (In non-integrable systems), Phys. Rep. C35 (1978) 1-128. p r i v a t e communication. 15. D.K. Campbe1.

PART 111 Reaction-Diffusion Processes

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 8.Nicolaenko (eds.) 0 North-Holland PublishingCompany, 1982

247

SOME CHARACTERISTIC NONLINEARITIES OF CHEMICAL REACTION ENGINEERING

Rutherford A r i s Sherman Fai r c h i 1d Distinguished Scholar C a l i f o r n i a I n s t i t u t e o f Technology A b r i e f survey i s made o f the d i f f e r e n t nonlinear forms which a r i s e as expressions o f chemical r e a c t i o n r a t e i n the general problem o f d i f f u sion and r e a c t i o n i n a s o l i d c a t a l y s t . These may be c l a s s i f i e d i n order o f increasing complexity as powers and polynomials, r a t i o n a l functions, and transcendentals. The c h a r a c t e r i s t i c s and some o f the t y p i c a l problems associated w i t h each o f these classes w i l l be surveyed.

INTRODUCTION I n attempting t o survey the c h a r a c t e r i s t i c types o f n o n l i n e a r i t y o f chemical r e a c t i o n engineering i t may be w e l l t o take examples from a general class o f problems which i s o f c e n t r a l importance i n the theory o f chemical r e a c t i o n processes. The complexity o f a c a t a l y t i c system requires a very thorough a n a l y s i s both o f the physical and o f the chemical processes t h a t are a t work. The porous catal y s t p e l l e t i s a network o f pore space o f various and varying diameters o f t e n l i n k e d t o a subsystem o f even f i n e r pores through a manufacturing process which involves the pressing together o f a f i n e powder. No exact model o f d i f f u s i o n i n so complicated a geometry i s possible and, i n e v i t a b l y , a s i m p l i f i e d model o f the geometrical s t r u c t u r e must be made. Even t h i s i s g e n e r a l l y n o t taken t o the second stage o f being t r e a t e d as a d i f f u s i o n model w i t h the r e a c t i o n as i n the boundary condition, b u t r a t h e r serves t o define an e f f e c t i v e d i f f u s i v i t y i n which both t h e geometry o f the pore s t r u c t u r e and physical p r o p e r t i e s o f t h e material which i s d i f f u s i n g are bound up together i n an e f f e c t i v e d i f f u s i v i t y . This d i f f u s i v i t y w i l l generally be dependent on the concentration o f the d i f f u s i n g species b u t f o r many p r a c t i c a l purposes t h i s dependence i s ignored. Jackson's monograph [lo] gives an e x c e l l e n t treatment o f t h i s t o p i c . But, w h i l e i t i s o f t e n possible t o make the assumption o f constant d i f f u s i v i t y and c o n d u c t i v i t y , i t i s n o t u s u a l l y f e a s i b l e , o r sensible, t o make any s i m p l i f y i n g assumption as t o the r e a c t i o n r a t e . This i s g e n e r a l l y present as a nonl i n e a r f u n c t i o n o f the dependent v a r i a b l e s i n a p a i r o f p a r a b o l i c o r e l l i p t i c equations. The f u l l d e r i v a t i o n o f these w i l l n o t be c a r r i e d o u t i n d e t a i l here f o r i t i s the burden o f t h e second chapter o f my t r e a t i s e on the "Mathematical Theory o f D i f f u s i o n and Reaction i n Permeable Catalyst.'' ( [ 4 ] ; t h i s w i l l be referred t o as MT, followed by a page o r equation number.) L e t i t here s u f f i c e t o say t h a t t h e equations are obtained by the usual mass o r energy balances and i n t h e i r time-dependent form, g i v e a p a i r o f equations f o r the dimensionless concentration, u, and the dimensionless temperature, v. L e t us f u r t h e r s i m p l i f y the assumptions by assuming t h a t only a s i n g l e r e a c t i o n i s t a k i n g place and t h a t t h i s may be characterized by the concentration o f a s i n g l e component o r by a v a r i a b l e which represents t h e extent o f reaction. The equation f o r the concentration then i s

240

R. A RIS

I n t h i s equation u i s the concentration, made dimensionless by t h e surface concentration u , t i s the dimensionless time equal t o a d i f f u s i o n c o e f f i c i e n t times t h e r e a l %ime d i v i d e d by the square o f some l e n g t h parameter; $2 i s t h e r a t i o o f the maximum r a t e o f r e a c t i o n t o t h e corresponding d i f f u s i o n f l u x and i s a measure o f t h e r e l a t i v e preponderance o f r e a c t i o n r a t e . R(u) i s t h e dimensionless r e a c t i o n r a t e which a t u = 1 has been normalized t o have the value R(1) = 1. Again, f o r s i m p l i c i t y , t h i s equation w i l l be used w i t h t h e O i r i c h l e t c o n d i t i o n t h a t u = 1 on the surface o f t h e r e g i o n f o r which i t i s t o be solved. I n t h e equation i t s e l f the r a t e o f change o f concentration i n any element i s necessarily due t o the n e t f l u x by d i f f u s i o n (the f i r s t term on t h e right-hand side) minus t h e r a t e a t which t h a t substance i s disappearing by t h e reaction. ( C f . MT. pp. 51-9.) When t h e r e are several r e a c t i o n s i t i s necessary t o i n s e r t more variables and, o f course, t o have more than one r e a c t i o n r a t e . When t h e r e a c t i o n i s n o t isothermal, i . e . , the temperature o f t h e c a t a l y s t p e l l e t i s n o t uniform o r kept c o n s t a n t ' b y some o t h e r means, i t i s necessary t o make R a funct i o n o f u and v.

A s i m i l a r equation can be obtained by an energy balance i n t h e p a r t i c l e and i t y i e l d s a second p a r t i a l d i f f e r e n t i a l equation f o r v, the dimensionless temperature. Again, t h i s i s a q u a s i l i n e a r p a r a b o l i c equation i n which t h e temperat u r e change i s the sum o f a c o n t r i b u t i o n from conduction d i f f u s i o n and one from reaction:

2 (2) a t = v2v + p$ ~ ( u , v ) i n R, v = l on aR . Here p i s a heat o f r e a c t i o n parameter and L t h e r a t i o o f the thermal and L

material d i f f u s i v i t i e s . We w i l l again keep t h i n g s simple by using t h e D i r i c h l e t conditions v = 1 on t h e boundary o f t h e region. The p a r t i a l d i f f e r e n t i a l equat i o n s o b t a i n i n a connected region o f t h r e e dimensional space, R , w i t h t h e boundary conditions asserted on t h e piece-wise smooth boundary o f t h i s region, an. ( C f . Fig. 1.)

b! 2

Figure 1: The c a t a l y s t p e l l e t

NONLINEARITIES OF CHEMICAL REACTION ENGINEERING

249

The theme o f the paper w i l l , therefore, concern t h e behavior o f t h i s sytem when R(u,v) takes on various forms. For t h e most p a r t we s h a l l be d e a l i n g w i t h t h e steady s t a t e where the parabolic nature o f the d i f f e r e n t i a l equations degenerates i n t o e l l i p t i c i t y . Thus we w i l l n o t be concerned w i t h t h e i n i t i a l condit i o n s although there s u r e l y are such. I n the steady s t a t e there i s a most important f u n c t i o n a l o f the s o l u t i o n which represents t h e effectiveness o f the c a t a l y s t p e l l e t under t h e conditions o f operation. I t i s , i n f a c t , the average r a t e o f r e a c t i o n throughout the p e l l e t d i v i d e d by the r a t e o f r e a c t i o n which w i l l o b t a i n i n t h e absence o f any d i f f u sion o r conduction l i m i t a t i o n whatsoever. B r i e f l y , i t can be given by t h e integral

By i t s d e f i n i t i o n , t h i s f u n c t i o n a l w i l l tend t o value 1 when t h e r e a c t i o n i s very slow compared w i t h t h e speed o f t h e d i f f u s i v e processes and w i l l tend t o zero as the r e a c t i o n r a t e dominates the possible r a t e o f d i f f u s i o n . The measure o f t h i s r a t i o o f r e a c t i o n t o d i f f u s i o n r a t e i s the parameter 0, which i s o f t e n r e f e r r e d t o as the T h i e l e modulus i n honor o f the American engineer who, almost simultaneously w i t h Zeldowich i n Russia and Damklihler i n Germany, recognized i t s importance and devised t h e effectiveness as a way o f expressing i t s consequences. (For a f u l l e r treatment o f the h i s t o r y see MT pp. 37-42 and [ 3 ] . ) Powers and Polynomials Let us assume t h a t the p a r t i c l e i s isothermal and t h a t we, t h e r e f o r e , have no need f o r t h e second equation derived from an energy balance. The simplest assumption i s , o f course, t h a t the r e a c t i o n i s o f t h e f i r s t order, which i s expressed mathematically by making R(u) = u. This gives a thoroughly l i n e a r equation and i t would be out o f order i n t h e present context t o discuss i t i n any d e t a i l . One remark may be made i n passing however, i t concerns the simplest case and i s almost t r i v i a l l y obvious. I f t h e r e g i o n R i s a p a r a l l e l - s i d e d s l a b o f i n f i n i t e e x t e n t o r sealed edges, then the equation becomes

The immediate s o l u t i o n o f t h i s i s

A graph o f q against Q i n l o g a r i t h m i c coordinates shows t h e features t h a t we should expect; namely, i t i s asymptotic t o 1 as I$ goes t o 0 and t o 0 as $ goes t o i n f i n i t y . This i s a basic feature of the problem and these asymptotic r e l a t i o n s h i p s h o l d i n a l a r g e number o f o t h e r cases. I n f a c t , t h e product o f r l and I$ tends t o a constant value and t h i s i s merely because a t i n c r e a s i n g l y high r e a c t i o n r a t e s the d i f f u s i n g and r e a c t i n g substance disappears by r e a c t i o n w i t h i n a very s h o r t distance o f the surface. The effectiveness o f the p e l l e t thus becomes proportional t o i t s e x t e r i o r surface area r a t h e r than t o i t s volume. We remark i n passing t h a t there are some important generalizations o f t h i s l i n e a r case t o more recondite shapes o f R and t o simultaneous f i r s t order reaction. The l a t t e r gives a r a t h e r s i m i l a r formulation i n terms o f vectors o f concentrat i o n and matrices representing d i f f u s i v i t i e s and r e a c t i o n r a t e constants.

R. ARlS

250

I f the f i r s t - o r d e r r e a c t i o n corresponds t o t h e l i n e a r case and may be dismissed a t a conference dedicated t o nonlinear problems, the next e a s i e s t form o f non1i n e a r i t y i s s u r e l y t h a t o f t h e pth-order r e a c t i o n which i n symmetrical bodies (slab, c y l i n d e r o r sphere) gives a form o f t h e Emden-Fowler equation:

I n the geometry o f the slab w i t h q = 0 we see t h a t t h e equation may be i n t e g r a t e d by C l a i r a u t ' s transformation i . e . , by m u l t i p l y i n g both sides by t w i c e t h e f i r s t d e r i v a t i v e and i n t e g r a t i n g . I f M2 i s the f i r s t i n t e g r a l o f t h e r i g h t hand side o f t h i s equation, U

[ u ' ( p ) I 2 = $*

2R(w)dw = ~$~[M(u;u,)]~

(7)

uO

Then, f o r the pth-order r e a c t i o n we merely have some s t r a i g h t f o r w a r d i n t e g r a l s t o calculate.

Whence a second i n t e g r a t i o n gives

and, since u(1) = 1,

I t i s best t o t u r n t h i s problem arsey-versy and l e t u be t h e parameter i n terms o f which both t h e effectiveness f a c t o r and t h e Thieye modulus are calculated. Figure 2 shows t h e T h i e l e modulus as a f u n c t i o n o f t h e dimensionless center conc e n t r a t i o n and Figure 3 the e f f e c t i v e n e s s f a c t o r versus t h e normalized T h i e l e modulus. By normalized T h i e l e modulus we mean t h a t t h e T h i e l e modulus i s m u l t i p l i e d by a f a c t o r such t h a t t h e asymptotic value o f qQ i s 1. I t i s c l e a r from Eq. 9 t h a t t h i s i s t h e square r o o t o f (p+1)/2. An i n t e r e s t i n g aspect o f the second f i g u r e i s t h a t f o r values o f p l e s s than o r equal t o 1 t h e i n t e g r a l def i n i n g $I i n terms o f u does n o t diverge as u tends t o zero. It f o l l o w s t h a t f o r values o f $ greatep than a c e r t a i n c r i t i c a ? value ( i t s e l f dependent upon p) the o n l y way i n which a s o l u t i o n can be obtained i s f o r t h e r e t o be a dead region i n the center o f t h e p e l l e t i n which t h e concentration i s i d e n t i c a l l y zero. Then t h e boundary conditions

NONLINEARITIES OF CHEMICAL REACTION ENGINEERING

Dimensionless centre eonccnlration u.

Figure 2:

The r e l a t i o n between the dimensionless c e n t r e concentration, uo, and t h e Thiele modulus $ f o r the pG order reaction.

Figure 3:

The effectiveness f a c t o r f o r the p g order r e a c t i o n as a f u n c t i o n o f t h e normalized Thiele modulus.

25 1

252

R. A R K

u(p) = u ' ( p ) = 0 a t p = IN

(12)

u(1) = 1 serve t o determine w.as w e l l as t h e constants o f i n t e g r a t i o n . For an i n t e r e s t i n g g e n e r a l i z a t i o n o f t h i s f r e e boundary phenomenon see N i c h o l s ' papers [13,14,15]; the geometry o f a f i n i t e c y l i n d e r has been s t u d i e d by Stephanopoulos [18]. The reduction t o the hypergeometric fu?F+tlion may be i l l u s t r a t e d i n cases p > 1 f o r then the s u b s t i t u t i o n U = l-(uo/u) t u r n s t h e i n t e g r a l i n eq. (11) i n t o an incomplete beta f u n c t i o n and t h i s can be expressed as

Now the hypergeometric f u n c t i o n converges on t h e c i r c l e 11-u;'l I = 1, and i t follows t h a t since the exponent o f uo i s negative (0 w i l l range from i n f i n i t y t o zero as uo goes from 0 t o 1. Also

showing t h a t $ i s monotonic decreasing and t h a t t h e r e l a t i o n between $ and u For -1 < p < 1 t h e hypergeometric f u n c t i o n must be transformed bg i s unique. the r e l a t i o n F(a,b;c;z)

= (l-z)C-a-bF(c-a,c-b;c;z)

,

giving

The s o l u t i o n f o r p = -1may be obtained i n terms o f Dawson's i n t e g r a l 2

x

F(x) = e-'

et2 d t

,

0

and i s

Since F has a maximum o f 0.541 the value o f $* i s 0.765. The various cases can best be summarized i n t a b u l a r form

253

NONLINEARITIES OF CHEMICAL REACTION ENGINEERING

,

m = l/lp+ll

4 = (2m)*(l-U)

U4F(cr,b;c;U)

= -(2m)-4(l-U)cr'U-4F(crl

, ,bl ;cl ;U)

.

When p = -m/(m+l) o r -(2m+3)/2m+l) and m i s a p o s i t i v e i n t e g e r t h e hypergeomet r i c functions are polynomials o f degree m. Table 1 Range o f m P

U

a

C

a'

a'

bl

c'

3

4+m

7

-4

-4

m-4

4

1-u: +l0

1

1-m

3

-4

-4

m-4

4

3

-4

-4

m-4

4

1-m

1

1-m

4

--

--

--

--

0,4

l - u p0 + l

l,o

4,l 3

b

4

m,l

1

a

m-4

1,z

l - u P0 + l

0

1

1-m

-3,-1 7,"

3

l-uP+l

0

1

1-m

3

-1,-

1-u-p-l 0

4

1

4-m

4

O q 1

0

m,O

Rational Functions The next most complicated n o n l i n e a r i t y a r i s e s i n t h e case o f k i n e t i c s which are governed by so-called Langmuir-Hinshelwood o r Michael is-Menten k i n e t i c s (actua l l y the names o f Hougen and Watson can be r i g h t l y associated with t h e engineeri n g a p p l i c a t i o n s o f t h i s form o f k i n e t i c s , as can those o f Briggs and Haldane w i t h the biochemical use). When t h e k i n e t i c s o f t h e r e a c t i o n depend on adsorpt i o n on the s o l i d surface o f the c a t a l y s t p e l l e t i t i s n a t u r a l and proper t o make the r a t e o f r e a c t i o n a f u n c t i o n o f the adsorbed concentration r a t h e r than t h a t o f the gas and the pores. This can g i v e a dimensionless r a t e o f r e a c t i o n o f the form o f a q u o t i e n t o f two polynomials: U(E0+E1U+. R(u) =

Eo

+

El

+

. .+up-l) ... + 1

(20)

A simple reduction allows t h i s f u n c t i o n o f u t o vanish a t u = 0 as w e l l as t o s a t i s f y the c o n d i t i o n o f being 1 a t u = 1. I n t h e s l a b geometry t h i s y i e l d s a s i m i l a r normalization o f t h e T h i e l e modulus and leads t o such curves as are seen i n Figure 4 f o r various cases o f the parameters. (For references see MT p. 170, Table 3.8.)

254

R. A R E

5

,

1

I

I

, , , , ,

I,

2

I 9

0.5

0.2

0. I 0.1

Figure 4:

0.2

I

2

10

The normalized effectiveness f a c t o r p l o t f o r second-order LangmuirHinshelwood k i n e t i c s . Fixed LO=l and various K.

The simplest form o f t h e Langmuir isotherm, i n which the adsorbed species tend t o form a monolayer on the adsorbing surface, has some h a p p i l y simple propert i e s . I n p a r t i c u l a r , i n the simultaneous adsorbtion o f two s o l i d s from a flowi n g medium the equations o f balance are r e d u c i b l e and thus amenable t o t h e hodograph transformation. Under t h e i r transformation t h e c h a r a c t e r i s t i c s o f the system i n the plane o f the two concentrations c o n s i s t s o f a p a t t e r n o f s t r a i g h t l i n e s . This f o r t u i t o u s l i n e a r i t y o f a p a r t i c u l a r nonlinear system i s t h e secret o f the way i n which i t i s p o s s i b l e t o get some very f a r reaching r e s u l t s i n t h e theory o f chromography o f such substances. [17] Transcendental Functions Once t h e assumption o f i s o t h e r m a l i t y i s abandoned i t becomes i n e v i t a b l e t h a t t h e k i n d o f n o n l i n e a r i t y t h a t w i l l a r i s e i s a thoroughly transcendental one. T h i s i s because temperature influences t h e r e a c t i o n r a t e constants i n a h i g h l y nonl i n e a r fashion so t h a t they increase w i t h temperature according t o t h e law dank/dT = E/RT2. Thus, t o take t h e very simplest o f r e a c t i o n s , t h e i r r e v e r s i b l e pth-order disappearance o f the key substance, t h e r a t e o f r e a c t i o n w i l l be k(T) cp and k(T) = k e-E/RT. Reduction t o dimensionless v a r i a b l e s thus leads t o a p a i r o f equationsoof the form

u py a t = ~ ~ u - + ~exp

t-l)

These equations are n o t as objectionable as one might a t f i r s t suppose f o r t h e n o n l i n e a r i t y enters i n t o one term o n l y o f t h e right-hand s i d e and does so i n p r e c i s e l y the same fashion f o r each. Thus i f we form a combination o f temperat u r e and concentration @u+v=z

NONL INEA RITIES OF CHEMICAL REACTION ENGINEERING

255

and i f L = 1, t h i s combination s a t i s f i e s the heat equation. I n t h e steady s t a t e i t s a t i s f i e s Laplace's equation. But z i s constant and equal t o (l+f!) on t h e surface and hence, by the well-known p r o p e r t y o f a p o t e n t i a l f u n c t i o n , i s cons t a n t everywhere. Thus pu + v

E

1 + p and

or

I f we choose u as the v a r i a b l e the various forms o f G(u) w i l l , f o r constant p , depend on t h e two parameters p and y. The same applies t o t h e f u n c t i o n F(v) and may, perhaps, be i l l u s t r a t e d r a t h e r more e a s i l y f o r t h i s function. I t has been normalized so t h a t F ( l ) = 1 and since u vanishes when v = l + p t h e curve must take one o f t h e forms shown i n Figure 5. I n p a r t i c u l a r , i f t h e p o i n t fi, y l i e s beneath the hyperbola By = 1 ( i . e . , i n region A o f t h e By-plane) then t h e funct i o n F i s monotone decreasing i n the i n t e r v a l o f i n t e r e s t which i s (1, 1+p). I f (ply) l i e s i n the region p then F has a maximum, b u t no p o i n t o f i n f l e c t i o n . On the other hand i f p , y l i e s i n t h e region C between t h e two hyperbolae, By/(l+p) i s greater than 2 b u t less than 4, then t h e curve has an i n f l e c t i o n i s monotone decreasing as v passes from 1 t o 1 + 0. Howp o i n t , b u t F(v)/(v-1) ever, i f p,y l i e s above t h i s t h i r d hyperbola then F/v i s n o t monotonic and Luss has shown t h a t t h i s i s a necessary c o n d i t i o n f o r m u l t i p l i c i t y o f steady s t a t e s 1121.

*I-

Q

Figure 5:

The forms o f t h e f u n c t i o n F(v) and P(w) f o r p = l and several values o f BlY.

256

R. ARlS

L e t me t u r n aside t o note t h a t even here t h e nonlinear expression has a very i n t e r e s t i n g s i m i l a r i t y p r o p e r t y and one which allows every p o i n t o f t h e calcul a t i o n t o give some i n f o r m a t i o n f o r a f a m i l y o f q,~$curves. V i l l a d s e n and Michelsen [19] observed t h a t t h e f u n c t i o n G(u) was p a t i e n t o f the f o l l o w i n g s i m i l a r i t y transformation. I f we w r i t e

and s e t u = ulU,

then a f t e r some a l g e b r a i c manipulation we have

S i m i l a r l y , w i t h the temperature f u n c t i o n

we have f o r v = vVl

where

,

B = (1+ P-vl)/vl

r

= y/vl

(31)

.

As we should expect, t h e l o c i o f p o i n t s B,T t h a t are t r a c e d o u t i n t h e plane by varying u1 o r v1 are t h e same, namely

(1 + p)r-yB = y

.

(B,r)(32)

This represents a s t r a i g h t l i n e through (p,y) and (-1,O). This transformation implies t h a t every p o i n t i n t h e i n t e g r a t i o n o f t h e d i f f e r e n t i a l equations (25) f o r symmetrical geometries can be i n t e r p r e t e d t o g i v e t h e effectiveness f a c t o r f o r some problem. Consider, f o r example, i n t e g r a t i n g t h e I t i s convenient t o s e t f = $p so t h a t symmetrical equation (6).

and t o i n t e g r a t e t h i s as an i n i t i a l value problem w i th adu4 --0 ,

u = u0

atf=O

.

(34)

257

NONLINEARITIES OF CHEMICAL REACTION ENGINEERING

When the p o i n t a t which u = 1 i s reached, t h e value o f +1du

5

i s Q and

@ du = +1du

4sq

Q=>+=eaf

(35)

By keeping p and y f i x e d and varying u , the (q,t$)-curve can be t r a c e d out, f o r u i s a parameter along it. But i f %he i n t e g r a t i o n i s stopped a t the p o i n t wRere u = u a problem i s solved f o r t h e parameters B and r given by eqn. (28). To see t h i s b e s u b s t i t u t e

E' 2 -- p2Q2G(ul;B,Y)/ul

u = ulu,

(36)

i n eqn. (6) so t h a t , by eqn. (27), i t becomes

$+hdU - G(U;B,T) . I f t h i s i s i n t e g r a t e d from

5'

= 0 u n t i l U = 1 then t h e value o f

(37)

5"

Q 2 -- 02P2G(ul;B,Y)/ul

is (38)

and

Thus the i n t e g r a t i o n of eqn. (33) t o t h e p o i n t a t which u = u,l gives

f o r the B and

r

say

u1 = u(El),

given by eq. (28).

It has sometimes been f e l t t h a t the nature o f t h e n o n l i n e a r i t y i n the r e a c t i o n r a t e constant i s mathematically complicated and t h a t , since t h e usual range of temperatures i s i n t h e region where t h e exponential i s increasing r a p i d l y , i t should be p o s s i b l e t o replace the Arrhenius f u n c t i o n by a p o s i t i v e exponential i n temperature. When t h i s i s done, as i s o f t e n t h e case i n combustion studies, a very i n t e r e s t i n g equation r e s u l t s . I n engineering c i r c l e s t h i s i s associated w i t h Franck-Kamenetskii [ 5 ] and i n more mathematical w i t h t h e name o f Gelfand [7]. See a l s o Joseph and Lundgren [ll].

Though the j u s t i f i c a t i o n f o r considering the problem v2u+eU = u = o

o

in

R ,

on aR

258

R. A R A

may be r a t h e r slender i f i t i s t o be regarded as an approximation t o t h e zerothorder exothermic r e a c t i o n , t h e equation remains o f g r e a t i n t e r e s t . FrankKamenetskii (1939) was the f i r s t t o apply i t t o combustion problems. He wrote the equation as +

6eU =

o

,

.

u(*l) = 0

(42)

I f the s o l u t i o n i s computed i n the usual fashion using u as parameter t h e r e i s a maximum value o f 6, which Frank-Kamenetskii took t o % e t h e measure o f a c r i t i c a l s i z e above which spontaneous i g n i t i o n should take place. T h i s i s t r u e i f spontaneous i g n i t i o n i s understood t o mean t h a t s o l u t i o n s o f eqns. (42) o n l y e x i s t i f there i s a dead core w i t h

This was f i r s t solved by quadratures f o r t h e slab ( q = 0) and l a t e r Chambr6 (1952) found an a n a l y t i c a l s o l u t i o n f o r t h e c y l i n d e r ( q = 1). For the sphere no a n a l y t i c a l s o l u t i o n i s obtained, b u t t h e problem becomes p a r t i c u l a r l y i n t e r e s t i n g as t h e r e are i n f i n i t e l y many s o l u t i o n s f o r a c r i t i c a l value o f 6. The paper o f Hlavicek and Marek [ 8 ] gives the most complete treatment from t h e chemical engineering viewpoint.

(i) The i n f i n i t e f l a t p l a t e . t o eq. (42) namely

For the s l a b t h e r e i s an a n a l y t i c a l s o l u t i o n

,

eu = A sech2{p&A6/2))

(44)

and the boundary c o n d i t i o n can be matched i f

.

A = cosh2{,/(A6/2))

(45)

This provides an equation f o r A which has two s o l u t i o n s when 6

6, = 0.878

.

(46)

I f 6 = 6 there i s j u s t one s o l u t i o n w i t h u(0) = En A = 1.188 and hence a maximum temphature r i s e w i t h i n o f 1.2RTZ/E. This c r i t i c a l value o f 6 i s h e l d t o be a c r i t e r i o n f o r the c r i t i c a l s l a b thfckness above which no steady s t a t e s o l u t i o n can be found and i s i n t e r p r e t e d as the explosion l i m i t .

However t r u e t h i s may be, i n p r a c t i c e i t i s based on an incomplete a n a l y s i s o f t h e problem. Steady s t a t e s o l u t i o n s can be found f o r any value o f 6 b u t t h e To see t h i s we boundary conditions (43) w i l l have t o be invoked f o r 6 > 6., r e t u r n t o eqn. (42) and w r i t e t h e boundary c o n d i t i o n s as u(1) = 0

,

u(w) = uo,

u'(w) = 0

Then i f w = o

,

O,U,,BY

we have eqn. (42) and i f

I

.

(47)

259

NONLINEARITIES OF CHEMICAL REACTION ENGINEERING

l > w > O,

uo=py

we have eqn. (43).

M u l t i p l y i n g eqn. (42) f o r q = 0 by 2u'(p) and i n t e g r a t i n g we

have U

[U'(P)12 = 2*(euo-e )

,

(48)

and hence

-4Uo

= 2e

(49)

sech-'[ expi-+( uo-u(p)))]

The boundary c o n d i t i o n a t p = 1 i s s a t i s f i e d i f

4

-4u,

-4U

(26) (1-w) = 2e

'sech-'(e

)

,

which since A = euO i s j u s t a form o f eq. (48) when w = 0. written 6(1-w)'

= 2e-uo{sech-1(e -4u0)]2

and observe t h e form o f t h e curve r e l a t i n g S(1-w)

Figure 6:

L e t t h i s equation be

(51) 2

and uo as shown i n Fig. 6.

The s o l u t i o n o f eq. (51).

26a

R. A RIS

Now t h e equation as we have w r i t t e n i t i n eq. (42) takes no account o f the f a c t t h a t a r e a c t i o n cannot proceed when t h e r e a c t a n t bas been completely exhausted, i.e., when u exceeds py. Thus i n place o f e we should r e a l l y have e {l-H(u-py)), where H i s the Heaviside step f u n c t i o n . I f py > 1.188 the s i t u a t i o n i s as i n d i c a t e d by the upper h o r i z o n t a l l i n e i n Fig. 6 which i s drawn through the p o i n t . 6 ( 1 - ~ )=~ 6* = 2e-pY{sech-1(e -4Bu)]2

.

Since u cannot exceed py t h e r e i s o n l y one s o l u t i o n o f eqn. (51) when 6 < 6 * , namely 1 p o i n t on t h e p a r t OA o f t h e curve. There can be no s o l u t i o n w i t h w > 0 f o r 6 < 6*. I f 6* < 6 < 6 t h e r e a r e two s o l u t i o n s w i t h UFO on t h e branches AB and BC r e s p e c t i v e l y , ht t h e r e i s a t h i r d s o l u t i o n w i t h u = py and w > 0 on the l i n e CD. I f A > 6 then t h e r e i s no steady s t a t e s o l g t i o n w i t h UFO, b u t there i s always a s o l u t f o n on DE w i t h w > 0. I n t h e l a t t e r cases t h e value o f w i s given by

I f py < 1.188 then the h o r i z o n t a l l i n e i n Fig. 6 would have t o be drawn from a p o i n t n: the branch OB. I n t h i s case, t h e s o l u t i o n i s unique. Hlavlcek and Marek [ 8 ] appear t o be the f i r s t t o have recognized t h i s f e a t u r e o f r e a c t a n t exhaustion. The effectiveness f a c t o r

when t h e s o l u t i o n s a t i s f i e s t h e boundary c o n d i t i o n s w i t h w > 0 then

These curves are shown i n Fig. 7, which i s based on t h e c a l c u l a t i o n s o f H1avicek and Marek. [9]

./ ,A 2

I

Figure 7:

0.01

0.1

0.2

0.5

1

10

Effectiveness f a c t o r f o r Frank-Kamenetskii's equation i n t h e slab.

261

NONLINEARITIES OF CHEMICAL REACTION ENGINEERING

(ii)The i n f i n i t e c y l i n d e r . Chambr6 (1952) showed how t o o b t a i n an anaI n case w0 t h e l y t i c a l s o l u t i o n o f eq. (42) f o r the i n f i n i t e c y l i n d e r , q=l. substitutions P du q - -V ,

p2eu = w

(56)

are made, reducing the equation t o dv 6 ai+2+V=O

(57)

Since v = w = 0 a t the center, the immediate i n t e g r a l o f eq. (57) i s v2 + 4v + 26w = 0

(58)

or

But eq. (42) can be w r i t t e n

so t h a t by s u b t r a c t i o n

The equation can be i n t e g r a t e d t o give eU =

8B 6(Bp2+1)'

where B has t o be chosen t o s a t i s f y the boundary c o n d i t i o n a t the surface, i . e . S(B+l)'

= 88

.

(63)

This equation again has two s o l u t i o n s provided t h a t 6 < 6 = 2. The same type o f analysis goes through t h e r e l a t i o n between uo and 6 an% i s shown as one o f the three curves i n Fig. 8. eu = (p*/b)' w =

where b s a t i s f i e s

cosh2[1n{b+(b2-1)*]

- i},

+ a I n p]

a = b(6/2)

4

(64) (65)

262

R. A RIS

;

I

‘1

‘

\

u,

Figure 8:

Centre ‘temperature’, u as a f u n c t i o n o f 6(1-w)2 f o r s l a b c y l i n d e r (C), and spherg (5).

(L),

The effectiveness f a c t o r i s then

(iii) The s here. The i n t e g r a t i o n o f eq. (42) i n t h e sphere requires, as we have s h e previous section, a numerical i n t e g r a t i o n and leads t o the isothermal f u n c t i o n o r i t s generalizations. However, a very s i g n i f i c a n t d i f f e r e n c e a r i s e s when the ( u ,&) curve i s continued t o l a r g e r values o f u as i n the curve 5 o f Fig. 8. I t o i s seen t h a t t h e curve does n o t merely go thr8ugh a maximum o f 6, the so-called c r i t i c a l value 6 = 3.32, b u t t h a t i t does n o t come back a s y m p t o t i c a l l y t o t h e u - a x i s b u t o g c i l l a t e s about t h e l i n e 6=2. Steggerda has given t h e f i r s t e i g h t t u r n i n g p o i n t s :

6

3.3220

1.6641

2.1083

1.9666

2.0095

1.9957

2.0002

1.9992

uo

1.6075

6.7409

11.3769

16.1363

20.9159

25.3332

31.7950

35.6683

This means t h a t we can o b t a i n an a r b i t r a r i l y l a r g e number o f s o l u t i o n s by making py l a r g e enough and 6 close enough t o 2. For &=2 t h e r e i s a s i n g u l a r s o l u t i o n which we can o b t a i n as follows. Let

263

NONLINEARITIES OF CHEMICAL REACTION ENGINEERING

,

v = p -du+ 2 dP

w=u+21np

(68)

then n

dv = 2-v-6eL dw

v

This ordinary d i f f e r e n t i a l equation has a s i n g u l a r p o i n t a t v=O, w=Pn(2/6). w=ln(2/6) i s a c t u a l l y a s o l u t i o n o f

But

which though i t does n o t s a t i s f y the boundary c o n d i t i o n a t p=O, w i l l s a t i s f y u(1) = 0 i f 6 = 2. This i s Emden's s i n g u l a r s o l u t i o n , and t h e s o l u t i o n s f o r 6 = 2 w i t h u corresponding t o successive crossing p o i n t s o f the curve S o f Fig. 8 w i t h the v g r t i c a l l i n e 6 = 2 approaching more and more c l o s e l y t o the s i n g u l a r s o l u t i o n , departing only i n the neighborhood o f p = 0. Since i t i s the sphere t h a t f i r s t shows the phenomenon o f i n f i n i t e m u l t i p l i c i t y , i t i s i n t e r e s t i n g t o i n q u i r e what behavior the s o l u t i o n f o r a f i n i t e c y l i n d e r might e x h i b i t . A t one extreme the c o i n t h i s i s a f l a t p l a t e ; a t t h e other the c u r t a i n rod i t approaches the i n f i n i t e c y l i n d e r . On t h e other hand, a c y l i n d e r w i t h equal h e i g h t and diameter has the same surface t o volume r a t i o as a sphere and one i s n a t u r a l l y l e d t o ask i f i t gives r i s e t o an i n f i n i t y o f solutions when the parameter i s

-

v2u

+ heU =

-

-

-

o

has some c r i t i c a l value. The most complete r e s u l t s o f t h i s k i n d have been given by V i l l a d s e n and Ivanov They showed [20] who t r e a t e d the exponent q i n eq. (42) as a "geometry factor.'' t h a t an i n f i n i t e number o f steady s t a t e s p r e v a i l e d when 1 < q < 9. No more than three e x i s t f o r q < 1 o r q > 10.44 and o n l y a f i n i t e number f o r 9 5 q 5 10.44. Thus i t would seem-that when the p e l l e t i s the l e a s t b i t more s p h e r i c a l ' than i t i s ' c y l i n d r i c a l ' ( i . e . , q = 1+&), t h e r e can be an i n f i n i t e number o f solutions. An exact p r o o f t h a t the p o s i t i v e exponential approximation does n o t change t h e p i c t u r e q u a l i t a t i v e l y has n o t been given f o r the case o f d i f f u s i o n and r e a c t i o n though there are two d i r e c t i o n s from which i t may be approached f o r the w e l l mixed reactor. Since there are grounds f o r b e l i e v i n g t h a t the q u a l i t a t i v e behav i o r o f the c a t a l y s t p a r t i c l e has some a f f i n i t y w i t h t h a t o f t h e s t i r r e d tank, i t may be o f i n t e r e s t t o comment on t h i s here [2]. One such discussion i s t o be found i n Poorels paper i n the Archive f o r Rational Mechanics and Analysis [16]. He studies the phase plane o f the s t i r r e d tank and shows t h a t i t i s closed i n both f o r the Arrhenius f u n c t i o n and t h e p o s i t i v e exponential. Another approach has been provided by Golubitsky and K e y f i t z [7], i n t h e i r " q u a l i t a t i v e study o f the steady s t a t e s s o l u t i o n s f o r a continuous f l o w s t i r r e d tank chemical reactor." Their argument turns on a c a r e f u l a n a l y s i s o f t h e p r o p e r t i e s o f the organizing center o f t h i s p a r t i c u l a r problem and they o b t a i n formulae appropriate t o a s i n g u l a r i t y they c a l l t h e 'winged cusp. ' T h e i r arguments t u r n on the behavior o f the Arrhenius expression and t h e i n e q u a l i t i e s which i t s d e r i v a t i v e s s a t i s f y . The p o s i t i v e exponential approximation could be obtained from t h e i r f u n c t i o n by r e p l a c i n g yy by v and t a k i n g t h e l i m i t o f y

264

R. ARlS

going t o i n f i n i t y . The i n e q u a l i t i e s which they f i n d necessary t o impose are s a t i s f i e d equally w e l l by t h i s l i m i t as by t h e more complicated f u n c t i o n . I t would, therefore, seem c l e a r t h a t the q u a l i t a t i v e behavior o f the two systems i s identical. ACKNOWLEDGEMENT This paper was prepared w h i l s t the author had t h e p r i v i l e g e o f being a Sherman F a i r c h i l d Scholar a t t h e C a l i f o r n i a I n s t i t u t e o f Technology. Anyone who knows Caltech need o n l y be t o l d t h a t i n the generosity o f i t s s t r u c t u r e , t h i s program f u l l y r e f l e c t s t h e excellence o f t h a t i n s t i t u t i o n . References and Schilson, R. E. (1961) " I n t r a p a r t i c l e D i f f u s i o n and Conduction i n Porous Catalysts," Chem. Engrg. Sci. 13,226, 237.

[l] Amundson, N. R.,

[2] A r i s , R. (1969), "On S t a b i l i t y C r i t e r i a o f Chemical Reaction Engineering,'' Chem. Engrg. Sci. 25, 149. [3] A r i s , R. (1974), "The Theory o f D i f f u s i o n and Reaction neering Symphony," Chem. Engrg. Educ. 8, 19.

-

A Chemical Engi-

[4] A r i s , R.

(1975), "The Mathematical Theory o f D i f f u s i o n and Reaction i n Permeable Catalysts" (2 vols.), Clarendon Press, Oxford.

[5] Frank-Kamenetskii , D. A. (1939), " C a l c u l a t i o n o f Thermal Explosion Limits," Acta phys. - chim. URSS 0 , 365. [6] Gelfand, I. M. (1969), "Some Problems i n the Theory o f Q u a s i l i n e a r Equations," Usp. Mat. Nauk XV, No. 2, (86), 87. Trans. Amer. Math. SOC. (1963), (2), 9,295. [7] Golubitsky, M.,

and K e y f i t z , 6. L. (1980), "A Q u a l i t a t i v e Study o f t h e Steady-State Solutions f o r a Continuous S t i r r e d Tank Chemical Reactor," S I A M J. Math. Anal. 316.

11,

, and Marek, M. (1968), "Heat and Mass Transfer i n a Porous C a t a l y s t P a r t i c l e . On t h e M u l t i p l i c i t y o f Solutions f o r t h e Case o f an Exothermic Zeroth-order reaction. 'I C o l l Czech. Chem. Commun. (Eng. Edn. )

[8] Hlavikek, V.

.

33, 506.

[9] Hlavicek, V.,

and Marek, M. (1968), "Zum Entwurf k a t a l y t i s c h e r Rohrreakt o r e n bei r a d i a l e n Warmetransport," Proc. I n s t . Chem. Engrs. and V.T.G.V.D.I. Meeting, Brighton 1968.

[lo] Jackson, R.

, "Diffusion

i n Porous Media" (1977), E l s e v i e r , New York.

and Lundgren, T. S. (1973), " Q u a s i l i n e a r D i r i c h l e t Problems Driven by P o s i t i v e Sources," Arch. Rational Mech. Anal. 9, 241.

[ll] Joseph, D. D.,

[12] Luss, D. (1971), "Uniqueness c r i t e r i a f o r lumped and d i s t r i b u t e d parameter chemically r e a c t i n g systems , I ' Chem. Engrg. Sci. 26, 1713. [13] Nicolaenko, B. (1977), "A General Class o f Nonlinear B i f u r c a t i o n Problems from a P o i n t i n the Essential Spectrum. A p p l i c a t i o n t o Shock Wave Solut i o n s o f K i n e t i c Equations," A p p l i c a t i o n s o f B i f u r c a t i o n Theory, Academic Press, New York, 1977.

NONLINEARITIES OF CHEMICAL REACTION ENGINEERING

265

[14] Nicolaenko, B., and Brauner, C. M. (1979), "Singular Perturbations and Free Boundary Value Problems," Proc. I V t h I n t e r n a t i o n a l Symposium on Computing Methods i n Applied Sciences and Engineering. I.R.I.A. V e r s a i l l e s , 1979, North-Holland Publishing.

[15] Nicolaenko, B. and Brauner, C. M. (1979) "Nonlinear Eigenvalue Problems which Extend i n t o Free Boundary Problems," Proc. Conf. on Nonlinear Eigenvalues. Springer-Verlag Lecture Notes i n Mathematics No. 782.

[XI Poore, A. B. (1973), " M u l t i p l i c i t y , S t a b i l i t y and B i f u r c a t i o n o f P e r i o d i c Solutions i n Problems a r i s i n g from Chemical Reactor Theory," Arch. Rational Mech. Anal. 52, 358.

[17] Rhee, H-K.,

A r i s , R . , and Amundson, N. R. (1971), "Multicomponent Adsorpt i o n i n Continuous Countercurrent Exchangers," P h i l . Trans. Roy. SOC. A.

269, 187.

[18] Stephanopoulos, G. (1980), "Zero Concentration Surfaces i n a C y l i n d r i c a l Catalyst Pellet,'' Chem. Engrg. Sci. 3, 2345. [19] Villadsen, J., and Michelsen, M. Spherical Catalyst P e l l e t s : Chem. Engrg. Sci. 3 , 751.

L. (1972), " D i f f u s i o n and Reaction on Steady-state and Local S t a b i l i t y Analysis,''

[20] Villadsen, J., and Ivanov, E. (1978), "A note on t h e m u l t i p l i c i t y o f t h e Weisz-Hicks problem f o r an a r b i t r a r y geometry factor,"

33, 41.

Chem. Engrg. Sci.

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Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko Ids.) 0 North-Holland Publishing Company, 1982

267

PROPAGATING FRONTS I N REACTIVE MEDIA Paul C. F i f e Mathematics Department U n i v e r s i t y of Arizona Tucson, AZ 8 5 7 2 1

Wavelike phenomena of many d i f f e r e n t t y p e s can be modeled w i t h t h e a i d of systems of p a r t i a l d i f f e r e n t i a l e q u a t i o n s of convection-diffusion-reaction type. Combustion t h e o r y provides a t y p i c a l s e t t i n g f o r t h e s e e q u a t i o n s , but t h e y a r e used i n a g r e a t number of o t h e r c o n t e x t s as w e l l . I shall concentrate on onedimensional steady-state A review ( n e c e s s a r i l y t r a v e l i n g f r o n t solutions. incomplete) w i l l be made of a number of important advances i n (1) asymptotic and o t h e r methods f o r r e d u c i n g t h e complexity of given systems, and ( 2 ) more r i g o r o u s r e s u l t s on s y s t e m of one o r two equations. INTRODUCTION

The s u b j e c t of my t a l k , propagating f r o n t s , h a s t o do w i t h a phenomenon observed i n a wide v a r i e t y of c o n t e x t s p h y s i c a l , chemical, and b i o l o g i c a l . The analogy i s more t h a n s u p e r f i c i a l : t o a g r e a t e x t e n t , s i m i l a r concepts and t o o l s can be used both i n modeling and i n a n a l y z i n g wave phenomena i n a l l t h e s e disciplines. I s h a l l of c o u r s e touch on some, p o s s i b l y t h e most b a s i c , of t h e "common ground" i n t h i s l e c t u r e ; but nuch mst n e c e s s a r i l y be l e f t unsaid.

-

F i r s t , t h e general picture. The p r o t o t y p i c a l example of f r o n t s i n r e a c t i v e media i s a flame, and a couple of my c a s e s t u d i e s w i l l be drawn from flame theory. But more a b s t r a c t l y , one can simply envisage a medium continuously d i s t r i b u t e d i n space which can e x i s t , a t each p o i n t i n space, i n a v a r i e t y of p o s s i b l e s t a t e s , d e s c r i b e d by an n-dimensional vector. These s t a t e s a r e e assume t h e r e are dynamic, i n t h a t they can change a c c o r d i n g t o known r u l e s . W a t l e a s t two r e s t s t a t e s a v a i l a b l e ; t h e s e a r e states which w i l l not change u n l e s s p e r t u r b e d by some i n f l u e n c e beyond t h e l o c a l dynamics. Suppose t h e e n t i r e medium e x i s t s i n i t i a l l y i n one of t h e a v a i l a b l e rest s t a t e s , and t h a t a l o c a l " i g n i t i o n " event happens: i n a l i m i t e d r e g i o n of space, t h e medium i s taken i n t o t h e domain of a t t r a c t i o n of a n o t h e r rest s t a t e . This p e r t u r b a t i o n , and t h e r e s u l t a n t a t t r a c t i o n t o t h e o t h e r s t a t e , may a f f e c t nearby regions i n t h e medium through some t r a n s p o r t mechanism, and t h e s e nearby r e g i o n s a r e induced t o undergo a s i m i l a r t r a n s i t i o n between rest s t a t e s . This i s t h e beginning of a chain r e a c t i o n , or domino e f f e c t . L e t us c a l l i t a propagating front. I n t h e above p i c t u r e , a f r o n t w i l l propagate i n a l l d i r e c t i o n s from t h e i g n i t i o n s p o t , but I would l i k e t o l i m i t t h e concept t o t h e case of a u n i d i r e c t i o n a l f r o n t i n a o n e d i m e n s i o n a l medium. With x denoting p o s i t i o n t h e s t a t e of t h e system, t h e state dynamics and t r a n s p o r t and u ( x , t ) e Rn mechanism a r e assumed t o be d e s c r i b e d by a system of p a r t i a l d i f f e r e n t i a l e q u a t i o n s of t h e t y p e ut

+

(f(u)),

-

( D ( U ) U , ) ~+ g ( u )

(1)

268

P.C. FIFE

(D i s a matrix). D e t e r m i n i s t i c continuous space-time models of t h e s i t u a t i o n described above almost always may be put i n t o e i t h e r t h i s form o r t h a t of a n i n t e g r o - d i f f e r e n t i a l equation. The second term on t h e l e f t of (1) accounts f o r "convection", and those on t h e r i g h t are t h e " d i f f u s i v e t r a n s p o r t " and " r e a c t i o n " terms. We c a l l D t h e d i f f u s i o n matrix. Fronts a r e now defined t o be s o l u t i o n s of t h e form v e l o c i t y c, s a t i s f y i n g

-

u(+)

uf

u = u(x-ct) = u(z)

f o r some

(bounded d i s t i n c t l i m i t s )

The f r o n t phenomenon is e s s e n t i a l l y nonlinear, because no such s o l u t i o n s e x i s t when f and g a r e l i n e a r f u n c t i o n s of u ( a t l e a s t when D i s a c o n s t a n t ) . I s h a l l consider only smooth s o l u t i o n s (and discontinuous shocks). Front s o l u t i o n s s a t i s f y -cuZ

+

(f(u)IZ u(*)

so

will

exclude

strictly

( D ( U ) U ~+) ~ g(u)

(24

=

(2b)

Uf

The following b a s i c q u e s t i o n s about f r o n t s arise: a. b. c. d.

For which u+, u-, and c do t h e r e e x i s t f r o n t s ? What i s t h e s t r u c t u r e of t h e f r o n t ' s p r o f i l e ? Is t h e f r o n t s t a b l e , (I) i n one space dimension (11) when imbedded i n a h i g h e r dimensional space? If u n s t a b l e , what o t h e r s t a b l e s t r u c t u r e s may appear?

The emphasis h e r e w i l l be on ( a ) and ( b ) , and t o s6me e x t e n t ( c ) . be i n two p a r t s :

The t a l k w i l l

I. Techniques ( p r i m a r i l y asymptotic) t o reduce t h e cornplexity of systems. 11. Some of t h e more d e t a i l e d and r i g o r o u s r e s u l t s t h a t have been obtained f o r t h e simplest systems (one o r two equations). I have not a t t e n p t e d t o c o q i l e complete r e f e r e n c e l i s t s f o r r e s u l t s similar t o those described here, o r even t o t r a c e t h e o r i g i n of many of t h e i d e a s o u t l i n e d . Moreover, t h e v a s t amount of e f f o r t on numerical approaches w i l l n o t be mentioned f u r t h e r . There a r e c e r t a i n immediate necessary c o n d i t i o n s f o r t h e e x i s t e n c e of a f r o n t which should be made c l e a r i n t h e s e i n t r o d u c t o r y comments. I n t e g r a t i n g (2a) from to -, one o b t a i n s

-

m

-

-CAU

+ Af(u)

g(u(z))dz

(3)

4

-

where Au u u-. lhis i s a condition r e l a t i n g u+,u-, and c, but i t i s not t o o usefuf i f t h e integrand g ( u ( z ) ) i s not known. A t y p i c a l occurence i s t h a t some components of of u accordingly:

g

vanish.

I s h a l l , i n f a c t , s e g r e g a t e t h e cornponents

ui

i s a p h y s i c a l v a r i a b l e i f g,(u)

f

0;

ui

i s a chemical v a r i a b l e i f gi(u)

1

0.

(The number of each k i n d i s not an i n v a r i a n t w i t h r e s p e c t t o transformations of (1) i n t o equivalent systems.)The s e t of p h y s i c a l v a r i a b l e s I denote by , t h e chemical ones by ; then

PROPAGATING FRONTS IN REACTIVE MEDIA

u

-

-

Af(u)

From (3) we o b t a i n A

d u and

A

-

(us u),

and s i m i l a r l y

f

-

1

269

-

(f, f).

1

(Rankine-Hugoniot

(4)

relations)

(5)

g(u*) * 0 In all, these a r e

n

equations r e l a t i n g

u+

and

c.

Though n e c e s s a r y , t h e i r

s a t i s f a c t i o n is by no means s u f f i c i e n t f o r t h e e x i s t e n c e of a f r o n t . I. REDUCING A SYSTEM'S COMPLEXITY I s h a l l proceed by o u t l i n i n g f o u r examples of asymptotic methods by which l a r g e r systems may be reduced t o a combination of smaller systems. This n o t o n l y r e s u l t s i n a n e a s i e r a n a l y s i s of t h e o r i g i n a l system, b u t i t a l s o c l a r i f i e s some q u a l i t a t i v e f e a t u r e s of t h e f r o n t ' s p r o f i l e . These f o u r examples are meant t o exemplify f o u r c a t e g o r i e s of approaches, each r e a l l y c o n p r i s i n g many p o s s i b l e And indeed i n some v a r i a t i o n s o r r e l a t i v e s which w i l l n o t a l l be mentioned. c a s e s many s i g n i f i c a n t r e l a t e d t r e a t m e n t s have appeared i n t h e l i t e r a t u r e . Again, my i n t e n t i o n i s n o t t o g i v e a complete survey.

I s h a l l attempt t o d i s r e g a r d d e t a i l s which do not bear d i r e c t l y on t h e mathematical s t r u c t u r e of t h e argument being presented; t h i s is done i n t h e i n t e r e s t of s a v i n g time and of e l u c i d a t i n g t h a t b a s i c s t r u c t u r e i t s e l f by doing away w i t h confusing s i d e i s s u e s . For example, I s h a l l n o t p o i n t o u t t h e v a r i o u s dimensionless c o n s t a n t s u s u a l l y a s s o c i a t e d w i t h combustion problems. F i n a l l y , t h e s i t u a t i o n s d e s c r i b e d i n v o l v e a small parameter E, and t h e methods E + 0. can be c l a s s e d a s lowest o r d e r formal asymptotic methods as Approximations which a r e h i g h e r o r d e r i n E can be o b t a i n e d , though sometimes with difficulty. 1. ZND and C J d e t o n a t i o n a n a l y s i s .

The names a s s o c i a t e d w i t h t h e T h i s i s t h e o l d e s t and s i m p l e s t of t h e examples. i n i t i a l s are Zeldovi& von Neumann, Doering, Chapman, and Jouguet. The f i r s t t h r e e developed t h e i r approaches t o d e t o n a t i o n s independently of one a n o t h e r , The CJ models go back f u r t h e r . There a r e around t h e t i m e of World War 11. s e v e r a l p o s s i b l e avenues t o t h e study of ZND d e t o n a t i o n s (see Williams (1965), f o r example). The approach I t a k e i s simply t o assume t h e t r a n s p o r t m a t r i x D is small, and proceed by formal asymptotics. Therefore r e p l a c e D by ED i n (21, where E << 1. A f r o n t can be c o n s t r u c t e d an a b r u p t p a r t (shock) a t z = 0 s e p a r a t i n g a f o r e and a n a f t region. We assume u+ (the s t a t e of t h e medium i n t o which t h e f r o n t advances) t o be given, and t h e d e t e r m i n a t i o n of c and u- t o be p a r t of t h e problem.

in t h r e e p a r t s :

(1) Ahead of t h e shock, s e t u z u+

,z >

0.

C l e a r l y , by (5) t h i s c o n s t a n t is a s o l u t i o n of (2a). 5

-

(2)

The shock has p r o f i l e determined by s t r e t c h i n g t h e v a r i a b l e u(z) = U ( C ) , so t h a t

Z/E,

2.

Set

270

P.C. FIFE

Neglecting t h e O(E) c o r r e c t i o n , one discovers t h a t t h e r e a c t i o n terms have vanished. The reduced problem is t h e r e f o r e l i k e a p h y s i c a l v a r i a b l e problem, and t h e Rankine-Hugoniot necessary c o n d i t i o n s apply: c(U+

- u-) + f(U+) -

f(UJ

-

0.

(7)

Since U+ = u+ is known, t h i s is a n equation r e l a t i n g U- t o c. Let u s assume t h a t t h e r e e x i s t s a t l e a s t one s o l u t i o n U- of ( 7 ) f o r each c i n some parameter range I, and moreover t h a t t h e r e e x i s t s an e x a c t s o l u t i o n of (6) connecting U+ t o U-. Then t h e r e w i l l be a one-parameter family of p o s s i b l e shock p r o f i l e s U(s) , with v e l o c i t y c being t h e parameter.

In cases of most p h y s i c a l i n t e r e s t , t h e system ( 7 ) can be reduced t o t h e w e l l known Rankine-Hugoniot system in t h e p h y s i c a l v a r i a b l e s d e n s i t y , momentum, and energy. In f a c t , assume t h a t t h e s e a r e t h z p h y s i c a l v a r i a b l e s i n q u e s t i o n , and t h a t t h e chemical v a r i a b l e s are given by U = pZ w i t h f = p a , where Z i s t h e vector of mass f r a c t i o n s of t h e c h e d x a l s p e c i e s , v A i s t h e f l u i d v e l o c i t y ; If p = U,, s a y , t h e n f l = pv. L e t us write and p i s t h e t o t a l d e n s i t y . h

f(U) as a f u n c t i o n of U and 2: f(^U,Z). TheJollowing f a c t can be e a s i l y v e r i f i e d : I f , f o r some constant Z+, t h e v e c t o r s U* s a t i s f y

then ( 7 ) is s a t i s f i e d w i t h t h e s e given

and w i t h

U+,

U* = p*Z+.

On t h e Rankine-Hugoniot l e v e l , then, t h e problem reduces t o (8) with Z+ given by t h e conditions ahead of t h e shock. This i s an elementary and well-studied problem. On t h e higher l e v e l of t h e f u l l equations ( 6 ) , l e t us simply assume t h e e x i s t e n c e of a s o l u t i o n connecting U+ and U-. If t h e r e is no s p e c i e s DUc d i f f u s i o n (more s p e c i f i c a l l y , i f t h e chemical components of t h e v e c t o r vanish whenever Z c o n s t a n t ) , then such an e x i s t e n c e r e s u l t follows from a r e s u l t of Gilbarg (1951), t o be mentioned l a t e r . A l l t h i s r e s u l t s in a onew i t h U(-) = U-, depending o n ' t h e parameter family of shock p r o f i l e s U(c) parameter c.

-

(3) Behind t h e shock, we r e v e r t t o t h e o r i g i n a l v a r i a b l e s , s o (6a) a p p l i e s w i t h D replaced by ED. This e q u a t i o n i s t o hold f o r z < 0 To match w i t h t h e shock p r o f i l e considered above, i t i s necessary t h a t U-. Now set E = 0 t o o b t a i n a n ordinary d i f f e r e n t i a l equation i n i t i a l u(0) value problem:

.

-

-cu

+

(f(u))2 = g(u), u(0) =

u-

.

2

<

0

(9a) (9b)

Since g f i g u r e s i n t o t h i s problem, t h e r e g i o n behind t h e shock i s t h e r e a c t i o n zone. In t y p i c a l c a s e s , t h e s o l u t i o n of t h i s problem w i l l approach a rest s t a t e u- a s z + This f i n a l s t a t e depends on c, but i n any c a s e w i l l s a t i s f y ( 4 ) and (5).

-.

I n summary, t h e problem of c o n s t r u c t i n g a f r o n t has been reduced t o (1) a shock problem f o r t h e p h y s i c a l v a r i a b l e s o n l y , p l u s (2) an i n i t i a l value problem f o r a f i r s t o r d e r ordinary d i f f e r e n t i a l equation in a r e a c t i o n zone behind t h e shock. This generally y i e l d s a one-parameter family of f r o n t s . The next problem would be t o decide which one of t h e s e a c t u a l l y occurs; t h e answer t o t h i s q u e s t i o n depends on o t h e r a s p e c t s of t h e phenomenon (experimental c o n d i t i o n s , e t c . ) being These o t h e r a s p e c t s , i n f a c t , u s u a l l y do s e r v e t o s e l e c t a unique modelled. Loosely r e l e v a n t candidate from among t h e c o l l e c t i o n w e have constructed.

PROPAGATING FRONTS IN REACTIVE MEDIA

271

speaking, i f t h e r e a r e no e x t r a physical c o n s t r a i n t s imposed a t z = 4 , t h e s o - c a l l e d Chapman-Jouguet values of u- and c are t h e r e l e v a n t ones.

then

A f u r t h e r s i m p l i f i c a t i o n of t h i s a n a l y s i s produces t h e e a s i e s t of t h e detonation models: t h a t of Chapman and Jouguet. For t h i s one assumes t h e r e a c t i o n rate g is l a r g e , but not s o l a r g e as t o overshadow t h e smallness of t h e t r a n s p o r t term. Then replace g by kg, where k >> 1 but kE << 1. The only e f f e c t of t h i s i n t h e above a n a l y s i s is t o a c c e l e r a t e t h e e q u i l i b r a t i o n process behind t h e shock. In f a c t , with t h i s parameter k introduced, one may d e f i n e a new space coordinate 11 kz, so t h a t (9a) becomes

-

-cu Equilibration w i l l magnitude 1 , hence Jouguet model takes a r e both compressed

n + (f(U)),

= g(d,

<

0.

occur over a d i s t a n c e i n t h e v a r i a b l e n, of t h e o r d e r of a distance, i n 2 , of t h e o r d e r l / k << 1. The Chapmant h i s d i s t a n c e t o be 0, so t h a t t h e r e a c t i o n zone and shock t o a s i n g l e point.

2. High a c t i v a t i o n energy asymptotics f o r flames and chemical r e a c t o r s .

Asymptotics of t h i s s o r t were apparently f i r s t used (independently) by BushFendell (1970) and Lican (1971). Many people, i n c l u d i n g Buckmaster, Kapila, Ludford, Margolis, Matkowski, Sivashinsky, van Harten and W i l l i a m s , have s i n c e done notable work i n t h e a r e a . The most e s s e n t i a l assumptions on which t h e method i s based involve t h e s t r o n g dependence of t h e r e a c t i o n f u n c t i o n g on t e n p e r a t u r e , which i s t a k e n t o be one of t h e "chemical" v a r i a b l e s . This dependence i s expressed by w r i t i n g g = g(+),

E

<<

1.

(10)

This, of course, says nothing i n i t s e l f , because any f u n c t i o n can be w r i t t e n t h i s way. (Xlr main assunption concerns t h e c h a r a c t e r of t h e dependence of g The form (10) suggests on t h e two arguments, and w i l l be s p e l l e d out l a t e r . that g, i n f a c t , depends s t r o n g l y on a l l t h e o t h e r s t a t e v a r i a b l e s a s w e l l . Again, t h i s form involves no l o s s of g e n e r a l i t y because of t h e ( a s y e t ) a r b i t r a r y dependence of g on t h e second v a r i a b l e , and I use i t f o r convenience.

My d e s c r i p t i o n of t h e method proceeds under s e v e r a l i n e s s e n t i a l ( o r only p a r t l y e s s e n t i a l ) assumptions, besides t h e p r i n c i p a l one. The method can be extended t o cases when these i n e s s e n t i a l ones, which a r e l i s t e d below, are relaxed t o some e x t e n t .

.

(1) The equations f o r t h e physical v a r i a b l e s u do not depend on u This assumption means, f o r i n s t a n c e , t h a t t h e h e a t generated by t h e chemical r e a c t i o n does not a f f e c t t h e d e n s i t y very mch. Under t h e s e conditions, G- any constant w i l l be a s o l u t i o n of t h e physical equations, and t h a t i s t h e s o l u t i o n we choose. Since 0 is a constant, w e disregard any dependence on 4 i n t h e considerations below. Thus u c o n s i s t s only of chemical v a r i a b l e s .

-

(2) The chemistry r e s u l t s from a s i n g l e exothermic r e a c t i o n A + B + C

Besides t h e tenperature T , t h e only relevant chemical v a r i a b l e s a r e t h e concentrations cA and cB of A and B. The s t a t e v e c t o r u has t h r e e conponents: (u1,u2,u3) = (T, cA, cB). Furthermore, t h e s i n g l e r e a c t i o n r e s u l t s i n a "g" of t h e s p e c i a l form

272

P.C. FIFE

where

$I

K a constant 3 ~ e c t o r .

is a s c a l a r r a t e f u n c t i o n and

( 3 ) D is i n v e r t i b l e and p o s i t i v e d e f i n i t e . diagonal, o r even c o n s t a n t . )

( I t is =necessary

that

D

be

-

N

f = f = 0. Usually f qui , q being t h e v e l o c i t y , supposed c o n s t a n t i n t h i s case. The c o e f f i c i k n t q can be absorbed i n t o t h e d e f i n i t i o n

(4)

of

c,

so t h e f o u r t h assumption i n v o l v e s no g r e a t l o s s of g e n e r a l i t y .

( 5 ) $I v a n i s h e s i f and o n l y if CA o r cB is zero. (Mathematically, i t may be convenient t o allow n e g a t i v e c o n c e n t r a t i o n s ; t h e n we assume $I = 0 f o r CA c 0 o r CB < 0).

-

It w i l l t u r n o u t t h a t uis c o n p l e t e l y determined from t h e knowledge of u+ -, t h e r e a c t i o n h a s gone t o completion i n t h e and t h e a s s u n p t i o n t h a t a t z s e n s e t h a t one of t h e cA o r cB is zero, t h e o t h e r > 0. I s h a l l return t o t h i s p a r t l a t e r ; f o r t h e moment, assume t h e u+ i s p r e s c r i b e d and u- t h e r e b y determined (we have y e t t o f i n d c ) .

The e q u a t i o n s are now

Preliminary t o t h e a n a l y s i s , one r e s c a l e s so t h a t t h e r e a c t i o n f u n c t i o n is t h e c o r r e c t o r d e r of magnitude. For t h i s , i t is a p p r o p r i a t e ( 1 ) t o e x p r e s s $I i n terms of u ur a t h e r than u , and (11) t o formally s e p a r a t e o u t t h e Edependence a s follows:

-

In t y p i c a l c a s e s , t h i s s e p a r a t i o n is s t r a i g h t f o r w a r d and d e f i n e s $ and II uniquely, up t o o r d e r of magnitude. 'Ihe d e t a i l s f o r Arrhenius t e n p e r a t u r e Now d e f i n e dependence w i l l be given a t t h e end of t h i s s e c t i o n .

z ' = n

$

I

d r o p t h e primes, and n e g l e c t t h e o ( 1 ) term i n (12):

11-1/2 c,

(11) becomes

This is t h e b a s i c e q u a t i o n t o be analyzed. We now come t o t h e h e a r t of high a c t i v a t i o n energy asymptotics, and t h e main as sunp t ion: E s s e n t i a l assumption: and u+,

-

For any state

-

-

0

for s

<

0.

For

conponentwise s t r i c t l y between

u-

A t z = -, t h e g a s i s burned, so CA o r Hence a c c o r d i n g t o assumption (5) above,

An i n t u i t i v e e x p l a n a t i o n i s needed. cB 0; i.e., u2or u30. $((uo-u-)s)

uo,

8

>

0,

$((uo-u-)e)

i s p a r t l y a measure of

t h e r a t e of t h e r e a c t i o n i n i t s dependence on t e n p e r a t u r e , r e l a t i v e t o i t s r a t e a t t h e maximm temperature ul-. I f one t h i n k s of s = 1 / ~ ,(14) and ( 1 5 ) simply s a y t h a t f o r f i x e d t e n p e r a t u r e less t h a n t h e maximum, t h i s r e l a t i v e r a t e f a l l s o f f " r a p i d l y enough" as E + 0.

273

PROPAGATING FRONTS IN REACTIVE MEDIA

The f r o n t is c o n s t r u c t e d in t h r e e p a r t s .

4

is z e r o when CA o r CB = 0, and t h i s is t h e c a s e when u = u+ uTherefore t h e c o n s t a n t u = u- (burned g a s ) is i t is t r u e t h a t 4 ( , ~E ) = 0. a s o l u t i o n of (13). Take t h i s as our s o l u t i o n f o r z < 0. ( a ) Since

-

( b ) Near z 0, t h e r e is burning. It occurs mainly near u = u-, because o f t h e high a c t i v a t i o n energy. Going under t h e assumption t h a t t h e flame is narrow, one s t r e t c h e s v a r i a b l e s t o analyze i t s i n t e r n a l s t r u c t u r e : u-uc ' TZ , v = - . I n t h e s e new v a r i a b l e s , t h e e q u a t i o n (13) becomes, t o lowest o r d e r ,

c c + $(v)K =

D v 0

0, v ( 4 ) = 0

,

(16)

where D = D(u-). There a r e two l i n e a r l y independent perpendicular t o K. Let wi = MioDv. 'hen (16) i m p l i e s

and s i n c e

wi(-)

-

=

0,

vectors

MI,

M2

O, wi E 0, i=1,2.

i t follows t h a t

I n s i d e t h e flame,

t h e r e f o r e , t h e r e a r e two l i n e a r l y independent r e l a t i o n s s a t i s f i e d by t h e couponents of V. This means t h a t t h e v e c t o r v is c o l l i n e a r w i t h a s i n g l e c o n s t a n t v e c t o r V, which is p e r p e n d i c u l a r t o both t h e v e c t o r s MID,. Clearly V

Thus

V = DO1 K.

may be t a k e n t o be t h e v e c t o r

v = a(c)V.

(17)

When (17) is s u b s t i t u t e d i n t o ( 1 6 ) , a s i n g l e e q u a t i o n f o r

a" This determines a ( c ) i n t e r e s t is a'(-), i n t e g r a t i n g from

-

+ $(aV)

-

a(-=) = 0

0,

a

-

uniquely, up t o t r a n s l a t i o n , f o r which can be found by n u l t i p l y i n g t o -: 0 a'(-) = ( 2 $(sV)ds)"2 E y.

-

I

emerges: (18)

<

r;

<

(18)

-.

by

Of g r e a t e s t

a'

and

(19)

This, t o g e t h e r w i t h (17). g i v e s t h e jump d i s c o n t i n u i t y s u f f e r e d by t h e d e r i v a t i v e of u a c r o s s t h e narrow flame. S p e c i f i c a l l y , u,(O+)

= v

c

= a'(-)V

(m)

= yV.

(20)

( c ) For z > 0, w e r e v e r t t o (13). Since i n t h i s r e g i o n u is s t r i c t l y assumption (14) s a y s t h a t t h e r e a c t i o n term i n (13) i s between uand u+, for E. There r e s u l t s t h e f o l l o w i n g boundary v a l u e problem:

(DUz),

+ cuz

-

u ( 0 ) = u-, u(-) U,(O)

0,

-

z

> 0;

u+,

= YV,

(21a) (2 1b) (21c)

t h e l a s t coming from (20). There being t o o many boundary c o n d i t i o n s , t h i s problem is overdetermined and h a s a s o l u t i o n o n l y if a c o p p a t i b i l i t y c o n d i t i o n is s a t i s f i e d . To determine what t h a t c o n d i t i o n is, i n t e g r a t e ( 2 l a ) once t o o b t a i n Lhz

+ cu

= c o n s t = cu+.

274

At

P.C. FIFE

z = 0,

t h i s gives

I n view of ( 2 1 c) , t h e c o n d i t i o n i s t h e r e f o r e

- u-).

yDoV = c(u+

( 2 2)

Although t h i s i s a v e c t o r e q u a t i o n , i t t u r n s o u t t h a t t h e r e i s a s i n g l e v a l u e of c f o r which i t i s s a t i s f i e d . To see t h i s , t a k e t h e scalar p r o d u ct of (13) w i t h t h e two v e c t o r s Mi o r t h o g o n a l t o K. After i n t e g r a t i n g once, one f i n d s cMii u

The

v al u es

M .(u+-u-) 1

at = 0,

z = f

rmst

m

which means t h a t

c known, t h e p r o f i l e

-

u

for

+

u

-

.

The p o s i t i v e d e f i n i t e n e s s of s o l u t i o n w i t h u(-) u+

hz = co n st .

be

same,

the

T h e r e f o r e (22)

K.

>

z

0

-

uz

0

there,

so

K:

y = cb.

becomes

This

c an be determined by s o l v i n g

-1 cD ( u ) ( u u + )

0,

u(0) = u-

and p o s i t i v i t y of

D

and

i s a m l t i p l e of

(u+-u-)

D,V

Recall t h a t by d e f i n i t i o n , de t er m i n es t h e v e l o c i t y C. With

+ Mio

c

.

e n s u r e t h e e x i s t e n c e of a

A t t h i s p o i n t I would l i k e t o do two t h i n g s : go back and j u s t i f y t h e claim t h a t u- can be determined beforehand, and s e c o n d ly go t h r o u g h t h e above c o n s t r u c t i o n of J, f o r Ar r h en i u s te m p e r a t u r e dependence.

a l l w e need i s The f i r s t is eas y . From (23) i t i s c l e a r t h a t t o d et er m i n e u-, b. But t h i s can be determined from t h e c o n d i t i o n t h a t one of t h e two c o n c e n t r a t i o n s cA = u2 o r cB u3 is z e r o . Suppose t h a t u2+ < u3+. 'Ihen i t is

u2

-

-,

t h a t v an i s h e s a t

so t h e second component o f (23) s a y s

b =

and we are done.

-u2-/K2,

For Ar r h en i u s k i n e t i c s , t h e r e a c t i o n f u n c t i o n can be t a k e n a s A

~ ( E ) c ~ c ~ ~ x ~ [,- ~ / E T ]

6 where

T

is a b s o l u t e t e n p e r a t u r e , and

a

may depend on

L

t o acco u n t f o r i t s

A

p o s s i b l y b ei n g l a r g e o r small. and d e f i n e a s c a l e d t e m p e r a tu r e

Let

be t h e t em p er at u r e of t h e incoming g a s ,

T+ . . a

TI

T-- T+ A

--

ul.

T+

0

The e x p l i c i t dependence of from (24) by s e t t i n g

CA =

u2,

1

on

-

CB

u1 u2 CA ,=, E E

u3,

1

and

L;

ET+

-CB E

and t h e exponent e q u a l t o 1

(Ul/E)

-I--A

E

*

T+

1

+E(~~/E)

can be found

PROPAGATING FRONTS IN REACTIVE MEDIA

275

+

The s e p a r a t e d form (12) i s found by r e p l a c i n g ui by u E V ~ and keeping only t h e l e a d i n g o r d e r term as E + 0. Doing t h i s w i t h tf;, knowledge v2- = 0 yields

,.

- 2 $ 4 ~ ) = v2exp[vlT+/(T-) I and

Since v1 < 0 f o r u s t r i c t l y between (14) and (15) follow immediately.

u+

and

u-,

our e s s e n t i a l assunptions

There i s a n a l t e r n a t e form f o r (13) which, being c o n c i s e , may be convenient f o r some purposes. Assume D i s d i a g o n a l and c o n s t a n t . For small E , t h e q u a n t i t y 1 u-u- $ (7) i s important only f o r u n e a r u-, where w e have s e e n t h a t v = aV,

so t h a t

-

u-u-

aV =-

u 1- u 1-

"1 '

With t h i s s u b s t i t u t i o n made i n t h e argument of of t h a t e q u a t i o n becomes

JI

The l a s t term can i n some s e n s e be approximated by

i n (131, t h e f i r s t component

B6(T-T-),

where

and we have -cT

DITzz

+

B6(T-T-).

-

'Ihis form may be immediately g e n e r a l i z e d t o h i g h e r dimensional problems f o r which T-, i s n o t knawn. I f t h e flame s h e e t h a s c u r v a t u r e t h e flame f r o n t , where T >> E . then t h e meaning of t h e G-function i s t h a t t h e square of t h e normal T a t t h e flame has a jump d i s c o n t i n u i t y t h e r e e q u a l t o 2B/D1. d e r i v a t i v e of An e x p r e s s i o n u s i n g a G-function i n t h e space v a r i a b l e s was used by Margolis and Matkowsky (1981).

3.

Weak shocks inducing a r e a c t i o n

The f r o n t s c o n s t r u c t e d i n Secs. 1 and 2 were such t h a t t h e i n t e r a c t i o n between t h e physics and t h e chemistry w a s of a very simple n a t u r e : i n t h e f i r s t case, t h e d o d n a n t f e a t u r e was a p h y s i c a l shock i n which no chemical r e a c t i o n s occur, and i n t h e second, t h e physics was u n a f f e c t e d by t h e chemistry. There a r e important s i t u a t i o n s , however, when t h e i n t e r a c t i o n is e s s e n t i a l t o t h e problem, and t h i s s e c t i o n c o n t a i n s t h e asymptotic a n a l y s i s of such a s i t u a t i o n . Models of t h i s The p r e s e n t treatment i s s o r t were s t u d i e d by F i c k e t t (1979) and Majda (1980). very roughly p a t t e r n e d a f t e r t h e asymptotic development i n Rosales and Majda (1980), which c o n t a i n s many o t h e r r e s u l t s as w e l l . The t h r e e main assumptions have t o do w i t h t h e f u n c t i o n s D, f , and g. ( 1 ) The t r a n s p o r t i s weak ( r e p l a c e D by ED) and decoupled: D - ( D

0).

(26)

O D (2) With energy a p h y s i c a l v a r i a b l e , r e p l a c i n g temperature (which was a chemical v a r i a b l e i n t h e preceding s e c t i o n ) , t h e e f f e c t of t h e chemical v a r i a b l e s

P.C. FIFE

276

on t h e p h y s i c s is weak, in t h e s e n s e t h a t ,.

f

-

2f(U,E u ) . A , .

(3) The a c t i v a t i o n energy i s high:

f o r some

uo,

with

g

~

and i t s d e r i v a t i v e s O(1) as

o + 0.

(u*

The s o l u t i o n s of (2) I s h a l l c o n s t r u c t have t h e appearance o f weak shocks is Go) which i n t e r a c t w i t h t h e chemistry by m a n s of g ' s s t r o n g approximately dependence on The formal expansion is as f o l l o w s , up t o t e r m s of o r d e r E.

c.

A

,

.

+ EV;

u = u

-

u0

c o n s t = u+; v

-

+ E V ~ ; vo

v

0;

f

(27)

,.

The f u n c t i o n

f

can be expanded t o second o r d e r i n

f(U,E 2-u) = f(uo,O)

A , .

+E

fiv

E

as f o l l o w s :

+ E 2fAl uN + T1 E 2 -f i l v v ,

(29)

2 t h e arguments of from t h e symbol

f;

e t c . always b e i n g

S u b s t i t u t i n g (26-29)

Below, I s h a l l omit t h e c a r e t s

i n t o t h e p h y s i c a l p a r t of (2) y i e l d s ( w i t h

I

= d/dz)

+$

+ flvt + ~

-cvl

(uo,O).

f.

[ f ~ u ' ( f l l w ) ' ] = E(Dv')'.

(30)

The terms of O ( 1 ) a r e C'

v

0 0

' + flVo'

= 0.

'.

T h i s is a homogeneous system of a l g e b r a i c e q u a t i o n s f o r t h e v a r i a b l e s v Since a n o n t r i v i a l s o l u t i o n is d e s i r e d , i t is n e c e s s a r y t o choose co t o b? a n eigenvalue of t h e m a t r i x fl(uo,O). I n t h e s t a n d a r d case, t h i s is t h e sound epee d. It a l s o f o l l o w s t h a t yo1 Therefore t h e same i s t r u e of

is a m l t i p l e of t h e a s s o c i a t e d e i g e n v e c t o r vo

Y.

itself: vo(z) = u(z) Y .

O(E) in (30) are

The terms of 'C

v ' 0 1

+ flVll

-

-

c v

l o

' - f 2 U" '

- 1

claY

- f 2 U - Aou'

- -*1 (fllvovo) +

1

-

1

+ (Dvo')

(U'ODY)',

where A = fll\YY. For a s o l u t i o n v1 t o e x i s t , i t is n e c e s s a r y t h a t t h e r i g h t hand s i d e be o r t h o g o n a l t o t h e n u l l v e c t o r Y* of t h e a d j o i n t o p e r a t o r fl* coI. We can assume t h a t t h e r e is only one such n u l l v e c t o r , and may normalize i t so t h a t YW 1; t h e n t h e c o n d i t i o n i s

-

-

clu' where

B =

m a t r i x of

A*",

-D = -D ' Y a y * ,

- V * -U ' - BUU'

and

V

-

* Y f2,

+

(5~')' 0

,the factor

,

( 3 1) f2

being the Jacobian

w i t h r e s p e c t t o i t s second argument.

f

.

I n t h i s way, variable u

t h e s e t of p h y s i c a l v a r i a b l e s has been reduced Let u s p a s s on t o t h e chemical p o r t i o n of (2):

t o a single

277

PROPAGATING FRONTS IN REACTIVE MEDIA

-,.- - c(Du')' -..

.

+ f(u,u)'

-cu' Terms of O ( 1 ) :

-

-c u'

+

-

A

f(UO'U)' = g(aY,u).

..-

,. In t h e ( t y p i c a l ) case t h a t

-

+ g(v,u).

f = F(u)u, u'

-

t h i s becomes

h(u,u)

where

The problem has been reduced t o a s i n g l e wave f r o n t e q u a t i o n ( 3 1 ) f o r a p h y s i c a l variable a, coupled w i t h a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n ( 3 2 ) f o r t h e chemical v a r i a b l e s 'ii,

-

From (26), the where g(O,u+) = 0.

boundary conditions The Rankine-Hugoniot

provide necessary c o n d i t i o n s on

c1

and

g(o-Y,u-)

- -t i )

at +oo are u(-) = 0, u(-) u r e l a t i o n s ( 4 ) and t h e r e l a t i o n s f o r a solution t o exist:

a-,u-

-

0.

I f t h e l a t t e r e q u a t i o n could be s o l v e d f o r u- as a f u n c t i o n of u-, then t h e f i r s t equation might provide a one-parameter family of p o s s i b l e end s t a t e s 0-, c1 being t h e parameter i n some range of values. The e x i s t e n c e of a s o l u t i o n of ( 3 1 ) , ( 3 2 ) w i t h t h e s e boundary c o n d i t i o n s i s a h a r d e r q u e s t i o n ; i f i s one-dimensional, t h i s problem has e s s e n t i a l l y completely been solved by Majda (1981). Computations have a l s o been performed (losales-Majda 1980). In summary, t h e s e f r o n t s c o n s i s t of weak shocks coupled w i t h h i g h l y a c t i v a t e d chemical r e a c t i o n s , w i t h weak t r a n s p o r t . The v e l o c i t y comes o u t t o be a p e r t u r b a t i o n of sound speed. The mathematical problem reduces t o a boundary v a l u e problem f o r a s c a l a r f r o n t e q u a t i o n coupled w i t h a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n f o r t h e chemical v a r i a b l e s .

4. Sharp Front Asymptotics f o r Chemical and B i o l o g i c a l Problems The u s e of (1) i n connection w i t h biology and chemical r e a c t o r t h e o r y i s u s a l l y c h a r a c t e r i z e d by t h e l a c k of a convection term f , and a l l v a r i a b l e s being chemical. I n Sec. 2 of t h e t a l k , I spoke of a c l a s s of problems w i t h no convection f o r which asymptotic a n a l y s i s produces a f r o n t w i t h a s h a r p c o r n e r a t one point. There i s a n o t h e r wide c l a s s of models, i n v o l v i n g r e a c t i o n f u n c t i o n s g of "sigmoidal" type, whose s o l u t i o n s a r e c h a r a c t e r i z e d by t h e appearance of s h a r p t r a n s i t i o n s r a t h e r t h a n s h a r p corners. These s o l u t i o n s may be genuine wave f r o n t s i n t h e s e n s e used h e r e , o r dynamical s t r u c t u r e s which l o c a l l y appear t o be conposed of s h a r p f r o n t s , but do not conform g l o b a l l y t o t h e s t r i c t d e f i n i t i o n . 'Ihese sigmoidal models have been used i n many c o n t e x t s . A very few of t h e r e f e r e n c e s a r e i n connection with: impulse propagation a l o n g nerve axons (Cesten e t a1 (1975)); o t h e r waves i n e x c i t a b l e t i s s u e ( O s t r o v s k i i and Yakhno (1979) 1;

(1975),

Keener

P.C. FIFE

270

chemical waves and p a t t e r n s (Tyson and F i f e (1980)); geographic d i s t r i b u t i o n of populations (Miaura and Murray (19781, Mirmra and Nishiura (1978); p a t t e r n s i n developmental biology (Mimra and Nishiura (1978)); more a b s t r a c t s e l f -organization models ( F i f e (1976) ). Here I w i l l go through one example t o i l l u s t r a t e t h e c o n s t r u c t i o n of a wave f r o n t composed of a s h a r p t r a n s i t i o n s e p a r a t e d by two regions of The example c o n s i s t s of two equations, with t h e components variation. g2(u) vanishing on t h e curves i n d i c a t e d , and p o s i t i v e and negative regions indicated:

i s assumed t o be l a r g e (making

The component replace i t by t h a t u1 t h e form

g1 1

F

gl, E

<<

1.

genuine slower gl(u). i n the

u1 a " f a s t " v a r i a b l e ) , so we The d i f f u s i o n matrix i s D ;), meaning

d i f f u s e s more slowly than

u2.

- [i

The f r o n t equation (3) t h e r e f o r e t a k e s

+ 71 g l ( u )

cull

+ EU1"

cu21

+ u21* + g2(u>

-

= 0

(33a)

0.

(33b)

There a r e t h r e e r e s t s t a t e s ; I s h a l l show how t o c o n s t r u c t a f r o n t passing between two of them, i n d i c a t e d by u- and u+ i n t h e f i g u r e . Eq'n (33a) and the fact that E << 1 would suggest t h a t g ( u ( z ) ) 9 0 u n l e s s u1' o r u1" i s l a r g e . For a f i r s t t r y , t h e r e f o r e , one migkt attempt t o f a s h i o n a f r o n t whose image i n t h e phase plane (Fig. 1) coincides (approximately) with t h e p o r t i o n of t h e curve between uand u+. But t h e p r o f i l e of such a f r o n t would be such that u has a minimum a t a p l a c e ( t h e bottom of t h e sigmoldal curve) where g > 0. Sfnce u ' I 0 t h e r e , (33b) would be v i o l a t e d . So t h i s attempt i s doom3 t o f a i l u r e , an$ i t i s necessary t o f i n d a way from u- t o u+ without following t h e sigmoidal curve a l l t h e way. W e can follow i t most of t h e way, however, i f a h o r i z o n t a l jump from one branch of i t t o another i s allowed a t some (as y e t unknown) value u2 = w a s i n d i c a t e d i n t h e f i g u r e . The d o t t e d l i n e i n t h a t f i g u r e , then, i s t h e proper " t r a j e c t o r y " . Such a jump n e c e s s i t a t e s a l o c a l r e s c a l i n g , so t h a t (33) w i l l s t i l l be satisfied. 'he f r o n t p r o f i l e experiences a sudden t r a n s i t i o n i n t h e u u2. As i n t h e examples i n Secs. 1 and 2, I s h a l l component, but not t h e c o n s t r u c t t h e f r o n t i n t h r e e parts: (a)

For

z

<

0,

(33) i s t o be s a t i s f i e d w i t h

E

-

0.

This means

gl(u)

-

279

PROPAGATING FRONTS IN REACTIVE MEDIA

-

0, and s i n c e u(-) = u-, t h e r e l a t i o n u1 h-(u ), u2 < w , h o l d s , h- b e i n g de f i n ed as t h e f u n c t i o n whose graph is t h e l e f t %and branch of t h e si g m o i d al curve. Now (33b) i m p l i e s ,

cu,' where

G (u,)

( b ) For hand branch def i n i t i o n :

= g2(h-(u2),u2)

+ u2" + G for

u2

= 0

(U,)

(34)

< W.

z > 0, t h e analogous c o n s t r u c t i o n i s t o t a k e p l a c e on t h e r i g h t u1 = h+(u2). W e can s t i l l u se ( 3 4 ) by j u s t completing t h e

These two p a r t s , t a k e n t o g e t h e r , c o n s t i t u t e t h e " o u t e r s o l u t i o n " . Gw(u2) h as t h e appearance:

The f u n c t i o n

It t u r n s o u t t h a t ( 3 4 ) , w i t h t h i s t y p e of n o n l i n e a r i t y , such t h a t Gw is ne g at i v e n e a r one rest s t a t e on t h e l e f t , p o s i t i v e near t h e o t h e r , and h as o n l y one change of s i g n i n the middle, h a s a unique (up t o t r a n s l a t i o n ) s o l u t i o n u(z;w) and a unique v e l o c i t y

c

-

(35)

C1(W).

w, which i s n o t y e t Of co u r s e I have shown t h e s e t h i n g s as depending on known. Fq. ( 3 4 ) is a c t u a l l y t h e e q u a t i o n of a scalar wave f r o n t w i t h o u t convection; most t h i n g s are known a b o u t them, as I s h a l l b r i n g o u t i n t h e n e x t section.

(c) The i n t e r n a l s g r u c t u r e of t h e shock-like t r a n s i t i o n n e a r found by r e s c a l i n g : 5 =; , ul(z) = U(c). This l e a d s t o

cu

5

+ u sc + gl(U,w)

-

0.

z

=

0

is

(36)

Here I have r ep l a c e d u2 by w , because i t s t a y s approximately e q u a l t o t h a t va l u e d u r i n g t h e t r a n s i t i o n . To match w i t h t h e o u t e r s o l u t i o n , U n e c e s s a r i l y s a t i s f i e s U ( h ) = h*(w). By o u r assumption, t h e n o n l i n e a r i t y h a s t h e ap p ear an ce

/

This h as t h e e s s e n t i a l p r o p e r t i e s of the p r e v i o u s f i g u r e , so, a g a i n , t h e r e i s a unique f r o n t p r o f i l e U(5;w) w i t h a unique

P.C. FIFE

280

c

-

C2(W)

(37)

W e now have t o hope t h a t (35) and (37) can be s o l v e d f o r t h e case, t h e formal c o n s t r u c t i o n i s f i n i s h e d .

w

and

C.

If t h i s is

The above One f u r t h e r comment about sigmoidal r e a c t i o n f u n c t i o n s i s i n order. c o n s t r u c t i o n h a s a s h a r p l o c a l f r o n t imbedded i n a g l o b a l f r o n t . The p r o p e r t i e s of t h e l o c a l one a r e only governed by t h e sigmoidal component g l , whereas t h e g l o b a l one depends a l s o on g2. I n a p p l i c a t i o n s , t h e r e are c a s e s i n which g 2 is such t h a t no g l o b a l f r o n t e x i s t s . Solutions with l o c a l fronts still e x i s t , however; they a r e j u s t n o t s t e a d y s t a t e s o l u t i o n s . A r e c e n t example i s t h e a n a l y s i s by Tyson and F i f e (1980) of t a r g e t p a t t e r n s f o r t h e Belousov-Zhabotinsky reagent. Our model i s based on t h e "Oregonator" k i n e t i c s , which h a s t h e f e a t u r e s d i s c u s s e d above. Crudely speaking, what our a n a l y s i s produces i s a s u c c e s s i o n of l o c a l f r o n t s , a l t e r n a t e l y of upjump and downjump t y p e s , t r a v e l i n g outward i n both d i r e c t i o n s from t h e o r i g i n . Other examples l i e i n t h e realm of s i g n a l (mainly p u l s e ) propagation a l o n g a nerve axon. The vanguard of t h i s approach was t h e work of Casten, Cohen, and Lagerstrom (1975) d e a l i n g w i t h t h e Fi&#ugh-Nagumo equations. 11. MORE RIGOROUS RESULTS FOR SIMPLER CASES

5. A S i n g l e Equation f o r a Chemical Variable; No Convection. With primes d e n o t i n g d i f f e r e n t i a t i o n w i t h r e s p e c t t o cu'

+ u" + g ( u )

-

0, u(*)

=

U*,

g(u*)

= 0.

z, t h e f r o n t e q u a t i o n i s (38)

' h e r e are t h r e e types of f u n c t i o n s g which a r e of most i n t e r e s t i n c o n n e c t i o n w i t h propagating f r o n t s , and I s h a l l b r i e f l y d i s c u s s t h e main known r e s u l t s f o r each. In t h e f o l l o w i n g , t h e f u n c t i o n g i s assumed t o be smooth, a l t h o u g h t h e v a r i o u s r e s u l t s probably hold f o r nonsmooth ones, such as t h e d i s c o n t i n u o u s g d i s c u s s e d i n t h e l a s t s e c t i o n . For d e f i n i t e n e s s , I s h a l l t a k e u- > u+. (a)

g

>

0

for

u+

u-

I n t h i s case, t h e r e is a (See blmogorov e t a 1 (19371, Uchiyama (1978).) p o s i t i v e minimal v e l o c i t y c* such t h a t a t r a v e l i n g f r o n t e x i s t s f o r e v e r y speed c > c*. These f r o n t s a r e s t a b l e only t o a very l i m i t e d c l a s s of p e r t u r b a t i o n s , i n c l u d i n g t h o s e of compact s u p p o r t ( S a t t i n g e r 1978). These f r o n t s have been used i n connection w i t h p o p u l a t i o n g e n e t i c s ( F i s h e r (19371, Aronson and Weinberger (1975)). flame t h e o r y , and chemical p a t t e r n s (Tyson and F i f e (1980)). In t h e f i r s t two a p p l i c a t i o n , t h e f r o n t w i t h minimal speed h a s proven t o be t h e most r e l e v a n t ; in t h e t h i r d a p p l i c a t i o n s , however, f r o n t s w i t h h i g h e r v e l o c i t i e s are e s s e n t i a l t o t h e c o n s t r u c t i o n . and appear i n t h e g u i s e of "phase f r o n t s " . I f g < 0 f o r u+ a r e now n e g a t i v e . (b)

<

g > 0 f o r u+ g

f

0

u

<

u-,

analogous r e s u l t s h o l d , e x c e p t t h a t a l l v e l o c i t i e s

< u < u- , w h i l e

f o r u near

u+, and g

>

0 n e a r u-:

28 1

PROPAGATING FRONTS IN REACTIVE MEDIA

I n t h i s c a s e , t h e v e l o c i t y c, and t h e f r o n t ' s p r o f i l e , are both unique (Kanel' 1962). To ny knowledge, t h e s t a b i l i t y of t h i s f r o n t has n o t been f u l l y investigated. The main relevance of t h i s c a s e is i n flame theory. In f a c t , t h e f i n a l model (25) derived i n Sec. 2 is a l i m i t i n g case, w i t h t h e "hump" i n t h e above f i g u r e becoming a 6-function. (c)

g'(u+)

<

0, g'(u-)

<

0 ( t h e b i s t a b l e case):

If t h e r e is only one i n t e r m e d i a t e z e r o (and i n c e r t a i n o t h e r c a s e s as w e l l ) , t h e r e w i l l a g a i n be a unique f r o n t w i t h v e l o c i t y C. The v e l o c i t y can b e This f r o n t is t h e most s t a b l e of a l l , g l o b a l p o s i t i v e , n e g a t i v e , o r zero. s t a b i l i t y having been proved i n ( F i f e and McIeod (1977)). These f r o n t s have relevance t o t h e study of nerve s i g n a l propagation, p a t t e r n s , population f r o n t s , and n o n l i n e a r t r a n s m i s s i o n l i n e s . Many of t h e r e s u l t s on s c a l a r population Weinberger t o e v o l u t i o n problems of t h e form

fronts

have

been

chemical

generalized

by

where t h e s u b s c r i p t n denotes d i s c r e t e t i m e , and un is a s c a l a r f u n c t i o n o f e i t h e r a continuous space v a r i a b l e r a n g i n g o v e r a l l space, o r a d i s c r e t e s p a c e v a r i a b l e ranging over a n i n f i n i t e equally-spaced grid. In p o p u l a t i o n g e n e t i c s c o n t e x t s ( f o r which t h e model was d e v i s e d ) , t h e g r i d p o i n t s would r e p r e s e n t c o l o n i e s , and t h e dynamics, given by t h e o p e r a t o r G, would be g i v e n by r u l e s of breeding, migration, and n a t u r a l s e l e c t i o n . S o l u t i o n s of

may i n many cases be f i t i n t o Weinberger's scheme (Weinberger 1 9 8 1 ) . He proves t h a t under n a t u r a l c o n d i t i o n s , f r o n t s e x i s t , and d i s t u r b a n c e s have, i n

a c e r t a i n s e n s e , a c h a r a c t e r i s t i c propagation speed. 6.

A S i n g l e Equation f o r a P h y s i c a l Variable:

-cut

+

-

(f(u))'

u",

u(*)

-

U*.

(39)

& r e t h e Rankine-Hugoniot c o n d i t i o n is a s c a l a r equation: d u

-

Af(u).

The q u e s t i o n of t h e e x i s t e n c e of f r o n t s is a t r i v i a l q u e s t i o n t o answer, so w e proceed t o t h e s t a b i l i t y of e x i s t i n g f r o n t s . A strong s t a b i l i t y result, with

282

P.C. FIFE

-

p e r t u r b a t i o n s r e s t r i c t e d t o approach 0 as z + , was proved by I l ' i n and Oleinik (1960). & h e r and Ralston (1981) have r e c e n t l y proved such a s t a b i l i t y r e s u l t when t h e term o n t h e r i g h t of (39) i s ( D ( u ) u ' ) ' .

7,

A Pair of Equations f o r Chemical Variables: T U '

= DU"

+ g(u),

u = (U1,U2),

p o s i t i v e and diagonal. ( a ) Conley and Gardiner (1981) have considered t h e c a s e when t h e n u l l c l i n e s of t h e gi a r e as follows:

+\

and t h e s i g n s of t h e f u n c t i o n s gi a r e a p p r o p r i a t e f o r t h e dynamics of competing s p e c i e s (ui being t h e d e n s i t i e s of t h e two s p e c i e s ) . When t h e r e i s none of s p e c i e s 1 p r e s e n t , a s c a l a r f u n c t i o n of b i s t a b l e t y p e e x i s t s connecting p o i n t s A and B; t h e r e is a l s o one connecting B and C when u2 0. Under c e r t a i n c o n d i t i o n s , Conley and Gardner proved t h e e x i s t e n c e of f r o n t s from A t o C , u s i n g Conley's c o n n e c t i o n index. Gardner (1980) had p r e v i o u s l y proved t h e e x i s t e n c e of f r o n t s under c e r t a i n o t h e r c o n d i t i o n s , u s i n g Leray-Schauder theory. S t a b i l i t y h a s n o t been i n v e s t i g a t e d .

-

283

PROPAGATING FRONTS IN REACTIVE MEDIA

( b ) Again co n p e ti n g s p e c i e s , b u t w i t h t h e e x i s t e n c e of a s t a b l e c o e x i s t e n t rest s t a t e assumed (ul # 0, u2 # 01, and w i t h t h e o r i g i n u n s t a b l e . Tang an d F i f e (1979) proved t h e e x i s t e n c e of a f a m il y of f r o n t s co n n ect i n g t h e o r i g i n t o t h e c o e x i s t e n t s t a t e , u s i n g 4 d i m e n s i o n a l phase s p a c e a n a l y s i s . (c) A t h r ee- s p e c i e s model of m i g r a ti o n w i t h (weak) n a t u r a l s e l e c t i o n h as been s t u d i e d by Conley and F i f e (1981). The e x i s t e n c e of f r o n t s was proved u s i n g Conley's index. ( d ) A two-equation model proposed by Murray i n co n n ect i o n w i t h chemical wave problems has been i n v e s t i g a t e d by Klassen and R o y (1979), who proved t h e e x i s t e n c e of f r o n t s , t o g e t h e r w i t h convergence r e s u l t s . ( e ) S. S. Lin (1980) s t u d i e d a model f o r flame t h eo r y , i n v o l v i n g a s i n g l e reaction A + B with constant pressure. The dependence of t h e d e n s i t y p p(T) on temperature is n o t assumed t o be weak. There r e s u l t s a system

-

Ut = (D(U)Ux)x

+ +(U)K,

U = (T,Z), K

E

R

2

,

=

with $(U) 0 f o r T n e a r T+ ( t h e te mp er at u r e of t h e unburned g a s ) , and f o r Z = 0. The m a t r i x D i s n o t a c o n s ta n t . The e x i s t e n c e and u n i q u en ess of a f r o n t was proved, u s i n g Ieray-Schauder t h e o r y . 8. Conservation Laws w i t h "Viscosity".

-cut

+

( f ( u ) ) ' = ( D ( u) u ') ',

u(+)

= uf.

(40)

The most b a s i c problems a r e t h e f o ll o w in g : ( a ) Find t h e s e t of s o l u t i o n s (u+,u-,c) ( b ) Given a Rankine-Hugoniot of (40).

of t h e Rankine-Hugoniot

t r i p l e (u,u-,c),

eq u at i o n s.

f i n d a solution (if it e x i s t s )

As mentioned b ef or e , t h e s e problems are t r i v i a l i n t h e scalar case, b u t f a r from t r i v i a l f o r systems. Problem ( b ) i s e a s i l y reduced t o f i n d i n g a t r a j e c t o r y between two c r i t i c a l p o i n t s of a n autonomous system of o r d i n a r y d i f f e r e n t i a l e qu at i o n s , t h e o r d e r e q u a l l i n g t h e number of e q u a t i o n s i n (40). I n t h e case of t h e s t a n d a r d e q u a t io n s of g a s dynamics, t h e r e are t h r e e e q u a t i o n s 'h en Prob. ( a ) is f o r t h e t h r e e d e n s i t i e s of mass, momentum, and energy. classical. The easiest m a t r i x D i n t h i s case is t h a t r e p r e s e n t i n g v i s c o s i t y and thermal c o n d u c t i v i t y . With t h i s D (not n e c e s s a r i l y c o n s t a n t ) a n e x i s t e n c e theorem f o r Prob. ( b ) was proved by G i l b a r g (1951). Re s u l t s on b o b . ( b ) f o r more g e n e r a l systems in which f s a t i s f i e s strict h y p e r b o l i c i t y and genuine n o n l i n e a r i t y c o n d i t i o n s have been o b t a i n e d mainly by Conley and Smoller (1970) and (most r e c e n t l y ) by K ey f i t z (1982). I a m i n d eb t e d t o Basil Nicolaenko. and Wildon F i c k e t t f o r s t i u u l a t i n g c on v er s at i o n s . Support from t h e National Science Foundation and Los Alamos N a ti o n al Labs is g r a t e f u l l y acknowledged.

REFERENCES

[I]

Aronson, D. G. and Weinberger, H. F., Nonlinear d i f f u s i o n i n p o p u l a t i o n g e n e t i c s , combustion and n e r v e p r o p a g a ti o n , i n : Proceedings of t h e Tulane Program i n P a r t i a l D i f f e r e n t i a l Equations and R e l a t e d To p i cs, L e c t u r e Notes in Mathematics, No. 446 (Springer-Berlin, 1975).

284

[2]

P.C. FIFE

Bush, W. R. and F e n d e ll , F. E., Asymptotic a n a l y s i s of l am i n ar flame p r o p ag at i o n f o r g e n e r a l Lewis numbers, Combustion Science and Technology 1

(1970) 421-428. [3]

Casten, R., Cohen, H. and Lagerstrom, P., P e r t u r b a t i o n a n a l y s i s of a n approximation t o Hodgin-Huxley t h e o r y , Quart. Appl. Math. 32 (1975) 365-

402. [41

Conley, C. and F i f e , P., C r i t i c a l m an i f o l d s, t r a v e l l i n g waves and an example from p o p u l a t i o n g e n e t i c s , J. Math. Biology, t o appear.

51

Conley, C . and Gardner, R., An a p p l i c a t i o n of t h e g e n e r a l i z e d Morse i n d ex t o t r a v e l l i n g wave s o l u t i o n s of a c o m p e t i t i v e r e a c t i o n d i f f u s i o n model, Math. Res. C e n t e r TSR 2144 (1981). Conley, C. and Smoller, J. A. , V i s c o s i t y matrices f o r two-dimensional n o n l i n e a r h y p e r b o l i c s y s t e m s , Comm. Pure Appl. Math. 23 (1970) 867-884. F i c k e t t , W.,

D e t o n a t io n i n m i n i a t u r e , Am. J. Phys. 47 (1979) 1050-1059.

F i f e , P. C . , P a t t e r n f o r m a t io n i n r e a c t i n g and d i f f u s i n g sy st em s, J. Chem. Phys. 64 (1976), 854-864.

[91

F i f e , P. C. and McLeod, J. B., The approach of s o l u t i o n s of n o n l i n e a r d i f f u s i o n eq u a ti o n s t o t r a v e l l i n g f r o n t s o l u t i o n s , Arch. R a t i o n a l Mech. Anal. 65 (1977), 335-361. Gardner, R. A , , Large a m p l it u d e p a t t e r n s f o r two competing s p e c i e s , Math. Res. Center TSR. (1980). Gilbarg, D., The e x i s t e n c e and l i m i t b eh av i o r of t h e one-dimensional l a y e r , Amer. J. Math. 73 (1951) 256-274.

shock

Kanel', Ya. I., On t h e s t a b i l i z a t i o n of s o l u t i o n s of t h e Cauchy problem f o r t h e eq u at i o n s a r i s i n g i n t h e t h e o r y of combustion, Mat. Sbornik 59 (1962)

245-288. Keener, J. P.,

Waves i n e x c i t a b l e media, SIAM J. Appl. Math. 39 (1980) 528-

548. Key f i t z, B., Bounds f o r v i s c o s i t y p r o f i l e s f o r 2 c o n s e r v a t i o n laws, Rocky Mountain J. of Math., t o ap p ear .

x

2

systems

of

Klaasen, C. and Troy, W., The a s y m p t o t i c b eh av i o r o f s o l u t i o n s o f a sy st em of r e a c t i o n d i f f u s i o n e q u a t i o n s which models t h e Belousov-Zhabotinskii chemical r e a c t i o n , J. D i f f e r e n t i a l Eq'ns, i n p r e s s . Lln, S. S., T h e o r e t i c a l s t u d y of a r e a c t i o n d i f f u s i o n model f o r f l am e p r o p a g a t i o n i n a g a s , P h . D . T h e s i s , U n i v e r s i t y of C a l i f o r n i a , Ber k el ey , Math. Res. Center TSR (1980). Ltnan, A , , A t h e o r e t i c a l a n a l y s i s of premixed flame p r o p ag at i o n w i t h a n i s o t h e r m a l c h a i n r e a c t i o n , Tech. R e p o r t , Inst. Nac. Tec. A e r o s p a c i a l "Eeteban Terradas", Madrid (1971). Majda, A. , A q u a l i t a t i v e model f o r dynamic combustion, SIAM J. Appl. Math., t o appear. Margolis, S. B. and Matkowsky, M. J., f u e l s , SIAM J. Appl. Math., t o appear.

Flame p r o p ag at i o n w i t h n u l t i p l e

285

PROPAGATING FRONTS IN REACTIVE MEDIA

Mirmra, M. and Murray, J. D., On a p l a n k t o n i c prey-predator e x h i b i t s p a t c h i n e s s , J. Theoret. Biology 75 (1979) 249-262.

model which

Mimra, M. and Nishiura, Y., S p a t i a l p a t t e r n s f o r i n t e r a c t i o n - d i f f u s i o n e q u a t i o n s i n biology, Proc. Symp. on Mathematical Topics i n Biology a t RIMS, Kyoto U n i v e r s i t y (1978). Osher,

S.

and

Ralston,

J.,

L1

stability

of

traveling

waves

with

a p p l i c a t i o n s t o convection porous media flow, p r e p r i n t (1981). O s t r o v s k i i , L. A. and Yakhno, V. G., The formation of e x c i t a b l e medium, B i o f i z i k a 20 (1975) 489-493.

pulses

in

an

Rosales, R. R. and Majda, A , , Weakly n o n l i n e a r d e t o n a t i o n waves, SIAM J. Appl. Math., t o appear. S a t t i n g e r , D., Weighted n o r m f o r t h e s t a b i l i t y of D i f f e r e n t i a l Equations, 25 (1978) 130-144.

t r a v e l i n g waves,

J.

Tang, Min Ming and F i f e , P. C., Propagating f r o n t s f o r competing s p e c i e s e q u a t i o n w i t h d i f f u s i o n , Arch. R a t i o n a l &ch. Anal, 73 (1979) 69-77. Tyson, J. and F i f e , Belousov-Zhabotinskii

P. C., Target p a t t e r n s i n a r e a l i s t i c model of t h e Reaction, J. Chem. Physics 73 (1980) 2224-2237.

Uchiyama, K., The behavior of s o l u t i o n s of some non-linear e q u a t i o n s f o r l a r g e t i m e , J. Math. Kyoto Univ. 18 (1978) 453-508.

diffusion

Williams, F. A., Combustion Theory, (Addison-Wesley, Reading, MA. 1965).

Weinberger, H.F., Long-time behavior of a c l a s s of b i o l o g i c a l models, SIAM J. Math. Anal., t o appear.

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PART IV Nonlinear Phenomena in Fluids and Plasmas

This Page Intentionally Left Blank

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 8. Nicolaenko (eds.) 0 North-Holland Publishing Company, 1982

289

REGULARITY RESULTS FOR THE EQUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC AT THE BRINK OF TURBULENCE Claude BARDOS Ddpartement de Mathhatiques Universite of Paris 13 Avenue J.B. Clement 93430 VILLETANEUSE

This work is divided in two parts firts we give the existing regularity results concerning the equations of incompressible fluids mechanic. These results are valid in a limited range of applications. It is conjectured that this limitation is due to the appearance of turbulence. Therefore in a second part we describe some recents numerical results obtained in situations where the abstract theorems are no more va1id;these computations give evidence of persistence of regularity for the two dimensional M.H.D. equations and of appearance of Turbulence after a finite time for the 3 dimensional Euler Equation.

I. INTRODUCTION The purpose of this lecture,ononehand,is to describe the classical results concerning the existence ofasmooth solution for the equations of incompressible fluids mechanic. These results have a rather limitated range of applications and it is conjectured that they cease to be valid when turbulence appears ; therefore, on the other hand, I will try to describe some recent numerical results concerning the behaviour of the fluid at the brink of turbulence (when the theoretical results are no more available). These numerical experiments would hopefully give some light on the process ; however they require a very sophisticated material both technical (size of the programs involved) and theoretical (use for instance of Euler Transform or Pad6 approximants) and in the future their interpretation may be subject to importants changes.

290

C. BARDOS

11. THEORETICAL REGUMRITY RESULTS

1.

The basic equations and the fondamental regularity results

The function u(x,t) defined in : R

x

will denote the velocity;it is a vector valued function

lRt (n = 1,2,3)

with value in lRn ; for numerical purpose, and

also to have at our disposal a problem in a bounded domain with no boundaries we will also consider the periodic case where x

E

Tn. Therefore X will denote either R:

Tn, in the first case we will be

concmnedwith functions u

or

Tn and u takes its values in

that decay at infinity (in some way) and in the second,

with periodic functions. The incompressible viscid Navier Stokes equation is then :

Incompressible refers to the relation V*u = 0 and this is equivalent to the following fact : the mapping J, defined by

is measure preserving. On the other hand the positive number u denotes the viscosity, and the relative strenght of the non linear term and of the viscous term is measured by the dimensionless Reynold number

R =

For

u

= 0

Typical lenght x Typical velocity viscosity

- Iv-vvl

I v=AvI

the equation (I) is called the Euler equation.

For a conductive fluid, when the variation of the electric field is neglected

the motion is governed by two set of equations involving the velocity u and the magnetic field b

aU + at

::

-+

u*Vu

:

-Vp +

#Au+ bVb , V*U

u*Vb = bVu+qAb

= 0

(V>O)

V*b = 0 (qb0)

(3) (4)

Some over simplified models of these equations can be given in one space variable

29 1

EQUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

they are the following :

(Burger Hopf equation) or

2. Regularity results for the Navier Stokes Equation

It is know that for n = 2 ( 4 ) with

v > 0

the equation (1) with v > 0 and the equations ( 3 ) ,

and q > 0 admit a smooth solution defined for all positive Lions [21]

times (Leray [I91

, Ladysenskaya [IS]

for the Navier Stokes Equation

for the M.H.D. equation) ; furthermore the solution is analy-

and C. Sulem [28]

(for t > 0 ) more precisely it can be extended in a complex neighbour-

tic in t 2

hood of lR

(or T2

for periodic functions) of the following shape :

i

Figure I

(cf. Kato Fujita [I41 or Foias Prodi [8]). The proofs of these results use basically the properties of the Laplace operator to balance (via Sobolev Theorem) the effect of the non linear terms. In dimension 3 similar results are proven only for a viscosity large enough (corresponding to a Reynold number of the order of one). For higher Reynolds numbers only the existence of a weak solution is proven. However one can show that the size of the domain (in x and t) where this solution may fails to be analytic is very small. Several results in this direction have been obtained startingwith Leray [I71 an then improved by Scheffer Cafarelli and Nirenberg [5]. 3

The result of [5]

D7Iy Foias Teman [9] and

is the following : the Hausdorff

dimension in IR xlRt of the set of pointswhere the solution fails to be smooth

292

C. BARDOS

is at most 1 . The proof it self is very interesting because it rests mainly on dimension analysis and therefore it is very cbre of the heuristic argument of Kolmogorov [I61 concerning the spectra of the Turbulence

.

3 . Euler Equation in 3 dimensions

For the Euler equation there is no proof of the existence of a solution (even a

weak one) for all time. The only result concerns the existence (and uniqueness) of a smooth solution for a finite time

It1 < Ti

depending on the size of the

initial data and mainly on the size of the initial vorticity (Lichtenstein [ZO] , Ebin and Marsden [7]),

and the following facts have been observed :

(i) If the initial data delongs to Ck then has long as the vorticity remains bounded

the solution will remain in the same space (Ebin and Marsden [7]

or

Foias Frisch and Teman [lo]). (ii) If the initial data is analytic then the solution remains analytic as long as the vorticity is bounded. However the complex domain where the solution is analytic may shrink and finally collapse when the vorticity cease to be bounded. This gives, for the domain of analycity the following picture. (Bardos and

The proof of all these results are done with the system :

and the introduction of the caracteristics curves of the motion i(t) = u(x(t),t)

,

x(T) = x

(9)

EOUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

293

For (ii) an extension of these curves to the complex domain is necessary and one has to control their distance from

R2 in term of the Holder norm of the

vorticity.

4 . Euler Equation in two dimension

In two dimension the vorticity w = V I\ u and therefore the term w V u

is orthogonal to the plane (x,,x2)

cancels. The vorticity is conserved along any tra-

jectory of the fluid particle and its norm in any Lp space

(I,
tant. From this observation one deduces by a compactness argument the existence of a weak solution for any initial data u ( O , * )

of finite enegy with

Vnu(O,*) = w(O,*)

one can also prove the uniqueness

bounded in Lp. When p =

of the solution (Youdovitch [33],

+co

Bardos [2]).

difficult because the unit ball of L' It has been noticed by Wolibner [32]

The case p = I

is slightly more

is not weakly closed. and Holder [13]

that from the uniform

boundedness of the curl one cannot directly deduce the regularity of the solution. One has to introduce an other argument concerning the pair dispersion. The problem is to control the distance beetween two fluid particles x(t) that the velocity field u(x,t)

and y(t)

knowing

is divergence free and has bounded curl. Due to

the divergence logarithmic singularity of the Green function, for the Laplace equation A$ = w

With u = V

A

$ =

, one has the estimate

(2, - $)

one deduced from (10) the following a priori

estimate :

Solving the ordinary differential equation z' =

? Clwl,

z Log

-1

and using a

comparison theorem, one obtains the estimate :

(

I

For sake of simplicity we consider

a

periodic domain of diameter 1 .

294

C. BARDOS

from which the Holder regularity of us can be deduced. Similar methods canbe extended to complex domain and one obtains (Bardos Benachour [3])

that if the initial data is analytic in a strip

the solution will be analytic in a strip of the following form :

5. Kelvin Helmoltz instability in two dimensions

The Euler equation can be written in conservative form :

therefore one can try to solve the initial value problem for some class of discontinuous initial data. In particular we will assume that w smooth density supported by the smooth curve

r

=

{(x,y(x,t))lx

0

=

V A U ~ is a

E

R) and that u

is of finite energy. p E

Let

D(lR2)

be a positive function satisfging the relation

We introduce the solution

E

The energy of u (*,O)

uE

I

p(x)dx

= 1.

of the Euler equation, with initial data given by

is uniformly bounded and we have, using the conservation

of the vorticity, the relation :

E

Therefore the curl of uo

L

1 2

).

remains uniformly bounded (with respect to m

2 2 2

E)

in

Since uE is bounded in L (lRt ; (L (IR ) ) ) one can extract a sub family

295

EQUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

which will converge (with ;u

to a function u

E

going to zero) in L m m t ; (L2 (IR 2 ) ) 2 )

is a solution of the equation

to prove the relation :

To obtain a weak solution one has

A s it is

weak star

usually the case one preves (17) using a compactness argument and for

this compactness argument we have at our disposal the estimate (15) which is the essential step. Therefore we obtain U. Frisch

[~oJ for

(see

C. Sulem, P.L. Sulem, C. Bardos and

the details) the following.

Theorem : Assume that uo

belongs to the space (L2@32))2

where r -

denotes a smooth curve and w (u)

estimate

:

0

2

Then the Euler equation in IRt XlR

aU at

+

uvu =

- vp

with initial data u ( * , O ) = u 2

2

( 0 )

,

and that we have

a smooth density satisfying the

:

v*u = 0

has a weak solution belonging to the space

2

Lrn(lR; (L (IR 1) ). 1

2

It is important to notice that V h u will not belong to L @ ) . This is related 1

2

to the fact that L (R ) is not reflexive. The introduction of the estimate in 1

2

L (lR )

is very similar to the method used to prove the existence of a weak solu-

tion for quasilinear equation (or systems) cf. Kruckov

[17]

or Glimm [I?]

it is also related to the introduction (cf. Temam and Strang [31])

;

of the space

296

C. BARDOS

of Bounded Deformation for plasticity problems. The situation described above givesan example of a weak solution for the Euler Equation. As long as the set where u

is singular remains a smooth curve r(t),

one can derive from (13) that u satisfies on r(t)

a Rankine Hugoniot condi-

tion ; at variance with the situation for quasilinear equations or systems we will prove (Theorem 2 ) the uniqueness of the solution with out any entropy condition. This is related to the fact that r(t) is a contact discontinuity with no loss of energy. The question of the existence and the smoothness of the curve r(t)

is called

the Kelvin Helmoltz instability. We will now describe this problem. We assume that the second coordinate of the curve r(t) = (x(A,t),y(X,t)

;h C R }

can be resolved in term of the first : y = y(x,t) and we denote by R(x,t)

-

S2(x,t)

the vortex density w(x,t)

n

d

x

u+(x,t) + U_(X,t) and by

,L.

v(x,t) =

the mean value of the velocity on the vortex

above and below the curve r(t>). line (u+ - denoting the limit of the velocity is a weak solution of the Euler equation :

From the fact that u

% +z at

a axi (u.1

and from the fact that u VAU

w

=

u.) J

0, v *u

=

(20)

0

satisfies the relation :

Q 6rct)

one deducesthat S2 and v satisfy the system 9at + % V I

a

+

a ax

-((nV,)

= v 2 =

0

Finally from the relations V*u = 0 and (21) we have

:

297

EOUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

v

I 2s

I - -

1

.=-I

!

(x,t) - y(x',t) (x-x'):+

x

1

2

2s

(x ' ,t) dx '

(24)

(y(x,t) -y(x',t))

- x'

2C2(x' ,t)dx'

(25)

(x-x') + (y(x,t) -y(x',t))

The equations (22), (23), (24) and (25) were directly derived by Birkoff [ 4 ] using mechanical arguments. This problem is known to be highly unstable and in fact we will show that the linearised problem may be badly posed (in f @ c )

for instance). We assume that

for t = 0 the vortex line is flat namely that we have

u((x, ,x2) ,O)

=

\

:

(-A,O) for x2 > 0 (26) ( A ,0) for x2 < 0

Therefore we have Q(0,x) f 2A and y(X,O) E 0 ; it is easy to verify that u given by (26) is a stationary solution of (20). Writing v = 0 + 6Vl,v2 = 0 + 6v2 1

n

= 2A +

6Q, y = 0+6y,

, we linearise the equations (22), (23), (24) and (25)

in the neighbourhood of this stationary state, to obtain :

a

- 6y = 6 v2 at

bv

1

=

a (SQ)+ 2A ,at 2A - I;;

a

6y(x,t)

(6~~) 0

-

6y(x',t)

(x - x')

dx'

2

-

6v = 1 2 2s Taking the

urier Transform of these equations with respect to x

5 the dual variable, we obtain

In (30) A(5)

:

denotes the matrix given by :

an

noting

C. BARDOS

298

The eigen values of A(6)

are

& therefore e tA(S)

t

will introduce in the 6

mode an exponential factor ettQ this implies that even for (6n,&y) ( 6 0 , 6y)

etA(')

E

(5(R))

2 1

( )

(?I'(IR))~ and this shows that the problem is

will not belong to

badly posed, except for initial data with exponentially decaying Fourier Transform. The fact that the Fourier Transform of the initial data decay exponentially implies that the initial data it self is analytic, and this means that the problem (22), (23), (24), (25) may be well posed only (even for a finite time) in the class of analytic function. Indeed we have the following.

Theorem 2

:

&

Assume that yo(x)

axis of functions yo(x + in),

I

in a strip S = {x + in

Ro(x)

are the restriction to the real

Ro(x + iq) which are holomorphic and bounded

< S]

ayo satisfies , and assume furthermore that ax

the estimate : aYO (x + in)I IIm -

ax

Then the problem (22)

time

It1 <

-

<

1/2

(25) ; with initial data

(32)

yo,

has, during a finite

0 -

T* , a unique analytic solution.

The proof of the Theorem 2 can be found in C. Sulem, P . L . Sulem, C. Bardos and U. Frisch [30]. (22)

-

The main idea of the proof is the following : the problem

(25) is (after elimination of v l , v2, using the equations (24) and (25))

of the following form

a

(') at n

: +

A

(i)

=

0

(33)

Where A denotes a first order non linear pseudo differential operator. Therefore one can applies the Cauchy Kowalevski theorem in scale of spaces (here we consider scales of spaces of analytic functions, cf. Ovsjannikov [26]

and Baouendi

and Goulaouic cl]. The main drawback of the method is the following we have to 2 control the quantity (x-x') + (y(x,t) - y(x',t))2 which appears in the deno-

1 ( )

3 denotes

the space of Schwartz functions.

EQUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

minator of ( 2 4 ) and (25). This quantity

299

may vanishes for y non real and for

x’ # x , to prevent the appearance of this type of singularity we assume that the hypothesis (22) is satisfied and we consider only time intervals during which remains bounded away from 1 . Therefore it seems that the theory will not be able to handle cases where the curve

r

rolls up before breaking. However

some numerical computations made by Meiron, Baker and Orszag [22]

r

cate a loss of analycity of the curve

seem to indi-

before the formation of rolls.

6 . Equations of Magneto-hydrodynamics

For the two dimension M.H.D. one can prove the existence for all times of a smooth solution for non zero fluid viscosity v (C.

Sulem

F8J). However when

and non zero magnetic viscosity

the fluid is inviscid only local existence is

proven. Tha situation is the same as in the case of the Euler equation in 3 dimension. In particular one can obtain a local existence result for analytic solution in a domain similar to the one given by the figure 2. On the other hand the one dimensional model

:

has been studied by C. Sulem, J.D. Fournier, U. Frisch and P.L. Sulem [29]. clear that if b equation

It is

is identically zero the system ( 3 4 ) reduces to the Burger

:

a”+,& at ax

3

0

(35)

and therefore, for any non increasing initial data, singularities will appear after a finite time. This property remains true for the system ( 3 4 ) when b(x,O)

is small compared to

the variation of u(x,O). Namely we have. Proposition I

: Assume that there exist a point

the initial data bo(x)

and

uo(x)

:

a

E

IR such that one has for

300

C. BARDOS

let x(t)

:

be the trajectory defined by i(t)

then

a ax u(x(t),t)

Proof

-

(cf.

C.

= u(x(t),t)

(37)

and x(0) = a

+ -ab (x(t),t) ax

will become infinite in a finite time.

Sulem, J.P. Fournier, U. Frisch and P.L. Sulem [ 2 9 ] ) .

+

defined by

One introduces the function z-

z+ =

u+b,

-

z = u-by with these new

functions the system (31;) becomes

- az+

a"+

a t + "-E

a"-

=

+

2

' 3-E

O

+

aZ% i = O

On the curve x(t) defined by ( 3 7 ) one has b(x(t),t)

and therefore using

= 0

the relation : ;r(t)

=

u(x(t),t)

= z+(x(t),t)

=

z-(x(t),t)

(39)

y

one obtains

:

( 4 0 ) can be solved explic ty a to give

-

2bA(a) ':2

+' zo (a)

(a)

- zo

-1

-2bA(a) t (ale

From ( 4 1 ) one deduces, using (39) the proof of the proposition 1 . On the other hand it is believed and known in some physical examples that a large magnetic field may stabilize the fluid and prevent the appearance of the Turbulence ; on our one space variable model this gives the

Proposition 2

[29]

:

:

Assume that the initial conditions u

and bo~Cm(ZR)

satisfy the relation : inflbol > 71 (sup u

-

inf u )

then the solution of ( 3 4 ) is smooth for all times.

(42)

301

EOUATIONS OF INCOMPRESSISLE FLUIDS MECHANIC

Proof : From -

the relations ( 3 8 ) we deduce the equation

which combined with the relation

gives

Assuming that z+(x,t)

-

z-(x,t)

=

Lb(x,t)

never vanishes and integrating ( 4 5 )

from zero to t we obtain

-

In the equation ( 4 6 ) a+ and a

denote the values at s

-

0 of the solutions

of the caracteristic equations

Finally the relation ( 4 2 ) implies that

ax

and therefore

z:(a-)

-

zi(a+)

is never equal to zero

remains uniformly bounded. Similar result yields for

ax

and

from these estimates one deduces that zf are smooth for all times.

Remark 1

n

: A similar result can be proven for the true M.H.D. equation in lR

(Bardos, Frisch, Sulem unpublished) in the following situation, one assume that 'L

b = B + b 0

0

0

where Bo is a constant vector field of large enough magnitude 'L

compared to bo

'L

and u and that bo

and u

decay fast enough for

1x1

'L

(eventually one can take bo

and u

of compact support), then the M . H . D .

equation

at

+ uvu=bVb-Vp,

written on the form

ab at +

uVb = bVu, V*u=V*b=O

(48)

-f

OJ

C. EARDOS

302

Can be viewed as small perturbation of the simple transport equation

+

azat

B0'VZt

whose solutions are given by &,t)

=

Zi(x

:

Bot)

+

and in particular for the curl of 2-

one has

:

(49)

at

where IT(Z+, Z-)

where a!11 . kll

is a term of the following form

:

denotes a suitable constant.

This terme is quadratic but the functions VZ'

and V2- propagate rougly in

apposite direction (of the order of ;Bo) therefore this phenomena of propagation balances the non linearity and may prevent the appearance of singularity. The idea of this proof is related to the proofs of regularity for non linear wave n equation which rely on the dispersion property of the linear wave equation in IR (cf. Klainerman [15] or Chadam [ 6 ] )

and they cannot be adapted to the case of a

bounded domain or of a periodic domain.

111.

NUMERICAL EXPERIMENT

In the previous section we have describe regularity results concerning the classical incompressible equations of Fluids Mechanic. We have shown that these results have a limited range of application. The purpose of these section is to give some comment on numerical computations which have been made outside this range of application, namely when turbulence may appear.

303

EOUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

1. Magneto hydrodynamic equations in two dimension

The computation described in this section have been made by Orszagand Tang [25] for the two dimensionnal periodic M.H.D.

equation, with the following initial

conditions :

b(x ,x ,0) = curl(a(x x ,O)) 1 2 1' 2

=

curl(cos x2 +

-21 cos 2x1)

(50)

there initial conditions are called the Orszag- Tang-Vortexp figure 3 and figure 4 show these initial conditions, figure 5

and

figure 6 show the beha-

viour of the energy spectrum which is proportional to the quantity

+ Where k denotes the wave number G(t,k)

and X

the corresponding Fourier coefficients

the wave length. The computation shows that E(X)

which implies that u(tpx)

-6(t)A

behaves like e

can be extended as an analytic function in the com-

plex neighbourhood of width 6(t) for several value of t

:

of the real domain. On the figure 7 is plotted

the expression Log 6(t).

6(t) = -Kt and seems to indicate that for any t lytic in a complex neighbourhood of width e

-Kt

This gives the estimate the solution will remain anaof the real domain. The mecanism

which prevent the formation of singularities may be explained by the numerical

+

results described on figures 8 and 9 , where 6- = curl Zf are plotted ; these terms become large only in the neighbourhood of neutral points ; around these points the phenomena becomes quasi one dimensional and therefore the Lorentz force

.rr(Z+,Z-)

and 5

-

introduced in the equation ( 4 9 ) becomes un effective even for 6'

large. Finally the oscillations of.the energy spectra in the figure 6

are related to the following fact

:

The small s-le

structures are located

around several distincts neutral points and this produces interferences in Fourier Spectra. On the figure 10 are plotted three curves describing the logarithmic decrement of the energy spectra

. The first one concerns the computations describedabove for

304

C. BARDOS

the M.H.D. equation, the second concerns some computations made by Frisch and Sulem for the 2 dimension incompressible Euler equation, by high resolution spectral methods on an initial data suggested by Meiron, The third one is related to the bound deduced from the complex version of the pair dispersion formula (Bardos and Benachour [3]).

This picture shows that the Euler equation is

analytic than the M.H.D. equation and that the theoretical results of [3] be sensibly improved.

Orszag- Tang

-

Vortex

Stream function contour u = curl@

T = 0.000

Figure 3

should

EQUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

-

(hirag- Tang Vortex Uagnetie Potential Contour for

T

-

(b

B

-

curla)

0 . m

Figure 4 Kinetic Energy Spectrum at time t with Onrag- Tang

- Vortex

as

-

.I.m

for the

ZD. U.H.D.

equetion

initial data.

Computation made by U. Ueneguzri on the C.R.A.Y.

I

of the N.C.A.R.

A

E(t.k)

10-20

5

erp(-d(t)k)

round off noise

. 0

.

20

40

,

60

80

l

100

I20

l

140

Figure 5

l

160

l

180

.

200

I

120

I

240

,

(kI

305

306

C. BARDOS

Kinetic Energy Spectrum at time with &.rag-

Tang

- Vortex

c

-

6.000

as initial data.

H. Uenegurri on the C.R.A.Y.

I

of t h e

for the

2D. H.H.D. equation

Computation made by

N.C.A.R.

I00

10-5

10-10

10-20

0

20

40

60

80

I00

120

Figure 6

140

160

I80

200

220

240

lkl

307

EOUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

Lo5 6(t)

1.2. I.

Logarithmic decrement of the kinetic energy spectrum for the 2 D. M.H.D. equation with Qszag- Tang Vortex a8

-

.

.3

,

.6

n

.4

.2

initial data.

Computation made at N.C.A.R.

on C.R.A.Y.

I

by U. Frisch, M. Meneguzzi. A. Pouquet and P.L. Sulem.

. .

> .4

.5

.6

.7

.8

.9

I.

1.1

1.2

1.3 1.4

1.5

1.6

1.7

1.2. -.4, -.6,

-.a, -I

-1.2

. -

-1.4,

Logarithmic decrement of the kinetic energy spectrum

-1.6,

for the 2D. H.H.D. with -1.8.

-2

,

-2.2,

-2.4, -2.6

I

-2.8. -3

-3.2"

-

a uniform initial magnetic field Bo

Figure 7

4.

time

C. BARDOS

Contour for E+

-

equation at time

curl(u+b) for the ZD. T I .55

n.u.o.

/

asi one imensionsl axis

Figure 8 Contour for at time

T

- E-

curl(u-b)

for the

.55

Figure 9

2D. H.H.D.

equation

308

EOUATIONS OF INCOMPRESSIBLE FLUlDS MECHANIC

I

2D Euler equation with initial stream function $ II

COB

x , +cod x 2t 1 I (COS 2 5 +

Theoretical ea t ima te Log 6(t) --wet

siven by Bledos and Benachour

Figure 10

COS

2x2)

C. BARDOS

310

2. Euler equation in three space variables The computation for the Euler equation in 3 space variables are made on a periodic solution with the following initial data u 1 (x1 ,x2 ,x3) = u2(-xl,x2,x3) = cos x 1 sin x2 cos x3 u3(x1 ,x2,x3) = 0

(52)

The quantity which is closely related to the appearance of Turbulence is the enstrophy Q,(t)

given by

The computation have been recently made in two different ways. ( I ) For t realnumerical computations are made using finite differences

and the Fast Fourier Transform of Orszag. (2) Keeping in mind that all the quantities m

n(t)

=

E lk12Pla(k,t)12 kEZ3

=

C )'(An

t2n

(54)

n=o

should blow up at the same type, one compute these expansions. We will describe with some details this second method following closely the work

of Morf

, Orszagand Frisch

only even terms in t Green, Vortex

c23]

; one noticesthat the expansion (54) contains

because the solution of the Euler Equation, with Taylor

for initial data is even in t

. To obtain

)'(A n

one needs o n l y

the derivatives of u at t = 0 ; one uses the formula =

n

d

which involves the successive derivatives of tives are given by the formula

aa

(55)

dt"

3

G(k,t)

for

t =

0 ; these deriva-

31 1

EQUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

Due to the particular form of the initial data the authors

p2l where able to

compute the serie ( 5 4 ) up to the order 4 5 . The coefficients :A satisfy the relation : )'(A n

=

(-I)

are real and

An+,. (PI The numerical computation shows the

existence of the limit

Which leads, for the complex extension of the Euler equation to a singularity at tL = - 5 .

To locate the first real singularity one uses the Euler transform

:

2 2 w = 6t /(t + 5 )

is transformed in the serie the serie R (t) = C An(P)t2n P 'L S (w) = C B F ) wn. The coefficient of this serie are positive and the radius of P convergence will determine the first real singularity. In the figure 1 1 we plot

the ratios 0.5,.

I < p < n . For p = 2 , 3 , 4 these ratios decrease monotically with increasing n and the extrapolation 0.1

to n =

m

is consistent

1/20 1/15 1/10

with a commun intersection Figure I I

for [ 2 3 ]

at

I/w(T*) = r- = 0 197

or t, = 5 . 2 . An asymptotic expansion of

:

in then computed giving

r(P)

c

rm ( I + (yp

-

I)/n + ...)

The formula ( 5 8 ) suggest for O (t) a pover law behavior near T* P (T* -t)-" with the critical exponents Yl

^.

0.8, y2 = 4 . 2 , y3 1 9 . 9 , Y 4

=

of the form

16.

To locate the first real singularity one can also use Pade approximts instead

of the Euler Transform, this is illustrated by the figure 12 where are plotted several curves concerning the computation of the enstrophy. The use of Euler

312

C. BARDOS

Transform or Pad6 approximants give also some informations concerning the complex singularities of the function s2 (t). 1

These singularities are plotted on the

figure 13. 40

Power serie

R

and Euler

8 0

Re(t)

TP

30 8 0

0

-5.01.

from [24]

Figure 12

Figure 13

-40

Remark 2 : The existence of complex

from

[24

singularities approaching the real domain

may be related (cf. Frisch and Morf [I I]

for a detailed discussion) to the appea-

rance, before the turbulence,of intermittent burst. In fact a high pass filtering process, in the presence of complex singularity t will give a signal of the j following form. I nR(t) = I / &

exp(-iR(t-Ret.)-in/4) c c e-R Im tj Re t - t j

In ( 5 9 )

j

)

(59)

j

R denotes the size of the filter (the Fourier transform of

been made equal to zero for IzI

C

1

(t)

has

R), the asymptotic behaviour is given for R

+

-

by a contour deformation around the half axis starting from the complex singulaI

rities of s2 (t). To be really valid such a computation would require a detail knowledge of the 1

behaviour of s2 (t) but in [ I l l

in the complex domain (This knowledge is not yet available)

one find some computationsconcerning other explicit examples like

the Langevin equation or the Burger Hopf equation (with the use of the Hopf Cole transformation).

313

EOUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

REFERENCES [I]

Baouendi, Cauchy

[2]

S.

and Goulaouic,

C.

: Remarks on the Abstract Form of Non linear

- Kovalevsky Theorems, Comm.

Part. Diff. Eq. (1977) 1151-1162.

Bardos, C. : Existence et Unicitl de la solution de l'Bquation d'Euler en deux dimensions, J. Math. Anal. and Appl. 4 0 (1972) 769-790.

[3J

Bardos, C. and Benachour, S. : Domaine d'AnalycitB des solutions de l'lquation d'Euler dans un ouvert de Rn, Ann. Scuola Norm. Sup. di Pisa, SBrie IV Vol. IV, no 4 (1977) 648-687.

[4]

Birkoff, G.: Helmoltz and Taylor Instability in Hydrodynamic Instability Proc. Symposium on Applied Math. 2 3 , A.M.S.

[s]

Cafarelli, G. and Nirenberg, L. : To appear.

[6]

Chadam, J. : Asymptotics for

2

u = m u +

G(x,t,u,uX,ut),

Global existence

and decay, Annali. Scuola Norm. Sup. di Pisa, Vol. 26 (1972) 31-65. [7]

Ebin, D. and Marsden, J. : Groups of Diffeomorphism and the motion of an incompressible fluid, Ann. of Math. 92 (1970) 102-163.

[8]

Foias,

C.

and Prodi, G. : Sur le comportement global des solutions non

stationnaires des dquations de Navier Stokes en dimension 2 , Rend. Sem. Math. Padova 39 [9]

Foias,

C.

(1967)

1-34.

and Temam, R. : Some analytic and geometric Properties of the

solution of the Navier Stokes Equations, Journal Math. Pure et Appl. Vol. 58

(1979)

339-368.

[lo] Foias, C . , Frisch, U. and Teman, R. : Existence des solutions C"

tions d'Euler, C.R. Acad. Sc. Paris 280 A (1975)

[Ill

des Equa-

505-508.

Frisch, U. and Morf, R. Intermittency in non linear dynamics and singularities at complex times, Phys. Rev. A . 2 3 .

[I21 Glimm, J. : Solutions in the large for non linear systems of equations,

Comm. Pure Appl. Math. 18

(1965)

697-715.

1131 Holder, E. : Uber die Unberchronkte Fortsezbarkeit einer Stetigen ebener

Bewegung. in einer unbegrentzen inkompressiblen Flussigkeit Math. Z ( I 933)

37

727-732.

[ I 4 1 Kato, T. and Fujita, M. : On the non stationary Navier Stokes Systems,

314

C. BARDOS

Rend. Sem. Math. Univ. di Padova 32

[IS] Klainerman, Appl. Math.

(1962),

243-260.

: Global existence for non linear wave equation, Corn. Pure

S.

33

(1980),

43-101.

[16] Kolmogorov, A.N. : The local strucuture of turbulence in an incompressible

viscous fluid for very large Reynolds number C.R. Ac. Sc. U.R.S.S. 30 301 also Soviet Physics Uspekhi 10 : 734.

:

[I71 Kruckov, N.S. : First order quasilinear equations in several space indepen-

dent variables, Math. Sbornik 81 (123) (1970), Math. U.R.S.S. Sbornik 10 (1970), 217-243. [I81 Ladysenskaia, O.A. : Mathematical Theorie of incompressible visquous fluids

Moscou 1961, Gordon

- Breach

New-York

(1963).

[I91 Leray, J. : Sur le mouvement d'un liquide visqueux emplissant l'espace,

Acta Math. 63 (1934). 193-248. [20] Lichtenstein, I. : Uber einige problem der Hydrodynamic, Math. 2.23 (1925). [21] Lions, J.L. : Sur l'existence des solutions de Navier-Stokes C.R. Acad.

Sc. Paris 248, (1959), 2847-2849. 1221 Meiron, D . I . ,

Baker, G. and Orszag, S . : Analytic structure of Vortex Sheet

Dynamics, Kelvin Helmoltz Instability, Preprint M.I.T. Department of Mathematics, Cambridge M.A. 012139. [23] Morf, R.,Orsazg, S. and Frisch, U. : Spontaneous Singularity in Three-

Dimensional, Inviscid, Incompressible Flow, Plys. Review Letters 44,9, (1980), 572-575. [24] Morf, R.,Orszag, S., Meiron, D., Frisch, U. and Meneguzzi, M. : Analytic structure of High Reynolds Number Flows, Seventh Conf. Numerical Methods in

Fluid Dynamics Stanford June 1980, Springer Lecture Notes in Physics. 1251 Orszag, S. and Tang : Small Scale Structure of two dimensional MagnetoHyrdrodynamic Turbulence, J. Fluid Mach. 90 (1979). 129- 136. [26] Ovsjannikov, L. : Singular Operator in Banach Scales, Dokl. Akad. Nank. .U.R.S.S.

163

(1965),

819-822.

[27] Scheffer, G. : Navier Stokes Equation in a Bounded Domain, Comm. Math.

Physic 73 (1980) 1-42.

EOUATIONS OF INCOMPRESSIBLE FLUIDS MECHANIC

315

[28] Sulem, C. : Quelques resultats de regularits pour les equations de la

Magneto-hydrodynamique, Comptes Rendus Acad. Sci. Paris 85 A (1977) 365-367. [29]

Sulem, C., Fournier, J.P., Frisch, U. and Sulem,, P.L. : Remarques sur un modele unidimensionnel pour la Turbulence Magneto-Hydrodynamique, C.R. Acad. Sci. Paris t. 288 (12 mats 1979) Ssrie A. 571-573.

1301 Sulem, C., Sulem,, P.L., Bardos, C. and Frisch, U. : Finite Time Analyticity

for the two and Three Dimensional Kelvin Helmoltz Instability, to appear in Comm. in Math. Physic. [31] Temam, R. and Strang, G. : Duality and relaxation in the variational pro-

blems of plasticity. Journal de Mgcanique, Vol. 19, 3 (1980) 493-527. [32] Wolibner, W. : Un thsorsme sur l'existence du muvement plan d'un fluide

parfait et incompressible pendant un temps infinhent long, Math. 2. 37 (1933), 727-738. l33] Youdovitch, 0.1. : The flow of a non stationary ideal and inviscid fluid

Math.

, Phys.

and Numer Math. J. 6 (1965) 1032-1066.

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Nonlinear Problems: Prerent and Future A.R. Bishop, D.K. Campbell, 8. Nicolaenko (eds.) 0 North-HollandPublishing Company, 1982

317

FINITE PARANETER APPROXIMATIVE STRUCTURE OF ACTUAL FLOWS C i p r i a n Foias and Roger Temam Analyse Numerique e t Fonctionnel l e CNRS e t U n i v e r s i t e Paris-Sud Bdtiment 425 91405 Orsay Cedex (France)

-

INTRODUCTION

Most approaches t o the mathematical understandinq o f the onset o f turbulence (e.q. 161, [4J , [lO] , [l]) assume t h a t the comnlexity o f the dynamical system {S(t)}t>o associated t o the i n i t i a l value problem f o r the Navier-Stokes equations i n some adequate f u n c t i o n a l space H (formed by divergence f r e e v e l o c i t i e s f i e l d s ) i s increasina i n an almost universal monotonic way, w i t h sudden jumps a t some threshold values of a p h y s i c a l l y s i g n i f i c a n t parameter. However t h i s universal p a t t e r n (although displayed by adequately constructed models) was n o t established f o r actual flows, for which even the d e f i n i t i o n o f {S(t))t.o i s n o t y e t s a t i s factory. The o n l y known f a c t , which i s r e l e v a n t t o t h i s problem, i s t h a t f o r Dlane flows ( w i t h s t a t i o n a r y boundary conditions and d r i v i n q forces) a l l i t s t r a j e c t o r i e s S(t)uo converqe, for t -+ m , t o a s e t X m a x c H o f f i n i t e Hausdorff dimension dmax , f o r which a r a t h e r e x p l i c i t , u m e r estimate d ( l / v ) , as f u n c t i o n o f the kinematic v i s c o s i t y v , e x i s t s ([9], c3J5.6). Althouqh d ( l / v ) f m f o r i t i s n o t y e t known whether dmax --t m f o r l / u --c On the o t h e r hand the change from a laminar t o a t u r b u l e n t f l o w o f a f l u i d i n motion i s o f t e n so f a s t t h a t i t i s hard t o conceive t h a t i t s mathematical understanding l i e s o n l y on the qsymptotic behaviour of u ( t ) f o r t j. m and thus on the s t r u c t u r e o f a ( s t i l l hypothetical, i f the flow i s n o t plane) a t t r a c t i n g s e t X c H A way by which nature could s a t i s f y a l l the guested approaches i s given i n the f o l l o w i n g

.

l/vpm

.

Conjecture. Any t r a j e c t o r y u ( t ) i s "very r a n i d e l y " converging ( i n a manifold M of f i n i t e dimension d such t h a t d t m f o r 1/v j.

-. H)

to

To prove t h a t t h i s conjecture i s " g e n e r i c a l l y " t r u e seems t o be outside the grasp o f the present mathematical techniques. Strangely enough, we can g i v e a rigourous p o s i t i v e answer t o an approximative form o f the conjecture. I n order t o s t a ' e t h i s answer l e t us r e c a l l t h a t H (see the next 9.1) i s a r e a l H i l b e r t

I

space and t h a t 1/2 o f the square U )* o f the norm o f u e H represents t h e ( A c t u a l l y we assume t h a t t h e density o f t h e k i n e t i c a l energy o f the f l o w u e H f l u i d i s = 1). This i s our Answer (see 9.3 below). manifold M of dimension i n i t i a l data (1)

l u ( 0 ) l 2 G Eo

.

For given

d<-

, the

O<El<
and O
such t h a t f o r a l l t r a j e c t o r i e s set o f the

distance ( i n H) from u ( t ) t o M o f length = 1 i n t e r v a l c (0,m)

.

< E:12

t's

u(t)

with

f o r which i s o f measure >, 1-T on any

318

C. FOIAS. R. TEMAM

A c t u a l l y t h i s answer i s n o t unexpected since on any s e t f i e l d s such t h a t

VRc H

o f velocities

j R ( c u r l u (2 dx c R

the map S ( t o )

(namely the deplacement

u(0)

~--t.u ( t o )

along the t r a j e c t o r y

u ( t ) ) i s a w e l l defined one t o one compact diffeomorphism ([2], we s h a l l show t h a t d (for

Eo, T

S(to)

, where

and v

Ch. 111). However

( l o g l/El)n

'L

f i x e d ) , which depends on a more p a r t i c u l a r s t r u c t u r e o f f o r plane flows and n = 3 otherwise.

n = 2

might represent the unavoidable experimental e r r o r i n the The q u a n t i t y El estimation o f the energy o f a f l o w o r even might represent a much more basic e n t i t y comparable w i t h the mean k i n e t i c a l energy o f a molecule o f t h e given f l u i d a t r e s t . I n t h i s case i t i s c l e a r t h a t d i f f e r e n t i a t i n g two flows u, v such t h a t l u - v I 2 < El

, although mathematical$meaningful i s v o i d o f any physical basis. But

because o f ( 2 ) even mical f u n c t i o n o f

t h i s microscopical

,T

Eo

El

w i l l y i e l d a reasonable non astrono-

.

and v

The c o n s t r u c t i o n o f M , given i n 5.3, a l s o provides some supplementary information on i t s p o s i t i o n i n H allowing us t o show ( i n 5.4) t h a t the behaviour o f d , as f u n c t i o n o f l / v , i s c o n s i s t e n t w i t h the conjecture. 1 5.1.

PRELIMINARIES ( We s h a l l consider t h e Navier-Stokes equations

(1.1)

ut

+

( u . 0 ) ~ = v Au

+

Vp

+

f

,

V.u = 0

in R

x

(0,~)

where

R i s an open connected bounded s e t c IRn (n=2 o r 3) such t h a t i t s 9 boundary r i s a compact manifold o f class C' o f dimension n-1 , and R i s l o c a l l y located on one side o f an I n order t o s i m p l i f y the e x p o s i t i o n we s h a l l f i r s t consider o n l y the boundary problem

.

As usual, l e t

H

and V

(see L3J .5.2 f o r more d e t a i l s ) denote t h e closure

i n L2(Q)n and H l ( ~ 2 ),~respectively, o f

v = Iv (')

See [31.55.1-3

:

V €

c p ) " : v.v = 0)

for more d e t a i l s .

,

FINITE PARAMETER APPROXIMATIVE STRUCTURE OF ACTUAL FLOWS

319

and AU

(1.3)

=

-

PA^

3 u G ~ ( A d) s f ~n H 2 (0)

for

B(u,v) = P[(u.v)v] where

Sobolev L -space o f order

.

the orthogonal p r o j e c t i o n o f { ~ ~ l of i = H~ such t h a t

a(A)

u,v E

2

~ = 1 , 2 ) denote the

H'(a)

for

R and P denotes

L2(n)n onto H There e x i s t s an orthonormal basis A wm = A, wm (n;=1,2 ), O<x1'<X2< ... and

,...

( 1 -4) (where

c1

as the others

c.'s

J

(j=2,3, ...) denote p o s i t i v e

i n the sequel

constants depending o n l y on 9 ) . The orthogonal p r o j e c t i o n i n H onto Rwl t + IRwm w i l l be denoted by Pm (m=1,2 ; Po = 0) We s e t

...

.

,...

IuI w i l l be t h e norm on H , IIvI( t h a t on V and (u,,~,) and ((v1,v2)) w i l l denote the corresponding s c a l a r products. The operator B i s continuous from d)(A) x V and V x a ( A ) t o H as w e l l as from V x V i n t o the dual V ' o f V. For many more f i n e estimates on B we r e f e r t o [3],5.1. Also B enjoys the f o l l o w i n g basic o r t h o g o n a l i t y property B(u,v),v)

(1.5)

= 0

whenever the l e f t hand s i d e makes sense. (1.2) can be With the above notations the i n i t i a l value problem f o r (l.l), w r i t t e n i n H , namely

(1.6) where

uOcH

du t v Au t B(u,u) = P f

for

t>O

,

u(0) = uo

i s given. For the o n l y sake o f s i m p l i f y i n g the presentation i n the

sequkl we s h a l l assume t h a t

Pf = 0

, i.e.

t h a t the forces are p o t e n t i a l .

The f o l l o w i n g are w e l l known c l a s s i c a l r e s u l t s (see, f o r instance, [5], o r [ll]) :

(')

The problem (1.6) has a (weak) s o l u t i o n

u( .)

A c t u a l l y we s h a l l use o n l y the f a c t t h a t where c2 i s s u i t a b l y chosen, O
A,,

.

[6]

such t h a t

>, c2 m2'n

for a l l

m = 1,2,

...

C. FOIAS, R. TEMAM

320

A c t u a l l y , because o f our assumption on

I f uo= V

f

we have

then there e x i s t s an i n t e r v a l

[O,t(u,))

, where

on which a l l (weak) s o l u t i o n coincide w i t h a ( s t r o n g o r r e g u l a r ) s o l u t i o n , i . e .

For such a strong s o l u t i o n t h e d e f i n i t i o n S(t)uo = u ( t )

(1.11)

makes sense. Moreover i n case has

t(uo) =

interval

[tO,m)

Ogt
,or

i n case

n=3 b u t if IIuo1/4_c c4v4

, one

and any weak s o l u t i o n i s unique and obviously i s r e g u l a r on any (to>O)

.

Among the other r e g u l a r i t y p r o p e r t i e s o f S(t)uo l e t us mention o n l y t h e f a c t t h a t S(t)uo i s a a(A)-valued a n a l y t i c f u n c t i o n o f t E (O,t(uo)) such that

(see [3],5.3).

5.2.

THE SQUEEZING PROPERTY

I n the sequel, a basic r o l e w i l l be played by the f o l l o w i n g squeezing p r o p e r t y of S ( t ) , which we obtained i n [3] ,§.5 :

321

FINITE PARAMETER APPROXIMATIVE STRUCTURE OF ACTUAL FLOWS

whenever

where

0 < ClY C Z y C3 <

A c a r e f u l perusat the estimates

where the constants

m

do n o t --

c8 and c9

or

m

.

are n o t dimensionless. The estimates (2.4) are

.

n=3 ( A c t u a l l y f o r n=2 2 and 4 , r e s p e c t i v e l y ) .

n=3 i t can be shown t h a t

some dimensionless

t,u,v

o f the p r o o f (given i n [3],5.5) o f t h i s property y i e l d s

v a l i d f o r both cases n=2 and 5 i n ( 2 . 4 ) can be decreased t o I n case

depend on

constants

coly

cll

the exponents 5 / 2 and

c8 3 0cl Xi/* and c9 d c Introducing the variables

with

.

the r e l a t i o n (2.3) can be w r i t t e n under the form

5

(2.6)

5 K

'R; Kd)

c12

(with

c12 dimensionless)

which can be i n t e r p r e t e d as showing t h a t thefrequencies K f o r which the squeezing property o f S ( t ) holds are l y i n g f a r i n s i d e the d i s s i p a t i v e spectrum o f the flow. This s t r o n g l y suggests t h a t the exponents i n ( 2 . 4 ) must be much nearer t o 1/2 and 1, respectively. 5.3.

THE APPROXIMATIVE STATIONARY MANIFOLD

L e t 0 < El << 1 << Eo

(3)

ci =

2 exp(2cg

Ci").

0<~<<, 1 be such t h a t

A c t u a l l y we can drop these conditions, b u t t h i s

would bake the expression (3.2) much more complicated.

C. FOIAS, R. TEMAM

322

Then we have the f o l l o w i n g r e s u l t , l o s e l y Theorem

1 - There e x i s t m p r o p e r K

and a

--

J, :

s t a t e d i n the I n t r o d u c t i o n : PmH

IJ,(PmU)-$(PmV)l C IPmu-PmV I

(3.3)

and such t h a t for any weak -----{ t c (0,m)

(3.4)

satisfying

(U,VE H)

u(.)

solution

M c H -i s the graph o f J, J, interval c (0,m) o f length = 1 , of R

)

to

set M

i s o f measure >/ 1

and

.

the

,

of (1.6)the

: distance ( i n H) from u ( t )

(where

-

(I-P,)H

I+

J,

-

,

< ]":E T

on any

i s the dimension

n ( = 2 o_r 3)

Before passing t o the proof, l e t us n o t i c e t h a t MJ, i s obviously a L i p s c h i t z manifold o f dimension m , which can be c a l l e d an approximative s t a t i o n a r y manif o l d o f (1.6). --

Proof-of-Theorem-lL e t u(.)

Thus f o r

any

be a (weak) s o l u t i o n o f (1.6)

r>O ,

and consequently i f (3.5) then

Let

to = t o ( r )

and choose a subset

M(m) o f

S(to)Cuo=V:l~uol~c r l maximal under the property

such t h a t

luol

2

< Eo

. Then

FINITE PARAMETER APPROXIMATIVE STRUCTURE

where

m

OF ACTUAL FLOWS

323

satisfies

(3.8) Then, by v i r t u e o f the squeezing property (see 5.2), we have t h a t t h e distance ( i n H) o f u(s+to) = S(to)u(s) t o M(m) i s bounded by

whenever

Ilu(s)ll 4 r

so t h a t i f to < 1

. But f o r

- 9 , then

~ { S C (t,t+l) :

any

, we

tat,

have obviously

by (3.6

t30 :

for all

\Iu(s)II 4 r } ] 3 1 - T / Z

-

to

.

I f we assume now t h a t

then i t f o l l o w s t h a t the s e t o f i s Q E:/' M(m)

has measure 3 1

-

T

t's

such t h a t the distance o f

on any i n t e r v a l

i s the graph o f the f u n c t i o n $,

: Pm M(m)

M(m) By (3.7),

u(t)

.

to

(t,ttl) c ( O p ) -(I-P,)H defined by

which moreover s a t i s f i e s the c o n d i t i o n (3.11)

lJI0(Pmu)

- $o(pmv)l <

IPmU

-

P,Vl

(Pmu, Pmv€ P, M(m))

.

can be extended on the By v i r t u e o f the Kirszbaum extension theorem (see [8]) $ whole HP, t o a f u n c t i o n JI s a t i s f y i n g (3.3). Thus i t 6Lemains o n l y t o v e r i f y (3.2). To t h i s aim we f i r s t note t h a t due t o the r e l a t i o n (1.9) and the f i r s t r e l a t i o n (3.1), the second c o n d i t i o n (3.10) i s a d i r e c t consequence o f (3.5). S i m i l a r l y , due t o the second c o n d i t i o n (3.1), (3.5), and the f i r s t c o n d i t i o n (3.10) imply (3.8). F i n a l l y we must have t o impose (3.5) and the f i r s t c o n d i t i o n (3.10). We define r by r e p l a c i n g (3.5) by an e q u a l i t y . Replacing the f i r s t r e l a t i o n (3.10) by an e q u a l i t y and t a k i n g (1.4) i n t o account, we g e t (3.2).

P

We remark t h a t i n case n=2 the approximative s t a t i o n a r y manifold can be chosen such t h a t any s o l u t i o n e v e n t u a l l y becomes and remains w i t h i n d i s ance

E:12

of

Ma

.

C. FOIAS. R. TEMAM

324

5.4.

THE BEHAVIOUR FOR w --+

0.

We want t o study the behaviour o f the dimension m o f approximative s t a t i o n a r y manifolds f o r w --+ 0 I n order t o avoid very d i f f i c u l t boundary l a y e r problems we make the remark t h a t a l l the previous considerations remain

.

v a l i d i f R i s replaced by Q = [O,LIn whew L>O i s f i x e d ,and the boundary problem (1.2) i s replaced by t h e p e r i o d i c c o n d i t i o n s (4.1)

u(O,..)

... , u(..,O)

= U(L,..),

= u(..,L)

.

S i n c e the Navier-Stokes equations are Galllean i n v a r i a n t , we can assume, by a change o f the reference frame, t h a t t h e s o l u t i o n s v e r i f y a l s o

.

I Q u dx = 0 Therefore

H and V become t h e closure i n L2 (9)n and H 1

o f the space o f a l l

IRn-valued trigonometric polynomials w(x)

;1

v.w

dx = 0 and

E

0

, respectively, such t h a t

.

A l l the other d e f i n i t i o n s and p r o p e r t i e s remain e s s e n t i a l l y unchanged. A c t u a l l y i n t h i s case much more concrete a l g e b r a i c s t r u c t u r e i s present. For instance the X,

are now o f the form 4a2 L-21k[

with

k c Zn

, k#O

and wAu

+

B(u,u)

has the f o l l o w i n g property. For any rn = 1, 2,

_.-

... t h e r e e x i s t s an i s o m e t r i c operator U

: H e(I-P,)H

such t h a t --

where X -

i s some adequate constant such t h a t

0<

(4.4) One defines

U w

j F i r s t choose t h e f i r s t

x 4 C14 x,2 .

( j = 1, 2, ...)

according t o t h e f o l l o w i n g scheme :

!2(=1,2, ...) such t h a t ~ 4 ' "

A . = 4rr L-21k12 chose J such t h a t AJ = 4a L-21Q'k12

L-2 > A,

, then if

and s e t Uw. wJ J J (Here some supplementary care i s necessary i n order t o avoid the complication due t o the f a c t t h a t the m u l t i p l i c i t y o f h o r XJ i s not simple. Since we i n t e n d J t o study t h i s s e l f s i m i l a r i t y phenomena elsewhere, we d o n ' t i n s i s t w i t h the d e t a i l s o f t h e proof.) F i n a l l y , f o r u0= V the s o l u t i o n s o f (1.6) s a t i s f y , i n case n=2 , besides the r e l a t i o n s (1.8), t h e r e l a t i o n s

FIN1TE PARAMETER APPROXIMA TI VE STRUCTURE OF ACTUAL FLOWS

--

o f (1.6). rf E1/Eo and v1/3 and El/Eo < 1/36 (4.6)

Proof _----

mv

(for instance

then

= dimension o f I$

--f

m

for

+

v

.

0

:

Mv

Let

=

M

where

J,,

= J,

J,V

I f u,v

in

are s u f f i c i e n t l y small

T

325

E

M

and t h e i r distance t o

H

enjoys the p r o p e r t i e s given i n Theorem 1.

$,

i s Q El

, then

there e x i s t

u*, vx

with

J,V

so t h a t

( u - v l & 6E:"

(4.7) Therefore i f satisfying v( . ) )

u(.)

t 21Pm ( u - v ) ) V

and v ( . )

luo19 l v o l \< Eo

.

are s o l u t i o n s o f (1.6) w i t h

i t f o l l o w s from (4.7) and (3.4)

u(0) = uo v(0) = vo ( f o r u ( . ) and

that I{tE ( 0 , l ) : l u ( t ) - v ( t ) I

.<

6E:"

t

ZIP,

(u(t)-v(t))l)l > 1

-

2.r

V

Since f o r v( .) we can take v ( t ) 0 , we o b t a i n t h a t f o r any s o l u t i o n of (1.6) such t h a t luol < Eo the f o l l o w i n g holds

(4.8)

I ( t e ( 0 , l ) : lu(t)l

,c 6E;l2

u(t)ll >

t 21P,

Now we assume t h a t there e x i s t s a sequence v L j a l l j=lYZ9. L e t now v be any o f these v j

O

operator considered i n (4.3) and l e t uo = UE:/2

w1

.. .

such t h a t

and l e t U

1

-

T

u( .)

.

mv

.(. m < 03 for j be t h e i s o m e t r i c

. We denote by

;(.)

the

C. FOIAS, R. TEMAM

326

s o l u t i o n o f (1.6) w i t h i n i t i a l data

Ei/2wl

dUu" t ~ [ U A U G+ B(UG,Uii)] and therefore

u ( t ) = (UG)(t/X)

.

. Then by

(4.3)

o

=

i s a s o l u t i o n o f (1.6) w i t h i n i t i a l data

.

Thus (4.8) i s v a l i d f o r t h i s u ( . ) But because uo(= E A j 2 Uwl) , luol = EA/2 UH i s orthogonal on PmH i t f o l l o w s t h a t P, u ( t ) E 0 and t h e r e f o r e (4.8) V

becomes

But, by (1.8) and (4.5),

so t h a t (4.9) i m p l i e s :

(4.10)

l{tc(0,l)

: Eo

Q

36E1

L2Xm

+

2 v t Eo

From (4.10) i t f o l l o w s t h a t there e x i s t s a Eo 6 36E1

+

t = (0,Z-r)

3 1- T

.

such t h a t

LLX, 2 v t Eo 71

and consequently E, .< 36E1

+

L2Xm 4 v ~Eo 71 f o r a l l

We conclude w i t h the c o n t r a d i c t i o n Eo

<

36E1

v = vj

(j=1,2 ,...)

.

. This proves t h e Theorem.

A c t u a l l y the Theorem i s a l s o v a l i d i f n=3 b u t t h e p r o o f must be modified i n order t o work w i t h small s o l u t i o n s i n V , and consequently the c o n d i t i o n Eo/E1 < 1/36 has t o be replaced by Eo/E1 < c15 w i t h an adequate constant c15 depending on t h e constant

c4

(see 5.1).

REFERENCES

[l] M.J. Feigenbaum, J . Stat. Phys. 21, 669 (1979); Phys. L e t t . 74J, see a1 so these proceedings.

375 (1979);

FINITE PARAMETER APPROXIMATIVE STRUCTURE OF ACTUAL FLOWS

S o l u t i o n s s t a t i s t i ues des 6 u a t i o n s Cours au c h a n c e

327

& Navier-Stokes

[2]

C. Foias,

[3]

C. Foias and R. Temam, Some a n a l y t i c and geometric p r o p e r t i e s o f t h e

s o l u t i o n s o f t h e e v o l u t i o n Navier-Stokes equations J. Math .pure e t appl , 58 ( 1978) pp.339-369.

.

141 E. Hopf, A mathematical example d i s p l a y i n g f e a t u r e o f turbulence Com.pure and appl. Math., 1 (1948) , pp.303-322. Ladyzenskaya, The mathematical theory o f viscous incompressible f l o w s Gordon-Breach , New York ( 1--

C5-l

O.A.

[6]

L.D. Landau and E.M. L i f s h i t z , F l u i d Mechanics Pergamon, Oxford ( 1 9 5 r

[7]

J.L. Lions, Quelques m6thodes de r e s o l u t i o n &probli?mes 11neai r e s Dunod-Gauthier-Vil l a r s , P a r i s (1969)

181 J.H. Wells and L.R. Williams, Imbeddin s and extensions Springer-Verlag, HeiaFEEKj&wTrk.

aux l i m i t e s non

~ I JA n a l y s i s

J. M a l l e t - P a r r e t , N e g a t i v e l y i n v a r i a n t sets o f compact maps and an e x t e n s i o n of a theorem o f C a r t w r i g h t J. D i f f . Equations, 22 (1976), pp. 331-348.

[9]

[lo] D. R u e l l e and F. Takens, On t h e n a t u r e o f turbulence rl01 Comm. Math. Phys., 20 (1971), pp.167-192 ; Corn. Math. Phys., 23 (1971, pp.343-344. I_

2

[Ill R . Temam, Navier-Stokes equations. Theory and Numerical A n a l y s i s North-Holland, A m s t e r d a m m o r k

m r

This Page Intentionally Left Blank

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North-Holland Publishing Company, 1982

329

THE ROLE OF CHARACTERISTIC BOUNDARIES I N THE ASYMPTOTIC SOLUTION OF THE NAVIER-STOKES EQUATIONS

F. A. Howes Department of Mathematics U n i v e r s i t y of C a l i f o r n i a , Davis Davis, C a l i f o r n i a 95616 U.S.A.

We d i s c u s s s e v e r a l boundary v a l u e problems f o r t h e s t e a d y Navier-Stokes e q u a t i o n s i n two dimensions when t h e approp r i a t e Reynolds number Re i s v e r y l a r g e . The e s s e n t i a l i d e a i s t h a t c e r t a i n curves i n t h e flow f i e l d on which t h e c o n d i t i o n s of n o - s l i p and impermeability a r e imposed a r e c h a r a c t e r i s t i c curves ( i n t h e mathematical s e n s e ) f o r t h e system of d i f f e r e n t i a l e q u a t i o n s governing t h e conservat i o n of momentum. This o b s e r v a t i o n , coupled w i t h some of t h e a u t h o r ' s r e c e n t r e s u l t s on s i n g u l a r l y p e r t u r b e d e l l i p t i c boundary v a l u e problems, l e a d t o s e v e r a l mathematically r i g o r o u s e s t i m a t e s on t h e b e h a v i o r of s o l u t i o n s of t h e Navier-Stokes e q u a t i o n s as R e -,0 . F o r example, o u r approach shows why t h e t h i c k n e s s of t h e boundary l a y e r i n P r a n d t l ' s theory i s of o r d e r w i t h o u t having t o res o r t t o t h e u s u a l h e u r i s t i c arguments. 1.

STATEMENT OF THE PROBLEM

Let us c o n s i d e r t h e two-dimensional, s t e a d y flow of a homogeneous, v i s c o u s , i n compressible f l u i d n e a r a p l a n e boundary. The e q u a t i o n s which govern t h e v e l o c i t y and p r e s s u r e f i e l d s , w r i t t e n i n dimensional form, are t h e Navier-Stokes equat i o n s ( c o n s e r v a t i o n of momentum)

+ v*u* +

vA*u* = u%*

9

5

p-'p*

5

s + v*v*9 + p-lp*1'

= u*v*

"A*V*

and t h e c o n t i n u i t y e q u a t i o n ( c o n s e r v a t i o n of mass) u*

+ v*

5 . 7

= 0;

c f . [ l ; Chap. 31. Here $(u*) and 9(v*) a r e t h e t a n g e n t i a l and normal components of d i r e c t i o n ( v e l o c i t y ) , r e s p e c t i v e l y , A* = a2/aS2 + a2/aT2, p is t h e c o n s t a n t d e n s i t y , p* is t h e dynamic p r e s s u r e ( t h a t i s , t h e d i f f e r e n c e of t h e a c t u a l p r e s s u r e from t h e h y d r o s t a t i c p r e s s u r e ) and v = p/p is t h e kinematic v i s c o s i t y . I n o r d e r t o s t u d y t h i s system more e f f e c t i v e l y , we r e n d e r t h e v a r i a b l e s dimens i o n l e s s by r e f e r r i n g d i s t a n c e s t o a r e p r e s e n t a t i v e l e n g t h L and v e l o c i t i e s t o a uniform speed U, t h a t i s , x = b/L, u = u*/u,

y = 7)/L 2 p = p*/(pu ).

v = v*/u,

These changes of v a r i a b l e allow us t o rewrite t h e e q u a t i o n s of motion a s cAu = uu SAV

x = uv X

u

+ vuY + pX + wY + pY =o,

+ v X

Y

F.A. HOWES

330

where A = a 2 / a x 2 + a2/ay2 and E = 1 / R e = v/(UL), f o r Re t h e d i m e n s i o n l e s s group known as t h e Reynolds number. The problem is completed by r e q u i r i n g t h a t u and at a solid v s a t i s f y t h e c o n d i t i o n s of n o - s l i p (u=O) and i m p e r m e a b i l i t y (-0) boundary i n t h e flow f i e l d , and t h a t t h e v e l o c i t y f i e l d (u,v) approach t h e u n i form flow away from such boundaries.

-

Our s t u d y of t h i s boundary v a l u e problem w i l l concern t h e c a s e of h i g h Reynolds ( o r e q u i v a l e n t l y , E -I0 ) , and so we t u r n now t o s e v e r a l approxnumber flow Re -.. imate forms of t h e system which u t i l i z e t h e s m a l l n e s s of E . 2.

EULERFL(XJS

I f we f o r m a l l y set c = 0 i n (N-S) w e o b t a i n t h e E u l e r e q u a t i o n s of motion f o r a n i n v i s c i d ( i d e a l ) f l u i d , namely

--

_-

-

+ pX = 0 -u v +sv + p = o X Y Y uu

+ v u

Y

X

(E)

-

-

u

(C 1

+ v

X

Y

so;

c f . [ l ; Chap. 61. The system (E) i s f i r s t - o r d e r , whereas t h e corresponding s y s t e m (N-S) is second-order, and consequently, t h e system (E), (C) t o g e t h e r w i t h t h e o r i g i n a l boundary c o n d i t i o n s i s a n over-determined problem. I n o r d e r t o obt a i n a meaningful s o l u t i o n of (E), (C), one t r a d i t i o n a l l y d r o p s t h e c o n d i t i o n of n o - s l i p a t a s o l i d boundary C on t h e grounds t h a t t h e t a n g e n t i a l component U of a n i d e a l flow c a n s l i d e , t h a t is, ii # 0 on C. C l e a r l y , t h e n , t h e s o l u t i o n fi,;,;) of t h e E u l e r problem cannot b e a uniformly v a l i d approximation t o t h e s o l u t i o n of t h e v i s c o u s (c > 0) problem, s i n c e i n t h e neighborhood o f a s o l i d boundary u and ii c a n d i f f e r by a n amount o f o r d e r one. A s t r i k i n g m a n i f e s t a t i o n of t h i s f a i l u r e of t h e E u l e r flow t o model t h e a c t u a l flow i s g i v e n by d'Alembert's paradox; c f . [ l ; Chap. 51.

3.

PRANDTL'S BOUNDARY LAYER THEORY

I n 1904 L. P r a n d t l [ l l ] (see a l s o [12; Chap. 71) was a b l e t o r e s o l v e d ' b l e m b e r t ' s paradox, and a t t h e same time, t o d e r i v e a s i m p l i f i e d form o f t h e Navier-Stokes e q u a t i o n s whose s o l u t i o n s modelled q u i t e c l o s e l y t h e b e h a v i o r of t h e a c t u a l flow n e a r a s o l i d boundary. The s t a r t i n g p o i n t of h i s i n v e s t i g a t i o n was t h e n o - s l i p boundary c o n d i t i o n , which l e d him t o r e a s o n t h a t t h e r e is a t h i n l a y e r n e a r t h e boundary i n which v i s c o u s f o r c e s outweigh i n e r t i a l and p r e s s u r e f o r c e s i n t h e d i r e c t i o n normal t o t h e flow. That is, n e a r a s o l i d boundary C t h e g r a d i e n t s ux, vx, and vy change v e r y l i t t l e r e l a t i v e t o t h e normal g r a d i e n t uy. A s a r e s u l t , some terms i n (N-S) could b e n e g l e c t e d , a t least n e a r C. T h i s t h i n l a y e r i n which P r a n d t l ' s s i m p l i f i e d e q u a t i o n s a r e v a l i d was termed t h e "boundary layer". F o r example, we can d e r i v e P r a n d t l ' s boundary l a y e r e q u a t i o n s i n t h e c a s e of a boundary p a r a l l e l t o t h e x - d i r e c t i o n and l o c a t e d a t y = 0 i f we proceed f o r m a l l y as follows. We begin by r e s c a l i n g t h e y and v v a r i a b l e s by s e t t i n g

u = ylc

Q

, 0

= vld,

i3

-

u,

0 = u X

6

8 = x,

f o r p o s i t i v e c o n s t a n t s ff and p t o b e determined. c o n t i n u i t y e q u a t i o n (C) and o b t a i n

= p,

Then we s u b s t i t u t e i n t o t h e

+ v Y = Q, + E $ - Q c u .

P h y s i c a l arguments s u g g e s t t h a t = p i s t h e o n l y a c c e p t a b l e c h o i c e of Proceeding next t o t h e momentum e q u a t i o n s (N-S) we o b t a i n sQ + p Q Q = BQ + + p^ 8, 00 e a e (N-S ' )

*

%If3

+

ouu = G e e + %U + a

-I*

PU*

and 8.

CHARACTERISTIC BOUNDARIES IN NA VIER-STOKESEQUATIONS

331

Once again, p h y s i c a l arguments r e l a t e d t o t h e matching of t h e flow n e a r t h e bound a r y w i t h t h e E u l e r flow suggest t h a t t h e c o n s t a n t cy must be 1/2. The second equation i n (N-S') implies t h a t t o lowest o r d e r i n s, $ = 0 i n t h e boundary 0 i n (N-S') w i t h l a y e r , t h a t is, $ = B, t h e i n v i s c i d p r e s s u r e . I f we now Pet s (Y = 1 / 2 and r e w r i t e t h e l i m i t i n g e q u a t i o n s i n t h e o r i g i n a l , dimensionless v a r i a b l e s , then we o b t a i n P r a n d t l ' s c e l e b r a t e d boundary l a y e r e q u a t i o n

-

-

= uu + vu mix. YY x Y This e q u a t i o n is p a r a b o l i c with y a s p a c e - l i k e and x a t i m e - l i k e v a r i a b l e . The choice of cy = 1 / 2 implies t h e formal r e s u l t t h a t t h e t h i c k n e s s of t h e boundary l a y e r is p r o p o r t i o n a l t o s1/2, t h a t is t h e s i z e of t h e r e g i o n of nonuniformity o f t h e Euler approximation is of o r d e r ,172. EU

4.

MATHEMATICAL BOUNDAUY LAYER THEORY

S e v e r a l mathematicians have been a b l e t o prove, under v a r i o u s assumptions on t h e flow f i e l d , t h a t t h e s o l u t i o n of t h e Navier-Stokes e q u a t i o n s (N-S) reduces t o t h e s o l u t i o n of t h e P r a n d t l e q u a t i o n (P) as E -+ 0 n e a r a s o l i d boundary,'and t h a t t h e s o l u t i o n of (N-S) reduces t o t h e "reasonable" s o l u t i o n of t h e E u l e r system (E) as c 0 away from such curves. We mention only t h e work of O l e i n i k [ l o ] , Nickel [91 and F i f e [41 which t h e r e a d e r can c o n s u l t f o r d e t a i l s and f u r t h e r r e f e r e n c e s .

-

5.

SOME SINGULAR PERTURBATION THEORY

I n o r d e r t o m o t i v a t e o u r d i s c u s s i o n of t h e Navier-Stokes e q u a t i o n s , which t o a c e r t a i n e x t e n t o b v i a t e s t h e use of P r a n d t l ' s approximation, l e t us c o n s i d e r s e v e r a l simple s i n g u l a r l y perturbed problems. The f i r s t example is t h e problem

1 where D(0;l) is t h e open u n i t d i s k i n R 2 , S is t h e u n i t c i r c l e , and m is a p o s i t i v e c o n s t a n t . This problem can be solved by s e p a r a t i o n o f v a r i a b l e s , and w e f i n d t h a t i t s unique s o l u t i o n u = u ( x , y ; c ) , t o t r a n s c e n d e n t a l l y small terms, can be w r i t t e n a s -1 112 2 2 U ( X , Y ; C )= B ( l v ( x , y ) ( e x p [ ( x +Y -I)(= ) I) I n o t h e r words, f o r (x,y) i n

m.

l i m u(x,y;c) = 0

-0 i n every compact s u b s e t of D ( O ; l ) , and u d i f f e r s from 0 by a n amount o f o r d e r one i n a r e g i o n (boundary l a y e r ) of o r d e r ,112 s i n c e t h e boundary d a t a Q is n o t n e c e s s a r i l y zero. We n o t e t h a t F = 0 i s t h e s o l u t i o n of t h e reduced (Euler) equation obtained from (I) by formally s e t t i n g s = 0, and t h a t t h e o r d e r of t h e d i f f e r e n t i a l e q u a t i o n drops from two t o zero. 5

The next example shows t h a t t h e boundary l a y e r can a l s o be of o r d e r

E A U = xu X

(11)

+ yuY + u,

u = Cp(X,Y), I f we s e t s = 0, we o b t a i n

c,

namely

(x,y) i n D(0;1),

(X,Y) on

s

1

xii + yTi = -Ti, x y which i s E u l e r ' s r e l a t i o n f o r homogeneous f u n c t i o n s of degree (-1) ( c f . [ 2 ; 0. From t h e e x a c t s o l u t i o n Chap. 11). I t s only s i n g u l a r i t y - f r e e s o l u t i o n is Ti

F.A. HOWES

332

o r from the theory of t h e author [5] we s e e t h a t the s o l u t i o n of (11) can be expressed as

-

u (x,y ;e

I

I

B( cp(x,y) exp [ (x2+y2- 1 ) (1-6) 16 I

f o r (x,y) i n where 0 < 6 < 1. Here we observe t h a t i n s e t t i n g 8 = 0 the order of the d i f f e r e n t i a l equation drops from two t o one. The r e s u l t i n g f i r s t order equation (*) has c h a r a c t e r i s t i c (base) curves given by the solutions of dx t h a t i s , they a r e the one-parameter family of s t r a i g h t l i n e s y = cx. It is c l e a r t h a t they e x i t D(0;l) along S1 nontangentially, a f a c t which implies t h a t t h e boundary layer i s of order e (cf. [8] o r [5,61) since the gradient terms i n (11) a c t u a l l y determine the boundary layer s t r u c t u r e . We consider next a problem i n which t h e right-hand s i d e contains gradient terms and y e t , the solution behaves near the boundary as i f these terms were absent. It i s (cf. [51) eAu = yu X

(1111

- xu

Y

+ u,

(x,y) in D(O;l),

(x,Y) on s

u = cp(x,y),

.

1

The c h a r a c t e r i s t i c curves of the reduced (6 = 0) equation a r e given by the solutions of the system d -dx= "I ds y8 2 Clearly the t h a t is, they a r e the family of concentric c i r c l e s x2 + y2 = c boundary 51 is i t s e l f a c h a r a c t e r i s t i c curve, and from the asymptotic form of the solution i n D(0;l)

f

-

u(x,y ;e

I

= O( lCp(x,y) exp [ (x2+y2-1)/s 1 / 2 ~)

.

,

we see t h a t t h e boundary layer i s of order s112. I n other words, near the boundary the solution u of (111) behaves l i k e t h e solutfon of problem ( I ) (with -1). The gradient terms in (111) have no e f f e c t s i n c e the boundary i s c h a r a c t e r i s t i c , a f a c t t h a t can be expressed a n a l y t i c a l l y by the r e l a t i o n (y,-x)

0(xZ+y2-1) = 0.

This s i t u a t i o n i s t y p i c a l i n cases where the boundary of the region is a characte r i s t i c surface of the f i r s t - o r d e r reduced equation; c f . [71 and [5,61. The f i n a l example of t h i s s e c t i o n has a s o l u t i o n which displays behavior analogous t o t h a t encountered in Examples (11) and (111). The problem is

z,, &. s-,

where R = 9 x 9, 9 = (-1,l) and a R = U U U f o r .& = [+1] X 9 and The boundary of the square R can a l s o be represented a s t h e s e t E+ = 9 x {+1]. Clearly we have t h a t (7x,y): F(x,y) = O] f o r F(x,y) = -(1-x2)(1-y2). along C-:

(-1,O)

along C+:

(-1,O)

OF

and

>

0

<

0.

However, along the s i d e s E+ and 2- we'have t h a t (-1,O)

OF

3

0.

Thus, in view of the two previous examples, we might expect t h a t the s o l u t i o n of Example (Iv) exhibits boundary layer behavior along t h e s i d e s C,, and E-, and that the thickness of the boundary layer region i s of order ~ ( 6 1 1 2 ) along Z-(& and E-). This turns out t o be t h e c o r r e c t asymptotic behavior of the solut i o n u = u(x,y;o); c f . [31, [6]. We note f i n a l l y t h a t along the s i d e 'Z+ where (-1,O) OF < 0 there is no boundary l a y e r , and t h a t t h e boundary data along Z+

.

CHARACTERISTIC BOUNDARIES IN NA VIERSTOKES EQUATIONS

333

a c t u a l l y determine t h e l i m i t i n g s o l u t i o n i n R. Indeed, i t follows from t h e theory that lim U(X,Y;E) = Cp(1,Y) 6 - 0

i n each compact subset of

R.

The examples of t h i s s e c t i o n are meant t o i l l u s t r a t e t h e i d e a t h a t i t is t h e i n t e r a c t i o n of t h e C h a r a c t e r i s t i c curves of t h e reduced ( E = 0) d i f f e r e n t i a l equat i o n with t h e boundary of t h e region t h a t determines t h e l o c a t i o n and t h e s i z e of t h e boundary l a y e r s . With t h i s as background, we r e t u r n f i n a l l y t o t h e NavierStokes system. 6.

CHARACTERISTIC CURVES ASSOCIATED W I T H THE NAVIER-STOKES EQUATIONS

The reduced system corresponding t o t h e Navier-Stokes equations is t h e Euler system uu uv

x

+ vuy -+w =

-px -py.

(We have dropped t h e overbars f o r s i m p l i c i t y . ) This i s a q u a s i l i n e a r system of f i r s t - o r d e r equations whose p r i n c i p a l p a r t is t h e v e l o c i t y f i e l d ( u , v ) ( c f . [2; App. 1 of Chap. 21); consequently, t h e c h a r a c t e r i s t i c equations a s s o c i a t e d with (E) a r e ~~

Thus, f o r small values of 6 we expect, based on t h e c o n s i d e r a t i o n s of $5, t h e boundary layer phenomena i n t h e Navier-Stokes system are r e l a t e d t o t h e a c t i o n of t h e s e c h a r a c t e r i s t i c curves w i t h t h e geometry of a s o l i d boundary flow f i e l d . I n o t h e r words, i f t h e boundary of a plane region is described equation F(x,y) = 0, then t h e s i g n of the i n n e r product

that interin t h e by t h e

(*) (U(X,Y), V(X,Y)) W(X,Y) should determine t h e l o c a t i o n and t h e s i z e of t h e boundary l a y e r s .

For example, t h e c o n d i t i o n s of n o - s l i p (u = 0) and impermeability (v = 0) make t h e inner product (*) vanish i r r e s p e c t i v e of t h e behavior of W, and so we know t h a t such a boundary i s a c h a r a c t e r i s t i c curve f o r t h e E u l e r system (E). We expect t h n t h a t t h e s i z e of t h e boundary l a y e r along such a curve 'is p r o p o r t i o n a l t o e l T 2 ; c f . Examples (111) and (IV). S i m i l a r l y , f o r p a r a l l e l flow from l e f t t o r i g h t in the r e c t a n g u l a r region D = (0,L) X (0,L) C R 2 , t h e entrance ( e x i t ) s e c t i o n x = 0 (x = L) i s noncharact e r i s t i c f o r t h e u-equation, and s i n c e t h e flow e n t e r s D along x = 0 we should p r e s c r i b e d a t a f o r u there. However, s i n c e t h e flow e x i t s along x = L we expect

a boundary l a y e r t h e r e of order c f o r t h e u-variable ( c f . Example (IV)). A s regards t h e walls y = 0, L, we note t h a t they a r e ch racteristic f o r t h e u - v a r i a b l e , and so we expect boundary l a y e r s t h e r e of o r d e r ~ l f 2f o r t h e u-variable. The s e c t i o n s x = 0, L are now The s i t u a t i o n i s reversed f o r t h e v-variable. c h a r a c t e r i s t i c f o r i t , leading us t o expect boundary l a y e r s of o r d e r ,112 t h e r e ; while t h e walls y = 0, L a r e n o n c h a r a c t e r i s t i c . Depending on boundary c o n d i t i o n s such as s u c t i o n o r blowing, we expect t o f i n d boundary l a y e r s of o r d e r E along y = 0, L f o r t h e v - v a r i a b l e . These considerations can be extended t o more complicated geometries in l R 2 and t o problems i n t h r e e dimensions. The i d e a i s e s s e n t i a l l y t h e same: i t is t h e i n t e r a c t i o n of t h e c h a r a c t e r i s t i c s of t h e a s s o c i a t e d E u l e r system with t h e s o l i d boundaries i n t h e flow f i e l d t h a t determines t h e l o c a t i o n and t h e s i z e of t h e boundary l a y e r s in t h e a c t u a l flow. ACKNOWLEDGMENTS. The a u t h o r wishes t o thank t h e National Science Foundation f o r i t s f i n a n c i a l support and Ida Mae Orahood f o r typing t h e manuscript.

F.A. HOWES

REFERENCES Batchelor, G. K., An Introduction to Fluid Dynamics (Cambridge U. Press, Cambridge, 1970). Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I1 (Interscience, New York, 1962). ECkhaUs, W. and de Jager, E, M., Asymptotic Solutions of Singular Perturbation Problems for Linear Differential Equations of Elliptic Type, Arch. Rational Mech. Anal. 23(1966), 26-86. Fife, P. C., Toward the Validity of Prandtl's Approximation in a Boundary Layer, ibid. 18(1965), 1-13. Howes, F. A , , Singularly Perturbed Semilinear Elliptic Boundary Value Problem, Corn. in P.D.E. 4(1979), 1-39. Hmes, F. A., Some Singularly Perturbed Nonlinear Boundary Value Problems of Elliptic Type, Proc. Conf. Nonlinear P.D.E.'e in Appl. Sci. and Engrg., ed. by R. L. Sternberg, pp. 151-166, Marcel Dekker, New York, 1980. Kamenomostvkaya, S. L., On Equations of Elliptic and Parabolic Type with a Small Parameter Multiplying the Highest Derivatives (in Russian), Mat. Sbornik 31(1952), 703-708. Levinson, N., The Firet Boundary Value Problem for cAu + A(x,y)ux+ B(x,y)u Y + C(x,y)u = D(x,y) for Small h , Ann. Math. 51(1950), 428-445. Nickel, K., Prandtl's Boundary Layer Theory from the Viewpoint of a Mathematician, Ann. Rev, Fluid Mech. 5(1973), 405-428. [lo] Oleinik, O., On a System of Equations in Boundary Layer Theory, U.S.S.R. Comp. Math. Math. Phys. 3(1963), 650-673. " [ll] Prandtl, L., Uber Flbsigkeitsbewegungenbei sehr kleiner Reibung, Proc. Third Int. Math. Congress Heidelberg 1904, pp. 484-494,Teubner, Leipeig, 1905. Translated as NACA TM 452. [12] Schlichting, H., Boundary Layer Theory, 2nd English ed. (McGraw-Hill, New York, 1960).

Nonlinear Problems: Present and Future A.R. Bishop. D.K. Campbell, B. Nicolaenko (edr.) 0 North-Holland Publishing Company, 1982

335

SOME APPROACHES TO

THE TURBULENCE PROBLEM

J. Mathieu and D. Jeandel Laboratoire de Mecanique des Fluides (L.A. 263) C. M. Brauner Laboratoire de Ma theiatiques Informatique-Systemes Ecole Centrale de Lyon 69130 Ecully FRANCE

-

This paper devoted to non-linear topics deals with a very specific case encountered in fluid mechanics. view.

The problem is examined from two points of

At first, the turbulent field is considered, this gives rise to

discussions on the crucial problem of closure which is examined in several ways. Connections between turbulence problems and mechanical problems treated in a statistical way are recalled. Even where a second order closure method is concerned, mathematical developments have to be introduced to solve a set of equations supplemented by boundary conditions. In the second part of this paper mathematical developments are presented, they correspond to fairly simple cases of closure, notwithstanding they bring to light the mathematical difficulties to treat these problems in an accurate way.

PRELIMINARY REMARKS This lecture is intended to discuss several aspects of a non-linear problem which is encountered in fluid dynamics and called "turbulence".

The basic

problem is conveniently settled; for an incompressible fluid the functions and P of

3

d

and t are linked to each other through the Navier-Stokes and

continuity equations. The existence and uniqueness of a solution for a set of boundary and initial conditions is conjectured, not demonstrated, except for two-dimensional space. It can also be postulated that the solutions exhibit a highly oscillating character which would lead to a statistical approach to this problem. Even in this form the problem is not trivial because we have to find, through a non-linear equation, statistical information at any point within a domain, starting from initial and boundary conditions which are also given statistically. Numerical approaches require very large computers with adequate nets; the relative influence of viscous and inertial terms strongly depends on the distance to a wall which tends to introduce several scalings.

On the other hand, the numerical scheme should not alter the properties of the basic equations whose solutions can be strongly oscillating. To clarify

336

J. MATHIEU, D. JEANDEL,

C.M. BRAUNER

this problem mathematical research has been initiated in several directions. L. D. Landau and Hopf have postulated that the basic system could generate sets of stable solutions from bifurcation points.

So

far, this theory cannot

globally be supported even though situations presenting some analogies have been encountered.

In the case of rotating cylinders (Taylor's experiments),

W. I. Yudowich and V. Velte have proved the existence of bifurcations which lead to new solutions which are not as symmetrical as the intial one.

D. Ruelle and F. Takens conjecture that the transition could be explained by the existence of "strange attractors" for unsteady solutions of the NavierStokes equations.

In fact, no accurate information is available where three dimensional space is concerned. We can also consider that when turbulence mechanisms appear, the physical meaning of deterministic solutions is lost, giving rise to stable statistical distributions.

So

far, we ignore the exact meaning of these

statistical solutions in three dimensional space. The equations for the joint characteristic function are known but mathematical difficulties previously encountered reappear in fairly similar forms. However, some recent findings should open the way to a more grounded theory of homogeneous turbulence. I.

AN OVERALL VIEW OF TURBULENCE PROBLEMS

1.1. Introduction Most physical phenomena are represented by mathematical functions (depending on space-time variables) which generally satisfy non-linear equations, however

in most cases a linear treatment can be considered as fairly convenient. For example,

classical

electromagnetic

phenomena

are

controlled

by

linear

equations. On the contrary, regarding fluid dynamic equations, the role played by the non-linear terms appears determinant. In some previous works very slow motions have been considered and the role played by the convective terms neglected, but so drastic an assumption concerned a very closed field of However, recent utilizations of the Stokes operator to solve

applications.

the Navier-Stokes equations are remarkable.

Fluid dynamic problems giving

rise to crucial non-linear mechanisms must be considered as most instructive in a conference typically devoted to non-linear topics. In fluid dynamics, turbulence is one of the most challenging effects in connection with non-linear mechanisms. In the subsequent chapters, we accept the fluid motion to be conveniently represented by the Navier-Stokes equation;

SOME APPROACHES TO THE TURBULENCEPROBLEM

337

in an Eulerian frame we have:

The fluid is supposed to be newtonian; if consequence of convective terms.

so,

the non-linearity is only a

Besides, it is conjectured that a large gap

exists between the smallest turbulent scales and the molecular characteristic sizes so that the above equation should be reliable for any turbulent structures (even for the smallest dissipative structures). During the last decade, a great effort has been made with sophisticated methods in order to understand turbulence mechanisms; this trend should not dissimulate historical development more specially devoted to find tractable ways for an unclosed p r ~ b l e m . ~ -Starting ~ from a deterministic equation for the instantaneous motion a statistical formalism has been introduced with the moments about the mean value of the velocity. These moments have been treated in a more or less sophisticated manner. For many years, such a theory prevailed;

so

far, it is still very useful for technical applications. On the

other hand, the increasing capability of large computers has led scientists to perform direct numerical simulations of turbulent fields using Navier-Stokes equations; computations conditions.

start

from both

adequate initial and boundary

The complete properties of the basic equations are obviously

considered with a numerical scheme which should not alter them.

Even though

the exact influence of such alterations is always open to debate, available results are interesting. Anyway, the turbulence problem can be considered in several different ways. We can first examine what may be origin of the random nature of many phenomena and, for example, shortly recall the physical equations which govern the associated density probability functions in several fields. Laying aside the Schrodinger equation, which is introduced in quantum mechanics

from a

deterministic approach, but whose solutions are interpreted in a probabilistic way, two main patterns of random phenomena may be considered. The first group is controlled by deterministic equations; however, random initial conditions generate random solutions (Liouville equations belong to this group).

In the

second group, initial conditions are formulated in a deterministic way but the random process is introduced through random coefficients in the basic equations (for example the Langevin equation for Brownian motion).

338

J. MATHIEU. D. JEANDEL, C.M. BRAUNER

We must immediately remark that turbulence exhibits a rather typical case of non-linear equations.

It can be conjectured that initial and boundary

conditions determine the flow field at any time; so far, the uniqueness of the solutions has not yet been ensured except for very restricted conditions which are not encountered in the problem under consideration which is typically located in three dimensional space.

From this point of view the role played

by the bifurcation theory proposed by L. D. Landau and E. Hopf could become illuminating but at the present time this approach, which is in progress, is not sufficiently worked out to permit a straightfoward analysis of the

In the case of unsteady solutions the role of "strange attractors" has been examined by D. Ruelle and F. Takens, but only partial conclusions have been given which cannot be extended to three dimensional space.

Navier-Stokes equations.

For the first group, with Hamiltonian systems, the Liouville equation is well adapted to predict the states of the system; a probability density function is introduced in phase space:

and it is found that: (1.3)

with (1.4)

H stands for the Hamiltonian of the system. For the second group, we must recall that Brownian motion leads to a Fokker-Planck equation provided that the memory of the system has faded enough. When we have to deal with a reactive medium, the concentration equation can be written as:

aci

at

'' j

aci

5= 'ci

a2ci

2 ax + i

ytc,

... c1

...I

.

SOME APPROACHES TO THE TURBULENCE PROBLEM

Through the production term y[C, occurs.

... C.]

330

a non-linear mechanism generally

Some connection can also be found between the Fokker-Planck equation

In fact, U. plays the role of a random coefficient in the

and this one.

J

concentration equations but

the Markovian

character of the

system is

invalidated. The problem under consideration can also be examined from the first point of view. We must treat a problem with deterministic equations and random initial conditions. The equations relative to kinematic properties of the fluid, and those concerning the concentration of the chemical species have to be considered as a blocked system having no random coefficient,

au . Aa x = o

.

j

Moreover, the existence and the uniqueness of a solution is supposed for sufficiently smooth initial conditions, but supported by mathematical considerations. been developed by Hopf (1952),' recently

by

Dopazo

and

so

However, this approach has first

Monin (1969),6

O'Brien (1975)

far, this assumption is not

.'

Lundgreen (1967),7 After

cumbersome

and more algebraic

computations a hierarchy of equations is found. At N given points in phase space they give the probability density functions to find fixed values of velocities and concentrations.

In order to simplify this section we propose classifying the existing methods for the modelling of turbulence into four groups.'

Developments which are

reported herein principally concern homogeneous fields, with constant mean velocity gradients (or even null:

isotropic turbulence) with adequate initial

conditions. From these theories, inhomogeneous effects can only be considered as small; moreover the Reynolds number is supposed large, so that each group of turbulent structures located in a range of the spectrum should play a typical role. Besides, for any statistical properties of the field, the departure from a normal law is rather moderate. In other words, the methods

340

J. MATHIEU, D. JEANDEL, C.M.BRAUNER

will be available for patterns of flows which can be relaxed to homogeneity and gaussianity.

Characteristic length and time scales are introduced; they

are necessary for the precision

of

the two concepts of homogeneity and

stationarity. 1.2. Physical Space Methods

1.2.1.

General Concepts and Second Order Closures

Disregarding what occurs in the spectral space where the role of turbulent structures is identified as a function of their sizes and directions, the -*

instantaneous velocity U is split into two parts

At first, using the Navier-Stokes equation it is found that:

u.u]

*

I f

At this stage, a first order closure can be introduced.

For instance we may

write:

-?

where vT is a predetermined function of X and t. The introduction of moments about mean values does not presume anything about the origin of random mechanism which is considered by its consequences only. Where

second

order

closures

are

concerned

the

previous

equation

is

supplemented by a rate equation for the second order moment only:

The whole turbulent flow is characterized by adequate length and time scales. In fact, a spectral equilibriun is assumed so that turbulent scales are linked to each other: turbulent

for example, the macro and microscales are related through the

Reynolds

number.

Moreover

for

large

Reynolds

numbers

small

SOME APPROACHES TO THE TURBULENCE PROBLEM

341

dissipative structures are supposed isotropic so that:

(1.10)

Finally we have:

The open nature of the problem is displayed by both the mean dissipation rate and the rate of strain pressure correlations. The equation which controls the dissipation rate can be written:

(1.12)

where for the sake of simplification, dimensionless forms are introduced

(1.13)

At this stage, no additional information is requested from the Navier-Stokes equation, and the closure is obtained by another means. been extensively developed by Lumley"

The method which has

in its more perfect form considers the

whole turbulent structure as a complex material, the properties of which have to be determined by experience.

Therefore Pij and Vij are supposed to be

dependent on conveniently selected arguments. The deviation tensor bij being introduced as

(1.14)

342

J. MATHIEU, D. JEANDEL, C.M. BRAUNER

it can be written that

2

“A

-

(1.15)

priori” the introduction of functional forms underlines a non-local

dependence, the memory of the structures being correlated with their sizes. For dissipative structures it is fairly convenient to admit that their characteristic time is small with respect to a characteristic time defined

=

aiQ/axm.

In fact, Lumley proposes that the two previous functiona:s/$ and >ij are considered as simple functions, which assume that the memory of turbulent structures has faded enough. It is a crude assumption which can only be supported by its consequences. from the mean flow, for example,

Besides, if we admit that the dissipative structures are not directly strained by the mean field (the amount energy flux is transformed without alteration through the inertial range, but during these exchanges it can be conjectured that turbulence structures lose most of their orientation), the anisotropy is considered as globally taken into account through b (disregarding the mean ij’ velocity gradient which generates such an anisotropy). We can then write:

I = I [bij,

2 q

, -E] .

(1.16)

With a suitable choice of arguments, we can admit I to be an isotropic

iz and E . Pipkin and Rivlin have demonstrated that such a ij’ function can only depend on the invariants (with respect to the orthogonal

function of b

group) which are determined from b.. , IJ

sz and z.

For a small degree of anisotropy the previous function can be expanded with respect to the anisotropic parameter:

I = f[E,

q2]

+

g

[Z,

q2] 11 t

...

.

(1.17)

More precisely we find that:

(1.18)

SOME APPROACHES TO THE TURBULENCE PROBLEM

where I1 is the second invariant of b.. 1J

experimental data.

'

343

the constants must be adjusted from

The second function Pij can be determined in a similar but more complex way, the pressure term having linear and non-linear relations to the fluctuating field.

(1.19)

These methods have extensively been developed during the last decade in several forms by Launder, Jeandel,

....

Computing Methods for Complex Flows

1.2.2.

We will take advantage of

this section devoted to modelling methods in

physical space to indicate some progress in computing approaches for complex turbulent flows. The patterns of bounded flows under consideration are strongly non homogeneous in more or less extended areas; the rate equation for the mean flow has to be coupled with a more or less complex turbulence model. Where a first gradient approach is used with VT as a given function of space and time variables, we must deal with both the momentum equation for the mean field and the continuity equation. The plane channel flow simplifications result in the dimensionaless basic equation: (1.20)

with the following given characteristic scales (2D channel width), 1 dP u =[---D]) 0

4

PdX

(1.21)

and the initial and boundary conditions (1.22)

J. MATHIEU, D. JEANDEL, C.M. BRAUNER

344

i

(0,

t) =

i

(1, t) = 0

(1.23)

Ro stands for uo 2D/v. Two kinds of viscosity models are retained leading to available predictions, namely (1.24)

(model 1 )

in which v

T

is dimensionless parameter and

vT (y, t) = R: ky &(o,

t) [ l

- exp - Y Ro]

;y

E

[O;

(1.25)

0, 1)

aY

vT (y, t) = vT (0, 1; t) VT

(yt t) =

VT (1

- Y; t)

y

i

i

y

(1.26)

[ O , 1; 0, 51

E

E [ot

5 1 11

-

(1.27)

(model 2)

Where second order closures are concerned such as previously proposed by

...

Donaldson, Launder, Lumley, Jeandel treated.

a partial differential system must be

The momentum equations and the continuity equation are supplemented

by four equations related to the components of the Reynolds stress tensor. The dissipation rate must also be considered as an unknown function. Where a channel flow is concerned, the problem may be formulated in a rather simple manner. Using dimensionless variables for the Reynolds stresses and dissipation a model of this kind may be written in the general form:

(1.28)

U is the vector the components of which are:

-2 U

2 -

uO

; U g = - u2 2

uO

(1.29)

We must notice that inhomogeneities greatly complicate the modeling of the velocity pressure correlations. In the model of Launder," a sophisticated

345

SOME APPROACHES TO THE TURBULENCE PROBLEM

expression inversely proportional to the incorporated in the non-linear operator

distance from the wall is [;I. Consequently, boundary

conditions for the mean and turbulent fields must be given in the well known semilogarithmic region. We also notice that the Reynolds stress equations contain gradient terms in the form:

(1.30)

that reinforces the elliptic character of the equations governing the turbulent quantities. A

numerical solution may be

searched by means

of an optimal control

When non-linear operators are concerned, they can be split into two parts: a linear operator A and a non-linear one B; and formulation of the problem. l2

we write: A [U]

=

B[U]

.

(1.31)

The associated problem can be written: A

[UAl =

B [A]

.

(1.32)

In that case, A may be considered as a control parameter, and the last state equation is solved minimizing a cost function of the form (1.33)

1.2.3

Provisional Comments

Even though these methods are carried out in physical space, references are often made to the size of the turbulent structures which are responsible for a phenomenum; in fact, an equilibrium state is always supposed which reinforces the role played by the energy flux which travels without alteration across the inertial range.

If a deterministic phenomenon is superimposed on the random field (with vortex shedding, the spectrum exhibits a peak at a rather low wave number) the amount of energy flux can be significantly altered so that many of previous assumptions are spurious. Launder, Cousteix take into account this disturbance by introducing an additional parameter (as is made in the

346

J. MATHIEU, D. JEANDEL. C.M. BRAUNER

thermodynamics of i r r e v e r s i b l e phenomena i n o r d e r t o account f o r a d i s e q u i l i -

I t i s a l s o p o s s i b l e t o d i r e c t l y work i n s p e c t r a l space.

brium). 1.3. Random

S p e c t r a l Methods motion

is

analysed are

accounting

referenced

for

both

by

the

structures

which

Non-linear

e f f e c t s d i s p l a y a n energy t r a n s f e r ;

r o l e played

their

sizes

and

by

turbulent

orientations.

t h i s energy f l u x g e n e r a l l y

t r a v e l s from t h e l a r g e s t t o t h e s m a l l e s t s t r u c t u r e s , b u t a r e v e r s e cascade process has been brought t o l i g h t very c l e a r l y (Gence13). Fourier transform of t h e two components u. and u

.i

S t a r t i n g from t h e

of t h e f l u c t u a t i n g v e l o c i t y

+

i t i s p o s s i b l e t o w r i t e t h e equation of t i e c o r r e l a t i o n a t two p o i n t s k and

of

spectral

space (which corresponds t o t h r e e p o i n t s

I n t e g r a t i n g over t h e wave number spectral

equation a t

a

1

it i s p o s s i b l e t o

unique p a i n t

+

2

i n physical space).

find the classical

k of s p e c t r a l space.

The g e n e r a l

s t r u c t u r e of t h i s equation can be represented by

For t h i s open problem it i s a l s o p o s s i b l e t o write a n equation f o r t h e t r i p l e c o r r e l a t i o n s a t t h r e e p o i n t s i n s p e c t r a l space

+

... + ... = Sib

+ +

[PI

q1

k,

t]

.

(1.36)

SOME APPROACHES TO THE TURBULENCE PROBLEM

347

To close this open set of equations Millionshikov (1941) and Proudman and Reid (1954)14 assumed that the probability law

-+

-+

-+

p [Ui' UQ' Url

$1

(1.37)

+ + which is parametrized by k, p,

l, 3 and

t is very similar to a gaussian law

so

that they write:

(1.38)

This assumption leads to: %

%

2

,

= o i l l r

.

(1.39)

Disregarding some contradictions 1 ese authors admit that a closure method can be given in the symbolic form:

Thereby, the role played by the cumulant terms if neglected.

In the previous

equation no limitation is imposed on the evolution of the triple correlations which increase too much.

These terms being responsible for the energy

transfer across the spectrum, the amount of energy located in some regions becomes negative which is irrelevant.

With the quasi normal hypothesis the

solution of the equation for the triple correlations

(1.41)

is given by

(1.42)

348

J. MATHIEU, D. JEANDEL, C.M. BRAUNER

where Siam must be considered as a known quantity. It is obvious that the memory of the turbulence is overestimated, the exponential term being only controlled by the kinematic viscosity.

In order to improve the method,

Orszag15 supposed that the role played by the cumulant terms can be introduced through a linear relaxation which has the symbolic form

This artificial viscosity contributes to reduce the memory time of turbulence. Moreover the integration is drastically simplified by supposing that the triple correlations can be taken at time t (in place of t').

From this

Markovian approximation the integration with respect to t is only carried out over the exponential term. The artificial viscosity pT plays a role in spectral space which could be compared to the turbulent viscosity introduced by Boussinesq in physical space. Starting from experimental data, adjustments can be made, but it is also possible to evaluate the characteristic time by considering, in the manner of Kraichman, a supplementary problem which is the advection of a passive scalar by a random field with coupled variables (Test Field Model TFM). The eddy damped quasi normal markovian model (E.D.Q.N.M.) has been extended to homogeneous fields with constant mean velocity gradients by Cambon. l6 spectral equation has previously been given by Mitchner and Craya:

The

( 1 .MI

with

;+;=g.

(1.45)

The action of the mean velocity gradient is explicitly represented by I 2ji whereas Raji accounts for non-linear effects. We suppose Reji to be evaluated in the form: (1.46)

We find that

(1.47)

SOME APPROACHES TO THE TURBULENCE PROBLEM

349

Besides it is written: 2

v(k + p

The

2

2

+ q 1

aQji

incorporation of

the

typically non-isotropic term Yaji is a

diagonalized form has extensively been discussed. Two facts can support this assumption. At first, the variation of @ with time should be specially Qji controlled by Secondly we are only concerned with a part of the triple correlations for the behavior of the equations which determine the evolution

Cljyi.

of the double correlations.

In order to simplify computing methods, the evolution of the turbulent structures is only considered as a function of their sizes, not of their orientation. so

-+

Consequently an integration is made over a sphere of radius lkl

that suitable functions are introduced such as:

+ An equation in terms of functions of modulus of k is obtained.

An angular

parametrization of the second-order spectral tensor is introduced in order to integrate analytically all the directional terms over a spherical shell. Modelling methods similar to those used in physical space are introduced; for example, the spectral deviatoric tensor €Iij plays a similar role to b ij' Finally, the terms of the closed equation for $ij exhibit only a tensorial anisotropy. The specific anisotropy related to directional effects disappears by integrating over the sphere.

At this stage, two scalar parameters a(k, t) and q(k, t) are to be adjusted. Exact linear solutions can successfully be used to determine a(k, t).

Complex evolution of spectral curves can be

predicted, for instance a peak can be introduced in a local range of wave number and direct and reversed cascades of energy can be brought to light Fig. 1 and 2 ) . Many extensions of this method have been made to turbulent fields including buoyancy effects, inertial forces and so on.

350

J. MATHIEU. D. JEANDEL, C.M. BRAUNER

1.4.

Pseudo Deterministic Approach (large eddy simulation)

Starting from the Navier-Stokes equations, direct simulations of isotropic turbulent fields have been made.

So

far, this simulation is not adequately

supported by mathematical considerations.

Bifurcation points, if they exist,

are detected and treated through an algorithmic scheme which should introduce its own mark so that it should be difficult to interpret the final results which include both the properties of the initial equations and those of the numerical scheme. We can suppose that a large eddy simulation is more typically convenient for turbulent fields subjected to overall effects capable of influencing more or less extended groups of turbulent structures. Except for isotropic turbulence which is only subjected to its own mechanism, in most of the flow patterns, selected turbulent structures are subjected to straining mechanisms or driving forces such as mean velocity gradients, buoyancy effects, etc. Therefore, it is possible to

conjecture

the

existence of coherent structures whose

development could be more suitably tackled in a deterministic way.

These

coherent structures, the sizes of which are comparable with those of the overall

flow, develop

in the presence of smaller structures which are

accounted for in the statistical coefficients.

For example, the pairing

process which is responsible to a large extent for the growth of turbulent areas can be considered as a deterministic mechanism whose developments are somewhat hampered by the fine grained turbulence which plays a role similar t o a viscous effect. So

far, the definition of such coherent structures is not precisely supported In these approaches, Deadorff17 and Leonard 18

by mathematical theories.

introduce a typical splitting of the overall field.

The "filter function"

used by Leonard can be linked to the spectral approach more directly. Therefore we use it preferentially. -*

Leonard separates the velocity field U.(x, t) into a grid scale component -

-9

*

li(x, t) and a subgrid component u i ( x , t)

U. = 1

ii.1 +

u!

1 '

'

0

[;/;I]

l

&'

=

1

.

(1.50)

The overbar precisely denotes the filtering operation which is not to be confused with either the mean value of the velocity, which has a statistical

J. MATHIEU, D. JEANDEL, C.M. BRAUNER

351

meaning, or with any statistical averaging, indicated by angular brackets. A filtered quantity has to be considered as a typical event which can occur under any circumstance.

This event, when it happens, develops in an

environment the properties of which have to be considered in connection with the existence of this principal event.

This should lead to conditional

samplings supplemented by averaging operations. If the physical properties of the fluid are constant, the result of filtering the Navier-Stokes equation is:

(1.51)

This equation is supplemented by the continuity equation:

(1.52)

For the advection term it is found that:

- u . u = U . U . + u!u. + u . u ! + u'u' i j 1 1 1 1 1 1 i j

(1.53)

'

By using a series expansion it is found that:

--

a

a u- .u- $1

(1.54)

The Leonard term can be written

-L i j = uiuj

-

-UiUj

.

(1.55)

The product of filtered quantities is considered without additional filtering. The "cross term"

cij =

u.u! 1 1

+

upj

SOME APPROACHES TO THE TURBULENCE PROBLEM

352

represents interactions between the residual field and the filtered field. "A priori" this sort of interaction should not be located in spectral space.

On the other hand, the subgrid Reynolds tensor

emphasizes the role of small structures. However, i

can

?.

prove1

t

at such

interactions can occasionally generate structures far outside the subgrid scale from a triadic analysis. We set

Tij = R i j

t Cij

and introduce the deviatoric tensor:

r i j = Tij

- 21 Tkk b i j .

(1.56)

On the other hand we write:

Finally we have:

aii

at t

a -uiuj - = j

-

- 1P * - &j [ L i j ax

t

r..] =I

.

(1.58)

Subsequent workers have followed Smagorinsky19 and write

aui

%Ii

Tlj

=

- V N 'ax

+

%I

j

for the "subgrid term". Such an assumption gives rise to several remarks:

(1.59)

SOME APPROACHES TO THE TURBULENCE PROBLEM

363

No tractable information has first been extracted for the subgrid term from the basic equations. With such a splitting, the subgrid terms are not objective terms. Therefore, the application of representative theorems to these quantities becomes questionable.* For a long time the energy cascade was considered as a one way street, the small structures being fed by the larger ones. Moreover it was conjectured that

interactions

were

only

approximately the same sizes.

possible

between

turbulent

eddies

having

This opinion was more or less supported by

experimental data extracted from isotropic turbulent fields.

In fact, the

energy cascade is a two way street and interaction occurs between structures of very different sizes. Applications of successive plane strains to grid generated turbulence has been illuminating (Gence 13-20).

At first the isotropic turbulence is subjected to

a straining process and turbulent structures are oriented by the mean field.

A second straining, the axes of which are different from the first ones by an angle

LY

is suddenly applied so that the turbulence is reoriented by this

second straining.

Gence13 has shown that the term describing the exchange of

energy can be written, after the change in position of the principal axes of the strain, as

-D 2 [b22

Q

- b33][2 cos2 - I ] .

(1.60)

LY

Either positive or negative values can be obtained depending on the angle a , D stands for aii/ax.. 3

The assumption of Smagorinsky supposes the principal axes of the subgrid turbulence to be the same as those of the filtered quantities.

This

hypothesis is not supported by Gence's experiments and by direct simulations carried out by Ferziger21 who states that "the two sets of axes appear to be essentially random with respect to each other."

The mean relative position of

the two systems is of course a most important parameter. Occasionally such a coincidence could occur.

* At

first, representation theorems require the introduction of additional argument in connection with, for instance, the rotation of the filtered field.

354

J. MATHIEU, D. JEANDEL, C.M. BRAUNER

Comparisons have been made between the computational results coming from an extended

E.D.Q.N.M.

method

(Cambon16)

and

from

large eddy simulations

including Smagorinsky’s assumption (Courty, Bertoglio, private communication), (Fig. 3 ) . Great discrepancies are shown in Fig. 3 , probably originating from the subgrid With the large eddy simulation, the evolution of the correlation term

model.

seems to be rather slow, a fact which should be in close connection with the evolution of the transverse velocity component.

1.5 Probabilistic Approach far, the probabilistic

So

properties of

the

flow have been

treated

disregarding the precise behaviour of the Navier-Stokes equation which controls any motion of the complex turbulent medium.

The motion is split into

two parts, the properties of the fluctuating part are supposed not to be far from a normal law.

At any point in the theory, it is necessary to clarify

whether bifurcation points exist or do not.

In fact, all the pervious

treatments are supported by very pragmatic considerations not typically related to the nature of the basic equation.

However, large eddy simulations

start from the Wavier-Stokes equation for a selected group of structures, but the deterministic computation is carried out up to an arbitrary solution no averaging process is introduced in order to give a statistical value to this approach. Besides the evaluation of the subgrid terms is very rough. Simple experience confirms that some statistical properties of turbulent fields can behave in a very typical way; experimental data show that the departure from a normal law can be very significant, so that is is not possible to account for the cumulant terms through a linear relaxation as is done in the case of the E.D.Q.N.H. assumption. Turbulent fields with chemical reactions display such typical behaviour.

If two non-reactive chemical

species are mixed together, it is found that the statistical properties of the concentration fluctuations more or less relax to a gaussian curve, but if the two species exhibit very strong reactive properties A and B cannot coexist at the same point, so

that the probability density function for the two

fluctuating concentrations should be represented by two planes. Even though asymptotic situations such as flames could be easier to tackle, it would be interesting to follow the evolution of the probability density function by controlling the reaction rate.

Experimental investigations carried out by Bennanni” are in progress to seize intermediate situations. The Damkholer Number compares two characteristic scales: the reaction time, which is directly connected with chemical properties, and a characteristic time linked

SOME APPROACHES TO THE TURBULENCE PROBLEM

to dynamical properties of the turbulent field such as

355

At first, it

seems that the Damkholer Number does not act on the probability density function in a continuous way

so

that it should be possible to encounter a

probability distribution not far from a Gaussian curve for large ranges of small chemical reaction rate. It is probably interesting to seize the topology of the reactive domain.

For flames where the rate of reaction is

high, reactive zones are complex surfaces; where diluted media are concerned, reactive zones can become more or less extended volumes. For the treatment of chemical reactions the trend should be to consider a probability density equation, a means which has recently been used. In the cases under consideration, the theory of moments having failed, scientists have introduced a fine probability function. The solutions have been developed in very simple cases.

For our purpose, we must underline the

role played by the Hopf equation which can be considered as the starting point for this theory as it has been shown by Gence (private communication).

The

Hopf equation can be considered as a probabilistic form of the Navier-Stokes equation. It is established in phase space and supposes the uniqueness of the solution when convenient initial conditions are given.

This restriction is

introduced through a punctual correspondence between the sets of points which represent the state of the system at two selected times.

In the original

paper by Hopf, this difficulty is perfectly brought to light. For the sake of simplicity, we consider an isotropic turbulence which develops in a reactive medium, and we indicate the equation governing the probability density function: (1.61)

1 1 1 that f [r,, Y1 t] d TA dTB represents the probability of having the 9’ + t) and cB(xl, t) respectively located in a range concentration c ( x 1 1 11’ 1 TA, rA + dTA1 andATB, r B + drB1 at a given time t. Finally we obtain: so

J. MATHIEU, 0. JEANDEL, C.M. BRAUNER

356

where ( I .63)

is the probability density of finding given values of concentration at two

+

separate points r.

In this form, the closure problem is clearly brought to

light.

At the present time, this approach is an unweeded path. Experimental data are unavailable and information is required to suggest suitable closures.

11. MATHEMATICAL STUDY OF A TURBULENT VISCOSITY MODEL

11.1. Introduction In this part, we shall present a mathematical study of a nonlinear parabolic equation including one of the above turbulent viscosity models. We wish to focus our attention on the more original one, namely model 2 (see Section 1.2.2). For a mathematical study of model 1, see Laink.23'24 In this paper, the proofs will be only outlined. For further details, we refer the 25 reader to Lain&,23 Brauner and Lain&. Our mathematical notations will be as follows: Q is the open interval ] 0 , 1 [ , x the space variable.

Let ] O , S [

be the time interval, t or

s

the time

variable, and Q = h ] O , S [ . We shall use some Sobolev spaces (see e.g., H2(Q), H-'(Q) (3), as well as the spaces namely Hi(Q), 1 L2(0,S; HO(R)), etc. Without any further specification, we shall denote by (

,)

2

the inner product in L (Q) and 1 1

.

1 1 the associated norm.

For u regular enough, let us define the "turbulent viscosity"

(2.1)

where G

2:

C1(fi),

G(x)

2

v1 > 0

.

We consider the following mathematical problem.

I

u ( 0 , t) = u(1, t) = 0 u(x,o)

=

Uo(X),

x e Q

Find u = u(x,t)

solution of

(2.2)

SOME APPROACHES TO THE TURBULENCE PROBLEM

where

V

>>

V

1

357

and u (u) is given by (2.1).

T

11.2. Existence of a Solution The existence of a solution to problem (2.2) is obtained via a standard Galerkin method. Specifically, we establish the following theorem: Theorem 2 . 1 :

2 1 L e t f given i n L ( Q ) , uo i n HO(Q).

solution

u

in

the

space

Then problem ( 2 . 2 ) has a

H~(Q))n ~ " ' ( 0 , s ;H~(Q)).

~ ~ ( 0 , s ;

Furthermore,

Proof:

Introduce the eigenfunctions w. of J w.(x) = sin jnx, j = 1, 2, .... One has w'! + J J look for a solution um(t) = gjm(t) wj

jll

2 2 the operator -d /dx , namely 2 A w = 0 where A . = (jn) . We j j J of the nonlinear differential

system:

(2.4)

u

m

(0)

= uom = p r o j . of uo on

vm ( 4 )

System (2.4) has a solution on an interval ]O,Sm[.

. The following estimates

will in particular yield that Sm = S. Lema 2.1:

2

The sequence {urn]i s bounded i n t h e space L ( 0 , s ;

n',(Q))

n L"'(0,S; L 2 ( Q ) ) , and

Proof:

Simply multiply (2.4) by g . (t) and add for j Jm

1 to m.

It follows

& = (f(t), um (t))

The integrate over ]O,S[ and apply Young's formula to the R.H.S.

358

J. MATHIEU, 0. JEANDEL, C.M. BRAUNER

Lemma 2 . 2 :

The sequence rum] is bounded in the space L 2 ( 0 , S ; €?(ill) and

n

~"(0,s; H,1(R)),

I-

( 0 , .) If

ax

2

a2Um

Sketch of the proof: and sum.

is bounded in L2 ( Q ) .

ax

I t comes:

In (2.4), replace w . by -1/A ", then multiply by g . ( t ) J j Jm

Integrating over lO,S[ leads t o

The l a s t term i n the R.H.S. may be bounded by

2 IG'

which can b e s p l i t i n t o two parts using Young's formula.

Finally we get:

2 (0,s)

If 2 a Il 2 ax

L~ (PI

359

SOME APPROACHES TO THE TURBULENCE PROBLEM

1

a

hence the requested estimates (for the estimate in L (0,s; HO(R)), integrate over ] O , s [ , 0 <

just

S).

s<

End of the proof of Theorem 2.1: As system (2.4) may be rewritten as 2

-= at

v p (-1a o m ax2

+P

(-a

v

m ax T

Cum)

2 where P is the projection operator from L (R) into Vm, it follows that the q sequence

is bounded in L2 (0,T; H-'(Q)).

Hence we can extract a converging subsequence m

weak and in L (0,s; Hi@))

{u ) such that u P P

+

in L2(0,S; H-'(Q))

weak. By a compactness theorem (see Lions [ 2 7 ] ) , u

L2(0,S; H2-&(Q)) L2(0,S;

C'(6)).

u in L2(0,S; H2(R))

weak star,

CI

-$

u in

0 < E < 4, and, by Soboler imbedding, in Then it is easy to pass to the limit in (2.4), to see that p strong,

is solution of (2.2).

11.3. Results on Regularity In order to prove the uniqueness, we must first establish further results on regularity,.namely: Theorem 2 . 2 : G( x )

2

2

1

Suppose now f e L (0,s; H O ( Q ) ) , v1 > 0 , and uo

tion i n L2(0,S;

E

$(R)

n <(R).

d(R))flLa(O,S;

G E

C2(6),

Gr(0)

= G ' ( 1 ) = 0,

Then problem (2.2) has a solu-

$(R)),

E

L2(0,S;

d(R)). Besides: (2.6)

I dw

Sketch of the proof: In (2.4), replace w by '/A: j

4: dx

360

J. MATHIEU, 0. JEANDEL, C.M. BRAUNER

Integrating by parts gives:

We refer to Brauner-Lain6 (1981) for the estimate of the R.H.S. Corollary 2 . 1 .

u is continuous in

6

and

au ax

( O l e )

E

Co([O,S]).

2

Proof:

From Theorem 2 . 2 , we know u E L2 (0,s; H3 (Q)), e L2 (0,s; H1(Q)). By an interpolation argument (Lions-Magenes [28]), u E Co([O,S]; H2(Q)), and by Sobolev imbedding, u

E

Co([O,S]; C1(fi)).

11.4. A Strong Maximum Principle for Weak Solutions In as much as (2.2) is a physical problem, we are only concerned in positive data.

What we need to prove is that any solution obtained via

Theorem 2.1 and 2.2

is not only non-negative, but positive in Q.

u0 (x) > 0 Vx

E

a

au

(0,t) > 0. ax The hypotheses are t h a t of Theorem 3.2.

consequence, we will establish

Theorem 2.3:

A

Q , f(x,t)

2

0

Besides suppose

a.c. in Q. Then solution of

(2.2)

in the sense of Theorem 2.2 verifies u(x,t) > 0 in Q. First it is easy to check u(x,t) 2 0 with the weak maximum principle. au Then uT(u) = GLet us now mention a result due to J. Hoser 129, 301. ax (o,t). 11.4.1 A Harnack Inequality for Parabolic Equations Let B denote an open bounded set in Rn and Q = Q x] O , T [ . the equation in Q:

We consider

(2.7)

SOME APPROACHES TO THE TUREULENCE PROBLEM

361

where a.. are bounded functions such that 1J

Let v be a weak solution of (2.7) such that au ,% ,

...,

i = I,

n,

belong to L2(Q). We suppose that v is non-negative in Q. We consider a compact and connected subdomain of n, A , and two subintervals I- = (t, tl < t < t2], I+ = it, t
Define Q- = I-x A , Q = I Lemma 2.3:

+

x A.

iiny non-negative solution of (2.7) satisfies:

sup ess v 5 c (1 +

inf ess v

Q

Qt

+.

where c > 1 is a constant which depends only on Q, Q- and Q 11.4.2 Positivity of u Now we can prove Theorem 2.3.

Let u 5 0 a solution of (2.2), and consider the

following auxiliary problem:

(2.10)

c U(X,O)

%

Lemma 2.4: u

= uo(x)

E Co(c)

. and u(x,t)

Proof: It is easy to verify u 2,

main point is to prove u > 0 .

2 ;(x,t) > 0 in Q. 2 cu 2 0 by the weak

maximum principle. The nd As the 2- member in (2.10) is zero, we can

382

J. MATHIEU, D. JEANDEL,

apply Lemma 2.3 to (2.10), with aij = a =

Now let xo

E

V

T

C.M. BRAUNER

(u) +

Vo.

Here A = v

and

R fixed. Since u > 0 in R, there exist a closed interval

containing x

0'

A C

R

and a time T, such that

'L

Suppose there exists t* > T such that u(xo, t*) = 0. With the notations of Lemma 2.3, we choose tl and t2 such that 0 < tl < t2 < t3 = t,

-

E

t4 = t, +

E.

1,

then we take

so

'L

.\I

inf u = 0, therefore sup u = 0

Q+

QOf course this is inconsistent with (2.11) and t* does

as a result of (2.9). not exist.

au 11.4.3 Positivity of -(O,t) ax Lemma 2.5: &(O,t) > 0 Y t E ]O,S]. ax The idea of the proof is to extend a demonstration by Protter-Weinberger [31] to positive weak solutions.

Corollary 2.2:

See the details in [23][25].

Suppose

Then exists a constant

01

> 0 such that (2.12)

11.5

Uniqueness of the Solution

The demonstration of the uniqueness is based

on

Gronwall's lemma, with

Corollary 2.2 as basic tool. Theorem 2.4:

The hypotheses are that of Theorem 2.3. a unique positive solution.

Then problem (2.2) has

SOME APPROACHES TO THE TURBULENCEPROBLEM

and u2 be two positive solutions of (2.2). 1 Clearly, u verifies

Sketch of the proof: u = u1

-

u2.

u(0,t)

363

Let u

= u(l,t) = 0

U(X,O) = 0

Set

(2.13)

. n

Multiply ( 2 . 1 3 ) by

- a% 7: ax

(2.14)

We remark that

( a given by Corollary 2.2) and

&o,

t) I

5 JZI

I%(t)

364

J. MATHIEU. D. JEANDEL, C.M. BRAUNER

Plugging this estimate in the R.H.S. of (2.14), it comes out, after several computations based on Young's formula, a relation of the form: (2.15)

where M

E

L1(0,s). By Gronwall's lemma, 1Ig(t)II

2

0 , hence

in Q and the uniqueness.

Acknowledgements:

The authors are

indebted

to Professor M. Crandall who

brought J. Moser's results to their attention.

FOOTNOTES See C. Lain6 [23].

A weighting process is introduced with the contraction kaeaji which minimizes the role played by the large anisotropic structures. 2 Recall that ff(Q) is the subset of functions of L (a) whose partial 2 derivatives up to the order are also in L ( Q ) . H:(Q) is the closure of 1 its dual space. If the set of smooth functions D(Q) in H ( Q ) , and H-'(Q) X is a Banach space, LP (0,s; X) is the set of functions of ]O,S[ into X of power p integrable, 1 5 p < +cD (essentially bounded if p = + V m = finite dimensional space spanned by wl, ..., wm'

a).

REFERENCES Glowinski, R., Mantel, B., Periaux, J., and Pironneau, O., "H-'

least

squares method for the Navier-Stokes equations,'' 1st Int. Conf. on Numerical

Methods

in Laminar and Turbulent Flow, Eds. Taylor, C.,

Morgan, K. and Brebbia, C. A. (Pentech Press, Swansea, 1978). Bradshaw, P., Ferris, D. H., and Atwell, N. P., "Calculation of boundary development using the turbulent energy equation," J.F.M. p. 593-616.

28 (1967)

365

SOME APPROACHES TO THE TURBULENCE PROBLEM

Launder, B. E., and Spalding, D. B., Lecture in "Mathematical Models of Turbulence," (Academic Press, New-York, 1972). Jeandel, D., Brison, J. F . ,

and

Mathieu, J.,

"Modeling methods

in

physical and spectral space," Physics of Fluids, December 1977. Hopf, E., J. Ratl. Mech. Anal. (1952), p. 1-87. Monin, A. S., and Yaglom, A. M., "Statistical fluid mechanics: mechanics of turbulence," The M.I.T. Press, Vol. 2, 1967. Lundgreen, T. S., Physics of Fluids 10, (1967), p. 969. Dopazo, _., and O'Brien, E. E., Combusion Science and Technology, Vol. 19 (1975) p. 99-122.

Mathieu, J., "IdCes actuelles sur la turbulence," CANCAM 79, Sherbrooke (1979), p. 465-502. Lumley, J., "Von Karman Institute," J. 1975, Lectures series 76. Launder, B. E., Reece, G. J., and Rodi, W., "Progress in the development of a Reynolds Stress closure," J. Fluid Mech. 68, p. 537-566. Cea, J., and Geymonat, G . , "Une mithode de liniarisation l'optimisation,"

Instituto

Nazionale

di

Alta

Matematica,

via

Symposia

Mathematica, Vol. 10 (1972). Gence, J. N., and Mathieu, J., "On the application of successive plane strains to guid generated turbulence," J. Fluid Mech. (1979), Vol. 93, Part 3, p. 501-513. Proudman, I., and Reid, W. H., "On the decay of a normally distributed and homogeneous turbulent velocity field," Phil. Trans. Roy. SOC. A , (1954), p. 163-189. Orszag, S. A., "Analytical theories of turbulence," J. Fluid Mech. 41, (1970), p. 262-386.

Cambon, C.,

Jeandel, D.,

and

Mathieu, J.,

"Spectral

modeling

of

homogeneous non isotropic turbulence," Journal of Fluid Mechanics, Vol. 104 (1981), p. 247-262. Deardorff, J. W., "A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers," J. Fluid Mech. 41 (19701, p. 453. Leonard, A., "Energy cascade in large-eddy simulations of turbulent fluid flows," Adv. in Geophys. A 18 (1974), p. 237. Smagorinsky, J. S., "General circulation experiments with the primitive equations. I: the basic experiment," Mon. Weath. Rev. 91 (1963) p. 99. Gence, J. N., and Matheiu, J., "The return to isotropy of an homogeneous turbulence having been submitted to two successive plane strains,'' J.F.M., Vol. 101, Part 3 (1980), p. 555-556. Ferziger, J. R., "Higher-level simulations of turbulent flows," V.K.I. lecture series 5 (1981).

J. MATHIEU, D. JEANDEL, C.M. BRAUNER

366

(221 Bennani, A., Alcaraz, E., and Mathieu, J., "An experimental setup for

interaction turbulence-chemical reaction research," To be published in Int. J. of Heat ans Mass Transfer. [23] Lain;, C., Etude mathematique et numerique de modiles de turbulence en

conduite plane, Thise, Lyon (1980). [24] Lain;, C., C. R. Acad. Sc. Paris, Sirie B, 291 (1980), p. 63-66. [25] Brauner, C. M. and Lain;,

C.,

"Mathematical and numerical study of a

turbulent viscosity model," to appear. [26] Adams,

-.,

Sobolev spaces, Academic Press, New York (1975).

[27] Lions, J. L., "Quelques me'thodes de re'solution de prob1;mes

aux limites

non lineares," Dunod, Paris (1968). "Prob1;mes aux limites non homoge'nes et applications," Dunod, Paris (1968). [29] Moser, J., "A Harnack inequality for parabolic differential equations," Comm. Pure Appl. Math. 17 (1964), p. 101-134, and correction in 20 (1967), p. 231-236. [30] Moser, J., "On a pointwise estimate for parabolic differential equations," Comm. Pure Appl. Math. 24 (1971), p. 727-740. [31] Protter, M. H. and Weinberger, H. F., "Maximum principle in differential equations, Prentice Hall (1967).

[28] Lions, J. L. and Magenes, E.,

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 8. Nicolaenko ( d o . ) @ North-Holland Publishing Company, 1982

367

INSTABILITY OF PIPE FLOW

Anthony T. P a t e r a and Steven A. Orszag Department o f Mathematics Massachusetts I n s t i t u t e o f Technology Cambridge, Mdss. 02139

The s t a b i l i t y of a x i s y m m e t r i c d i s t u r b a n c e s i n HagenP o i s e u i l l e ( p i p e ) f l o w t o i n f i n i t e s i m a l non-axisymmetric p e r t u r b a t i o n s i s s t u d i e d by numerical s o l u t i o n o f t h e Navier-Stokes e q u a t i o n s . I t i s shown t h a t , i f t h e axisymmetric flow i s allowed t o evolve according t o t h e Navier-Stokes e q u a t i o n s , t h e non-axisymmetric i n s t a b i l i t y i s s t r o n g f o r Reynolds numbers R 23000 b u t weak f o r RL1000, t h e c u t - o f f b e i n g due b o t h t o i n c r e a s e d decay o f t h e a x i s y m m e t r i c f l o w and reduced g r o w t h o f t h e nona x i s y m m e t r i c p e r t u r b a t i o n . The non-axisymmetrlc i n s t a b i l i t y can be i s o l a t e d from t h e v e r y f a s t a x i symmetric decay i f t h e axisyrnmetric f l o w i s f o r c e d i n such a way t h a t i t s energy i s m a i n t a i n e d c o n s t a n t , b u t phase r e l a t i o n s a r e a l l o w e d t o develop. I n t h i s case, i t i s found t h a t t h e Reynolds number a t which t r a n s l t l o n o c c u r s i n experiments, R 2 2000, d e l i m i t s t h e r e g i o n s o f i n v i s c i d (1.e. R-independent) growth and v i s c o s i t y dominated decay. INTRODUCTION Experiments i n d i c a t e t h a t , u n l e s s g r e a t c a r e i s t a k e n t o remove background d i s t u r b ances, H a g e n - P o i s e u i l l e ( p i p e ) f l o w undergoes t r a n s i t i o n t o t u r b u l e n c e a t Reynolds 2000 numbers R (based on p i p e r a d i u s and c e n t e r l i n e v e l o c i t y ) o f t h e o r d e r o f (Wygnanski 8 Champagne 1973). Experimental work has s u c c e s s f u l l y addressed t h e problem o f r e l a t i n g t r a n s i t i o n s t r u c t u r e s (e.g. p u f f s , s l u g s ) t o f u l l y - d e v e l o p e d t u r b u l e n t f l o w (Rubin e t . a l . 19791, b u t t h e r e has been l i t t l e p r o g r e s s experimenta l l y o r a n a l y t i c a l l y towards f i n d i n g and c h a r a c t e r i z i n g an i n s t a b i l i t y which i s s i g n i f i c a n t a t t h e low Reynolds numbers a t which t r a n s i t i o n o c c u r s n a t u r a l l y . I t i s now g e n e r a l l y accepted t h a t H a g e n - P o i s e u i l l e flow i s s t a b l e t o a l l l i n e a r axisymmetric and non-axisymmetric d i s t u r b a n c e s (Davey 8 D r a z i n 1969; M e t c a l f e 8 Orszag 1973; Salwen e t . a t . 1980). Thus n o n l i n e a r i n t e r a c t i o n s must be i n v o l v e d i f e x p e r i m e n t a l o b s e r v a t i o n s a r e t o be e x p l a i n e d . Most p r e v i o u s f i n i t e - a m p l i t u d e s t a b i l i t y work has been a x i s y m m e t r i c i n n a t u r e . The a m p l i t u d e expansion techniques o f S t u a r t (1971) [extended t o problems where no n e u t r a l c u r v e e x i s t s (Reynolds 8 P o t t e r (196711 have been used t o suggest t h e e x i s t e n c e o f a x l s y m m e t r i c e q u i l i b r i a and, hence, axisymmetric s u b c r i t i c a l i n s t a b i l i t y (Davey 8, Nguyen 1971; l t o h 1977). However, P a t e r a 8 Orszag (1981) have shown n u m e r i c a l l y t h a t t h e s e s o l u t l o n s do not exist. I t was found t h a t , i n t h e r e g i o n o f wavenumber ( a ) , Reynolds number ( R ) space i n which t h e search was conducted, p i p e flow i s s t a b l e t o a l l f i n i t e a m p l i t u d e axisymmetric d i s t u r b a n c e s . We suspect t h a t p i p e f l o w i s g l o b a l l y s t a b l e t o a l l l i n e a r and n o n l i n e a r a x i s y m m e t r i c d i s t u r b a n c e s a t a l l (a,R). Thus n o t o n l y f i n i t e - a m p l i t u d e b u t a l s o non-axisymmetric d i s t u r b a n c e s must be considered b e f o r e an l n s t a b l l i t y can be o b t a i n e d i n H a g e n - P o i s e u i l l e f l o w . We

A.T. PATERA, S.A. ORSZAG

368

d i s c u s s t h i s s i t u a t i o n b r i e f l y i n t h e c o n t e x t o f p l a n e channel f l o w s . I n plane P o i s e u l l l e f l o w , two-dimensional e q u i l i b r i a do e x i s t (Zahn e t . a l . 1974; H e r b e r t 1976); by themselves, t h e y can n o t e x p l a i n t r a n s i t i o n inasmuch as t h e y o n l y e x i s t f o r R 22900 ( w h i l e t r a n s i t i o n may o c c u r a t R = 1000) and t h e y r e s u l t i n s a t u r a t e d s t a t e s , n o t c h a o t i c m o t i o n . However, we have p r e v i o u s l y shown (Orszag 8, P a t e r a 1980, 1981), t h a t t h e two dimensional s t a t e s a r e s t r o n g l y u n s t a b l e t o three-dimensional p e r t u r b a t i o n s . Furthermore, v e r y slowly-decaying two-dimensional s t a t e s ( q u a s i - e q u i l i b r i a ) e x i s t down t o R "= 1000. To t h e e x t e n t t h a t three-dimensional f i n i t e - a m p l i t u d e f l o w s a r e t y p i c a l l y c h a o t i c i n shear f l o w s (Orszag 8, # e l l s 19801, t h i s i s o l a t e s t h e b a s i c mechanism o f t r a n s i t i o n I n terms o f f i n i t e - a m p l i t u d e two-dimensional d i s t u r b a n c e s and i n f i n i t e s i m a l three-dimensional p e r t u r b a t i o n s . I n c o n t r a s t t o plane P o i s e u i i l e flow, i n plane Couette f l o w t h e r e a r e n e i t h e r e q u i l i b r i a n o r q u a s i - e q u i l i b r i a . However, i t was found (Orszag & P a t e r a 19811 t h a t t h e same mechanism t h a t operates i n p l a n e P o i s e u i l l e f l o w a l s o h o l d s i n p l a n e C o u e t t e flow, i . a . twodimensional decaying f i n i t e - a m p l i t u d e s t a t e s a r e s t r o n g l y u n s t a b l e t o t h r e e dimensional i n f i n i t e s i m a l p e r t u r b a t i o n s . The three-dimensional i n s t a b i l i t y i s n o t y e t c o m p l e t e l y understood, a l t h o u g h i t i s c l e a r from numerical experiments t h a t i t i s i n v i s c i d i n n a t u r e , I n c o n t r a s t t o t h e two-dimensional e q u i l i b r i a (and t h e i r l i n e a r c o u n t e r p a r t , T o l l m i e n S c h l i c h t i n g waves), which r e q u i r e f i n i t e v i s c o s i t y . Note, however, t h a t t h i s i n v i s c i d three-dimensional I n s t a b i l i t y i s n o t a c l a s s i c a l i n f l e c t i o n a l I n s t a b i l i t y . For, i f t h e i n s t a n t a n e o u s l y i n f l e c t i o n a l two-dimensional p r o f i l e s (generated by t h e e q u i l i b r i a ) d r i v e t h e i n s t a b i l i t y , t h e n one would a l s o e x p e c t two-dimensional p e r t u r b a t i o n s t o t h e e q u i l i b r i a t o be u n s t a b l e , which i s n o t t h e case. The success o f t h e two-dimensional f inite-amp1 i t u d e / t h r e e - d i m e n s i o n a l i n f l n i t e simal s t r u c t u r e i n e x p l a i n i n g s u b c r i t i c a l t r a n s i t i o n i n p l a n e channel f l o w s suggests t h a t such a mechanism may a l s o prove r e l e v a n t i n p i p e P o i s e u i l l e f l o w , e s p e c i a l l y s i n c e p i p e P o i s e u i l l e f l o w and p l a n e C o u e t t e f l o w a r e v e r y s i m i l a r i n terms of b o t h l i n e a r t h e o r y and two-dimensional f i n i t e - a m p l i t u d e b e h a v i o r . I n t h i s paper, we show t h a t t h i s i s indeed t h e case. Furthermore, new i n s i g h t s a r e p r o v i d e d i n t o s u b c r i t i c a l 1 i n e a r t h r e e - d i m e n s i o n a l i n s t a b i I i t y , and new t o o l s a r e presented f o r c a r r y i n g o u t t h e s t a b i l i t y a n a l y s i s when two-dimensional e q u i l i b r i a do n o t e x i s t . THREE-DIMENSIONAL LINEAR PROBLEM The problem t o be s t u d i e d here i n v o l v e s t h e s t a b i l i t y o f H a g e n - P o i s u i l l e flow ( d e f i n e d as f l o w i n a c i r c u l a r p i p e o r r a d i u s I d r i v e n by t h e c o n s t a n t pressure g r a d i e n t -4/R) t o f i n i t e - a m p l i t u d e a x l s y m m e t r i c and i n f i n i t e s i m a l t h r e e dimensional (non-axisymmetric) p e r t u r b a t i o n s :

Here x,r, and 0 a r e t h e a x i a l , r a d i a l , and a z i m u t h a l c o o r d i n a t e s , r e s p e c t i v e l y , and E i s a ( s m a l l ) parameter r e l a t e d t o t h e a m p l i t u d e o f t h e three-dimensional component. W i t h t h e decomposition ( 1 ) (and E < < I ) , t h e Navier-Stokes equations a r e sepqrable i n 0 so t h a t 'it i s c o n s i s t e o t t o assume t h a t t h e If I s a l s o expanded i n a &dependence o f v i l ) i s o f t h e form e+ime F o u r i e r s e r i e s i n x (keeping a l l modes i n t h s d i r e c t i o n because o f n o n l i n e a r i n t e r a c t l o n s ) , t h e v e l o c i t y i s represented as

.

e

where

2a/a

i s the periodicity interval i n

i a n x Iqme e

he x - d i r e c t i o n .

R e a l i t y of t h e

INSTABILITY OF PIPE FLOW

369

v e l o c i t y f i e l d implies t h e f o l l o w i n g symmetry:

where * denotes complex conjugate. Note t h e assumption o f p e r i o d i c i t y i n x and 0 corresponds i n c l a s s i c a l stability theory t o a temporal s t a b i l i t y analysis. The incompressible Navier-Stok$s equations govern t h e e v o l u t i o n o f To O ( E ~ ) , t h e equations f o r v i 9 ) are

-f

v

i n time.

at

where 6 . i s t h e KroneckeSdeIta f u n c t i o n and t h e c a r a t denotes a u n l t v e c t o r The bounhdry c o n d i t i o n s on v are

;Aq)

(r = 0)

bounded

the l a t t e r being t h e n o - s l i p c o n d i t i o n . No boundary c o n d i t i o n s a r e r e q u i r e d i n t h e o t h e r d l r e c t i o n s due t o t h e assumption o f p e r l o d i c l t y . Note t h a t c l a s s i c a l l i n e a r s t a b i l i t y theory i s a s p e c i a l case o f t h e above formI n t h i s case, t h e r e l e v a n t parameter u l a t i o n i n which v ( o ) ( x , r , t ) = ( l - r * ) x . space i s (R,a,m), and I t i s g e n e r a l l y agreed t h a t a l l disturbances decay. The s o l u t i o n t o t h e l i n e a r c h a r a c t e r i s t i c - v a l u e problem w i l l be denoted as CR,a,ml. With n o n l i n e a r e f f e c t s allowed, t h e parameter space must be extended t o i n c l u d e a measure o f amplitude, such as t h e modal energies: (4a)

(The f a c t o r of 6 i s so t h a t ener l e z f o r n # 0 a r e measured r e l a t i v e t o t h a t ( I) represent t h e and o f the laminar p a r a l l e l flow ( I - r 1 x 1. E# axisymmetric and non-axlsymmetric p e r t u r b a t i o n energies>%spectlvely.

9

370

A.T. PATERA. S.A. ORSZAG

NUMERICAL METHOD The numerical technique used here f o r s o l v i n g t h e Navler-Stokes equations i s very s i m i l a r t o t h a t discussed i n d e t a i l i n Patera 8 Orszag (1981a) f o r s o l v i n g t h e f i n i t e - a m p l l t u d e axisymmetric problem. The e v a l u a t i o n o f t h e n o n l i n e a r terms, the time-stepping method, and t h e m a t r i x i n v e r s i o n techniques a l l c a r r y over d i r e c t l y t o t h e non-axlsymmetric problem. I t i s o n l y I n t h e treatment o f t h e a x i s , and, i n p a r t i c u l a r , i t s e f f e c t on t h e choice o f s u i t a b l e s p e c t r a l expansions and boundary c o n d i t i o n s i n which s p e c i a l care must be taken i n s o l v i n g t h e t h r e e dimensional problem. We l i m i t discussion here t o these more s u b t l e p o i n t s .

-+

is expanded i n As i n t h e axisymmetric problem, t h e r a d i h l dependence o f v a Chebyshev polynomial s e r i e s , where t h e p t h Chebyshev polynomial i s d e f i n e d

T

P

(cos 8 ) = cos p e

52

The f a c t t h a t t h e domain o f i n t e r e s t i s 0 r 2 I whereas t h e argument o f t h e Chebyshev polynomials, z, satisfies -I z <_ I can be handled I n one o f two ways. F i r s t , a simple s c a l i n g map can be used t o s t r e t c h t h e [ O , l ] interval. Even b e t t e r , a given v a r i a b l e can be expanded i n an even o r odd Chebyshev s e r i e s , t h e p a r i t y o f t h e expansion being determined by t h e behavior o f t h e v a r i a b l e a t t h e axis, r = 0. I n p a r t i c u l a r , we r e q u i r e expansions f o r t h e a x i a l , r a d i a l , and azimuthal v e l o c i t i e s , denoted u,v, and w, respectively. as r + O we examine t h e To determine t h e leading behavior o f u,v, and w most s i n g u l a r p a r t o f ( 3 1 , namely t h e viscous terms. (The c o n d i t i o n s d e r i v e d i n t h i s way w i l l a l s o i n s u r e boundedness o f t h e v o r t i c i t y and divergence a t r = 0). I n c y l i n d r i c a l coordinates t h e viscous s t r e s s term Is n o t diagonal, i.e. t h e s t r e s s i n t h e Xk ( X I = x,x2 = r,x3 = 8 ) d i r e c t i o n cannot be expressed s o l e l y i n terms o f vk ( k here again denoting d i r e c t l o n a l comDonent). However, a simple t r a n s f o r m a t i o n o f t h e independent v a r i a b l e [u,v,w]+ E.?,;] def ined by

-U ' U

-v = v + i w

(5)

-

w = v - i w For a v e l o c i t y component o f t h e form t h e viscous term can now be expressed as

+diagonalizes t h e - v i s c o u s operator. v(x,r,e)

=

Grim

elCInx e l m e

2 0

0

0

r 0

L

0

where

L='L ar

a

( r - ) - a

ar

2

2

n

(m-I) 2 -r 7

nm

-.

W

nm

je

ianxe ime

INSTABILITY OF PIPE FLOW

37 1

Note t h e change o f v a r i a b l e ( 5 ) i s advantageous f o r t h e s o l u t i o n o f t h e NavierStokes equations: i n t h i s form t h r e e separate Laplacians must be inverted, whereas using [u,v,w] t h e r e I s a c o u p l i n g o f two o f t h e equatlons which would r e q u i r e ( i f t h e s o l u t i o n were f u l l y I m p l i c i t ) a d d i t i o n a l computing time, storage, and complexity. Analysis o f l o c a l behavior near t h e s i n g u l a r p o i n t o f ( 6 ) a t a t t e n t i o n t o m 2 0 since r e a l i t y r e l a t e s m and -m) gives

-u

-

r

w

-

r

fm

,

k(m-1)

m>O

,

in

;

I, Rnr

# I;

I , Rnr

r = 0 (restricting

m=O

m = I

(Batchelor 8 G i I I 1962; Orszag 1974). Consistent w i t h t h i s behavior t h e complete s p e c t r a l expansion o f t h e v e l o c i t y f i e l d can be w r i t t e n as

R

where

= 2p

+

b(k,m

191)

( 0 (r = 1 b(r,s) =

and

k

I

I (r = I

and

and

s

even)

or

( r = 2,3

and

s

odd)

and

s

odd 1

or

(r = 2,3

and

s

even)

r e f e r s t o d i r e c t i o n a l component.

R e a l i t y imposes t h e c o n d i t i o n

on t h e f i n i t e F o u r i e r s e r i e s . F i n a l l y , we discuss i m p o s i t i o n o f boundary c o n d i t i o n s a t r = 0. When t h e equation t o be solved can be regularized, no boundary c o n d i t i o n i s r e q u i r e d (1.e. t h e s i n g u l a r i t y i n t h e equation i s removed by r e j e c t i n g t h e unbounded s o l u t i o n ) . I f t h e equation cannot be regularized, a boundary c o n d i t i o n a t r = 0 i s r e q u l r e d A boundary c o n d i t i o n i s e a s i l y obtained from ( 7 ) f o r e i t h e r t h e s o l u t i o n o r i t s derivative. We w i l l comment b r i e f l y here on t h e s o l u t i o n o f t h e l i n e a r - theory c h a r a c t e r i s t i c value problem as l i n e a r modes provide t h e i n i t i a l c o n d i t i o n s f o r t h e f i n i t e

372

A.T. PATERA, SA. ORSZAG

amplitude runs. As i n t h e simulation, a pseudospectral Chebyshev method i s used t o s e t up t h e m a t r i x equations, and e i t h e r a c u b i c a l l y convergent ( i n v e r s e i t e r a t l o n / R a y l e i g h q u o t i e n t ) l o c a l method o r a g l o b a l (OR) a l g o r i t h m i s used t o s o h e t h e a l g e b r a i c eigenvalue problem. RESULTS Before presenting f i n i t e - a m p l i t u d e r e s u l t s , we comment t h a t t h e numerical code a c c u r a t e l y simulates l i n e a r behavior. Test runs i n d i c a t e t h a t a t R = 4000, a = 1.0, w i t h P = 32, A t = 0.025, t h e energy o f b o t h m = 0 modes, {R,a,m) = {4000, 1.0,O) , and m = I modes, 14000, I .O, I ) , decay a t a r a t e i n a d d i t i o n t o these l i n e a r which agrees t o w i t h i n about 0.1% o f l i n e a r theory. t e s t s , axlsymmetrlc f i n i t e amplitude runs were a l s o made t o c o n f i r m t h e c o n c l u s i o n s o f o u r previous work on p i p e f l o w (Patera & Orszag 1981). I n the f i n i t e - a m p l i t u d e a x l s y m m e t r i c / l l n e a r non-axisymmetric runs, i n i t i a l c o n d i t i o n s are o f t h e form

+

where t h e l i n e a r mode v I s chosen t o be t h e l e a s t s t a b l e w a l i mode a t t h e g i v e n i n i t i a l ehergy o f t h e axisymmetrlc d i s t u r b a n c e i s s p e c i f i e d as ( t = 0 ) . The i n i t i a l amplitude o f t h e non-axisymmetric component as t h e theory i s l i n e a r I n t h e azimuthal d i r e c t i o n . Note t h a t t h e r e s u l t s discussed below concern f l o w s developing from i n i t i a l c o n d i t i o n s c o n s i s t i n g o f w a l l modes w i t h m = I . The behavior o f c e n t e r modes and modes w i t h general m ( # I ) i s n o t important and w i l l be discussed b r i e f l y a t t h e end o f t h i s section. For t h e range o f R,a run here, t h e numerical parameters P = 32, N = 8, A t = 0.025 p r o v i d e f u l l y converged r e s u l t s as was confirmed by doubling ( o r h a l v i n g ) t h e values g i v e n f o r P,N, and A t above and comparing the resulting solutions. The n a t u r a l way t o proceed i n seeking a t r a n s i t i o n - r e l a t e d i n s t a b i i i t y i s t o e s t a b l i s h t h a t an i n s t a b i l i t y e x i s t s f o r Reynolds numbers B l l g h t l y l a r g e r than RT, t h e t r a n s i t i o n a l Reynolds number, and f o l l o w l n g t h e behavior o f t h e instabI n Fig. I we demonstrate t h a t a threei l i t y as R passes through RT. dimensional i n s t a b i l i t y o f t h e type found p r e v i o u s l y i n plane channel flows a c t u a l l y e x i s t s i n Hagen-Poiseuille flow; t h e wavenumber dependence o f t h e l n s t a b l I i t y a t R = 4000 i s i I l u s t r a t e d by a p l o t o f En E+& f o r q = 0, I vs = 0.04). For a = 0.5 t h e r e i s o n l y weak c o u p l i n g o f t h e time. (Here E(0) axlsymmetric a n ~ o b 8 - a x i s y m m e t r l cf i e l d s and three-dimensional growth i s less than f o r a = 1.0 o r a = 2.0. The growth o f non-axisymmetric disturbances a t a = 2.0 i s approximately t h e same as t h a t a t a = 1.0, b u t t h e axisymmetric s o l u t i o n s decay much more q u i c k l y . Thus, t a k i n g i n t o c o n s i d e r a t i o n both a x l symmetric decay and non-axlsymmetrlc growth, a = 1.0 appears t o be n e a r l y t h e most dangerous wavenumber. I n Fig. 2 we agaln p l o t En E# f o r q = 0,i vs t i m e b u t now a t a = 1.0 l e t t i n g t h e Reynolds number v a r y from 500 t o 4000. Although i t i s c l e a r from t h i s s e r i e s o f runs t h a t t h e I n s t a b i l i t y i s dangerous a t R = 3000 and I n s i g n i f f c a n t a t R = 500, a Reynolds number of 2000 ( o r s l i g h t l y l a r g e r ) I s n o t s i n g l e d i n t h e p l o t as d i s t i n g u i s h i n g growth from decay. T h i s problem o f i n t e r -

INSTABILITY OF PIPE FLOW

373

t

100.

C

a=i. 2.

In E .5

Fig. 1. A plot of the growth of non-axisymmetric (three-dimensional) perturbations on (decaying) axisymmetric states at R = 4000. When the axisymmetric flow is of sufficiently large amplitude, the nonaxisymetric perturbation grows exponentially. Observe that when (I 1.0 the pipe flow is most unstable. Lower wavenumbers significantly decrease the growth o f non-axisymmetric perturbations, while higher wavenumbers result in sharply increased axisymmetric decay.

-5c

t

0. 0.

% x R=4000

-

InE

'

-50.

Pig. 2. A plot of the growth of non-axisymmetrfc perturbations on decaying axisymmetric states with a = 1.0 for various R. The instability is strong for R 2 3 0 0 0 and weak for R S 1000. the cut-off due primarily to increased axisymnetric decay.

100.

374

A.T. PATERA, S.A. ORSZAG

p r e t a t i o n a r i s e s from t h e f a c t t h a t , I n p i p e flow, f i n i t e - a m p l i t u d e a x i s y m m e t r i c d i s t u r b a n c e s decay on a t i m e s c a l e even s h o r t e r t h a n t h o s e i n p l a n e C o u e t t e f l o w , and t h u s t h e d e t a i l s o f t h e i n s t a b i l l t y a r e obscured by t h e two-dimensional damp i ng

.

To i s o l a t e t h e three-dimensional i n s t a b i l i t y r e q u l r e s f o r c i n g o f t h e axisymmetric f l o w i n such a way t h a t i t e v o l v e s n a t u r a l l y w h i l e m a i n t a i n i n g a f i x e d a m p l i t u d e . (amplitude -1 frozen, We term such a f o r c i n g o f t h e two-dimensional f l o w Ameaning t h a t o v e r a l l a m p l i t u d e s a r e c o n s t r a i n e d , b u t phase r e l a t i o n s a r e n o t . To w i t , t h e f o l l o w i n g scheme i s used: a f t e r eachE ( 8 ) and E ( O ) are mode i s t h e n s c a l e d by and a l I n a r e ?he d e s i r e d ( c o n s t a n t i , where and F$gi and p e r t u r b a t i o n e n e r g i e s , r e s p e c t i v e l y . Note t h a t , w i t h t h i s scheme, harmonics can develop and waves can propagate. We a r e o n l y imposing a weak e n e r g e t l c c o n s t r a i n t on t h e a x l s y m m e t r i c f l o w .

$8)

%$I n F i g . 3 we show a p l o t s i m i l a r t o F i g . 2 e x c e p t h e r e t h e a x i s y m m e t r i c f i e l d i s A-frozen. Note f i r s t t h a t t h e growth o f three-dimensional modes i s v e r y c l o s e l y pure e x p o n e n t i a l as would be expected i f one a c t u a l l y had a x i s y m m e t r l c e q u i l i b r l a ; t h u s we have e f f e c t i v e l y separated t h e e f f e c t s o f axlsymmetric decay and non-axisymmetric growth. More i m p o r t a n t l y , n o t e t h a t i n t h i s p l o t R = 2000 delimits regions of i n v i s c i d (i.e. R-independent) growth ( R L 2 0 0 0 ) and viscous-dominated behaviour ( R $2000). T h i s i s r e f l e c t e d by t h e c l o s e spacing o f the- Rn E(;At c u r v e s f o r R 22000 and t h e a b r u p t change t o wide spacing a t R = 2000. The f a c t t h a t f o r c i n g i s r e q u l r e d t o h i g h l i g h t t h e most s i g n i f i c a n t aspects o f the i n s t a b i l i t y i n Hagen-Poiseuille f l o w b u t n o t i n plane P o i s e u i l l e f l o w i s c e r t a i n l y l i n k e d t o t h e d i f f e r e n t n a t u r e G f t r a n s i t i o n I n t h e s e two f l o w s . The appearance o f p u f f s and s l u g s I n H a g e n - P o i s e u i l l e f l o w experiments i s evidence t h a t e x p l o s i v e l y u n s t a b l e t r a n s i t i o n s t r u c t u r e s may u l t i m a t e l y decay ( b a r r i n g v e r y l a r g e d i s t u r b a n c e s ) , j u s t as t h e b e h a v i o r o f o u r I n s t a b i l i t y i n t h e u n f o r c e d case i s numerical evidence o f t h e same phenomenon. F u r t h e r i n s i g h t i n t o t h e p r e s e n t non-axlsymmetric i n s t a b i l i t y i s o b t a i n e d by i n v e s t i g a t i n g t h e requirements on t h e f o r c i n g p r o t o c o l i n o r d e r t h a t t h r e e dimensional I n s t a b i l i t y be maintained. I n p a r t i c u l a r , we d e f i n e an a x i s y m m e t r i c f i e l d t o be $-(phase - ) f r o z e n I f , a t every t i m e step, t h e axisymmetric f i e l d I s r e s e t t o i t s f r o z e n v a l u e i n t h e l a b o r a t o r y c o o r d i n a t e frame; note the d i f f e r e n c e between $ - f r o z e n (where phase r e l a t i o n s a r e n o t a l l o w e d t o e v o l v e f o r A-frozen and $ - f r o z e n n a t u r a l l y ) and A-frozen. I n F i g . 4 , ]In E ( I ) f i e l d s a r e compared. Note t h a t t h e r e i s v i r k b l l y no growth i n $ - f r o z e n case. Furthermore, i t i s n o t t h e l a c k o f harmonic g e n e r a t i o n i n t h e $ - f r o z e n f i e l d t h a t r u l e s o u t growth, f o r , i f t h e a x i s y m m e t r i c f i e l d i s $ - f r o z e n a t some t i m e T a f t e r i t has e v o l v e d n a t u r a l l y f o r 0 ( t < T , three-dimensional g r o w t h r a p i d l y t u r n s o f f . The d r a s t l c a l l y d i f f e r e n t r e s u l t s o b t a i n e d a r e no doubt due t o t h e s y n c h r o n i z a t i o n phenomenon found i n p l a n e P o i s e u i l l e f l o w ( P a t e r a 8 Orszag 1981b), where b o t h t h e two-dimensional wave and t h e most dangerous t h r e e dimensional p e r t u r b a t i o n t r a v e l a t t h e same phase speed (1.e. t h e temporal e l g e n v a l u e r~ i s r e a l ) . I n t h e $I - f r o z e n case t h i s s y n c h r o n i z a t i o n i s inhibited. We have n o t y e t t h o r o u g h l y i n v e s t i g a t e d t h e i n s t a b i l i t y o f a x i s y m m e t r i c d i s t u r b ances t o h i g h e r azimuthal-wavenumber (m> 1 ) p e r t u r b a t i o n s . However, i t appears (as i n l i n e a r t h e o r y ) t h a t t h e e f f e c t o f i n c r e a s i n g m Is s t a b i l i z i n g . Also, r u n s have been made i n which t h e i n i t i a l c o n d i t i o n s I n v o l v e c e n t e r modes r a t h e r t h a n w a l l modes. I t i s found t h a t , f o r t h e same I n i t i a l maximum a m p l i t u d e o f t h e axisymmetric d i s t u r b a n c e , w a l l modes a r e t h e more dangerous. (Note t h a t i f one compares non-axisymmetric growth due t o a w a l l and c e n t e r mode o f t h e same i n i t i a l energy, t h e l a t t e r appears more dangerous. However, t h i s r e s u l t i s an a r t i f a c t of t h e energy d e f i n i t i o n s ( 4 ) I n which t h e energy o f t h e

INSTABILITY OF PIPE FLOW

375

t

100.

0

1

/ R=4000 2-D , A - FROZEN

3000

2000

In E

1000

500

1

-5c

Pig. 3. A plot of the growth of non-axisynmetric perturbations on A-frozen axisymmetric states with a = 1.0 for various R. In A-freezing the twodimensional field we prevent axisymmetric decay but rllow non-axisymmetric qrowth, thus isolating the mechanism of three-dimensional instability. It is seen that at R G 2 0 0 0 there is a change from viscous behavior (strongly R-dependent) to inviscid growth (approximately R independent).

-

t

100.

0,

/

A- FROZEN

3-D

In E

50

?ig. 4. A comparison of the growth of non-axisymmetric perturbations driven by A-frozen and +-frozen axisynmetric fields. In the +-frozen case synchronization between the axisymnetric and non-axisymmetric flow is inhibited and there is no three-dimensional growth. These runs are at R 4000. a = 1.0.

-

A.T. PATERA. S.A. ORSZAG

376

d i s t u r b a n c e I s weighted w i t h d i s t a n c e from t h e a x i s . Thus v e r y l a r g e a m p l i t u d e c e n t e r modes r e s u l t i n small e n e r g i e s . F o r example, an i n i t i a l energy o f 0.04 corresponds t o r o u g h l y a 10%. maximum amp1 i t u d e f o r a wal I mode, b u t t o almost a 40% maximum a m p l i t u d e f o r a c e n t e r mode.) T h i s work was supported by t h e O f f i c e o f Naval Research under C o n t r a c t s No. NOOOI4-7FC-0138 and No. N00014-79-C-0478. Development o f t h e s t a b i l i t y codes was supported by NASA Langley Research Center under C o n t r c a t No. NASI-16237. REFERENCES: [I1

B a t c h e l o r , F.K. 8 G i l l , A . E. 1962 axlsymmetric j e t s . J . F l u i d Mech.

c21

Davey, A. 8 Drazin,P. G. a pipe. J. F l u i d Mech.

C31

1971 F i n i t e - a m p l i t u d e s t a b i l i t y o f p i p e Davey, A. 8 Nguyen, H.P.F. flow. J . F l u i d Mech. 701.

C41

H e r b e r t , T. 1976 P e r i o d i c secondary m o t i o n s i n a p l a n e channel. I n Proc. 5 t h I n t . Conf. on Numerlcal Methods i n F l u i d Dynamics (ed. A . I . van de Vooren and P. J . Zandbergen), p. 235, S p r i n g e r .

C5l

I t o h , N. 1977 N o n l i n e a r s t a b i l i t y o f p a r a l l e l f l o w s w l t h subc r i t i c a l Reynolds number. P a r t 2. S t a b i l i t y o f p i p e P o i s e u i l l e f l o w t o f i n i t e axisymmetric d i s t r u b a n c e s . J . F l u i d Mech. 82, 469.

c61

M e t c a l f e , R. W. 8 Orszag, S. A. l i n e a r s t a b i l i t y o f p i p e flows. Washington.

C7l

1969

2,209.

Analysis o f the s t a b i l i t y o f

14,529.

The s t a b i l i t y o f P o i s e u l l l e f l o w I n

45,

Orszag, S. A. 56.

1973 Numerical c a l c u l a t i o n s o f t h e Flow Research Report No. 25, Kent,

1974 F o u r i e r s e r i e s on spheres. Mon. Wea. Rev.

102, -

C8l

Orszag, S. A. 8 K e l l s , L. C. 1980 T r a n s i t i o n t o t u r b u l e n c e i n p l a n e P o i s e u i l l e and p l a n e C o u e t t e flow. J . F l u i d Mech. 96, 159.

C9l

1980 S u b c r i t i c a l t r a n s i t i o n t o Orszag, S. A . 8 Patera, A. T. t u r b u l e n c e i n p l a n e channel f l o w s , Phys. Rev. L e t t e r s , 989.

[I01

198J Three dimensional i n s t a b i l i t y Orszag, S. A. 8 Patera, A. T. o f p l a n e channel f l o w s . To be p u b l i s h e d .

C I I1

Patera, A. T. 8 Orszag, S. A. 1981 F i n i t e - a m p l i t u d e s t a b i l i t y of axisymmetric p i p e flow. To appear i n J . F l u i d Mech.

c121

Reynolds, W. C. 8 P o t t e r , M. C. 1967 F i n i t e - a m p l i t u d e I n s t a b i l i t y o f p a r a l l e l shear f l o w s . J . F l u i d Mech. 278 465.

[I31

Rubln, Y., Wygnanski, I., 8 H a r l t o n l d l s , J. H. 1979 F u r t h e r o b s e r v a t i o n s on t r a n s i t i o n i n a p i p e . I n Laminar-Turbulent T r a n s i t i o n , IUTAM Symposium (ed. R. E p p l e r 8 H. F a s e l ) , p. 17, Springer.

C141

Salwen, H., Cotton, F. W . , 8 Grosch, C. E. 1980 L i n e a r s t a b i l i t y o f P o i s e u i l l e f l o w i n a c i r c u l a r pipe, J . F l u i d Mech., 98, 273.

45,

INSTABILITY OF PIPE FLOW

[I51

S t u a r t , J . T. Mech. 3, 341.

[la]

Wygnanski, I . 8 Champagne, F. H. 1973 On t r a n s i t i o n I n a pipe, P a r t I . The o r i g i n o f puffs and s l u g s and t h e flow I n a t u r b u l e n t s l u g . J . F l u i d Mech. 2, 281.

[I71

Zahn, J.-P., Toomre, J . , S p i e g e l , E. A. 8 Gough, D. 0. 1974 N o n l i n e a r c e l l u l a r rnotlons i n P o i s e u i l l e channel flow, J . F l u i d Mech. 2, 319.

1971 N o n l i n e a r s t a b i l i t y t h e o r y , Ann. Rev. F l u i d

377

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Nonlinear Problems: Present arid Future

379

A.R. Bishop, D.K. Campbell, 6. Nicolaenko (eds.) 0 North-HollandPublishing Company, 1982

SOME FORMALISM AN0 PREDICTIONS OF THE PERIOD-DOUBLING ONSET OF CHAOS

M i t c h e l l J. Feigenbaum Theoretical D i v i s i o n Los Alamos National Laboratory Los Alamos, New Mexico 87545 We i n i t i a l l y consider one-parameter f a m i l i e s o f maps fA w i t h t h e p r o p e r t y t h a t f o r a convergent sequence o f parameter values, the map possesses s t a b l e p e r i o d i c o r b i t s o f successively increasing period: An

+

A,

;

fA has a s t a b l e o r b i t o f p e r i o d 2n n

.

I n an i n t e r v a l o f parameter values around ,A,

a f i x e d - p o i n t theory provides uni-

versa1 l i m i t s f o r these maps when a p p r o p r i a t e l y i t e r a t e d and rescaled.

The

operator whose f i x e d p o i n t i s considered i s

where f has a quadratic extremum conventionally l o c a t e d a t t h e o r i g i n .

IY i s

determined by r e q u i r i n g T t o have a f i x e d p o i n t g: Tg = g; g(0) = 1, g(x) = $(x2), $ r e a l a n a l y t i c . About g, DT has a unique eignevalue, 6, i n excess o f 1. Numerically, 9 IY = 2.5029..

6 = 4.6692..

I n terms o f t h i s f i x e d p o i n t g,

where v i s an f-dependent magnification.

We assume f has been s u i t a b l y magni-

f i e d so t h a t we can take v = 1; the l i m i t i n (2) i s i n t h i s sense universal. (Observe t h a t the o b j e c t on the l e f t i s T n f

hat

.)

380

M.J. FEIGENBAUM

Consider

(An(p))

i s a sequence o f parameters f o r which fA has a period-doubling sequence

o f o r b i t s a l l o f s t a b i l i t y p. superstable values,

etc.).

(p = -1 determines b i f u r c a t i o n values, p = 0

For I p I < 1 xn(p)

i s an element o f a s t a b l e Zn-

cycle, which we choose as t h a t element o f smallest modulus. For l p l > 1 xn(p) i s the smallest element o f an unstable c y c l e which i s c o e x i s t e n t w i t h an a t t r a c t o r o f i n t e r e s t . For example, there i s a value o f p < -1 which i s t h e slope o f fZna t i t s smallest f i x e d p o i n t when t h e r e i s a superstable 2"'

cycle:

Figure 1: A schematic p l o t o f fZn showing t h e p o i n t a t which the slope equals p < -1.

F p r a more negative value o f p , we are considering an unstable f i x e d p o i n t o f f 2" when t h e r e i s a superstable 2"' such t h a t a Zn+'-cycle

cycle.

f Zn when period doubling has accumulated. A,

where

S i m i l a r l y , t h e r e i s a sequence o f pr < -1

i s superstable, u n t i l a t

fi

we consider t h e f i x e d p o i n t o f

Clearly,

381

PERIOD-DOUBLING ONSET OF CHAOS

(5) g(2) =

R ,

g'(2) =

;.

Thus, C c p c - 1 correspond t o stable p e r i o d i c behaviors below A,, an i n t e r v a l about the o r i g i n

while p <

fi

f o r which f2nmaps

onto i t s e l f : I f2"

Figure 2: (Actually,

p* such t h a t f2" maps an

t h e r e i s a p:

"Misiurewicz-points" [1,2]

-j

i n t e r v a l about the o r i g i n onto i t s e l f . I n t h i s case An(p*)

p*.)

will

determine t h e

a t which ergodic and mixing behavior occurs.

Accord-

i n g l y p serves as a "universal" parametrization o f maps and t h e l a r g e n behavior o f (3) and (4) s h a l l a l l o w a uniform presentation o f b o t h p e r i o d i c and "chaotic"

behaviors. Returning t o ( 3 ) , consider

= Tnf Am

+ (An+r(p)

-

h,)OTn*aAf +

M.J. FEIGEMBA UM

382

where

Since as r

+

lim

i s g, i t follows t h a t

w

and

k(p) i s independent o f f since

and g k(p).

0,lJ

must have slope p a t i t s smallest f i x e d p o i n t , which then determines

Equation (8) [3] i s t h e strongest consequence o f T ' s f i x e d p o i n t , since

i t determines

which

l i m i t s o f i t e r a t e s e x i s t and then t h a t these l i m i t s are

universal. For a l l a p p l i c a t i o n s , t h e s t r a t e g y i s t o approximate i t e r a t e s by these universal l i m i t s , where parameter values, amount o f magnification, and order o f i t e r a t i o n are "massaged" i n t o t h e form o f t h e l e f t - h a n d s i d e o f (8). I n t h i s way Lyapunov exponents, t r a j e c t o r y s c a l i n g f u n c t i o n s [4], termined.

etc. are de-

For example by ( 3 ) ,

Accordingly, denoting iC(p) as t h e smallest f i x e d p o i n t o f g

*

OYP'

Then we have,

demonstrating t h a t a s c a l i n g applies whatever t h e s t a b i l i t y . follows from (4) t h a t

(It immediately

PERIOD-DOUBLING ONSETOF CHAOS

303

p = giJ,p(2(ld)

(12)

which, as mentioned above, i s the f a c t used t o determine k(p)).

which,

Also, by (7),

o f course, demonstrates t h a t t h e parameter convergence r a t e 6 i s inde-

pendent o f s t a b i l i t y .

( I n p a r t i c u l a r , t h e Misiurewicz values a l s o accumulate t o

,A a t the r a t e 6.) I t remains t o determine k(p).

For p s u f f i c i e n t l y near

C;

(8) can be used even

f o r r = 0.

For a l a r g e r range o f p (8) i s used f o r r = 1 and then (9) i s used t o reach r = 0. (A precise computation o f g i s u l t i m a t e l y deeply nonlinear 0,c1

so t h a t numerics must be employed i f approximations are unacceptable.)

demonstrate t h e simplest case:

i.e.,

Next, by (10) 2(p) 2 2 + (p

or

-

P)2'(c)

=

g(2 + (p

- P)

2'(fi)) + k(p)g6(i()

We

304

M.J. FElGfNBAUM

Since g and

are determined (by s o l u t i o n o f t h e f i x e d p o i n t equations), every-

t h i n g i n (16)

i s universal.

(Together w i t h (131,

it i s also clear t h a t i f

period-doubl i n g sequences o f p e r i o d i c behavior converge t o A, f o r a1 1 p <

p,

from below, then

sequences must converge from above. )

As an example o f the u t i l i t y o f (8), we compute t h e Lyapunov exponent i n the

v i c i n i t y o f A,.

a

limIn w n

A t An(p),

k fnl

according t o (8) fZncan be approximated by go,p.

Define

Then, by the chain r u l e ,

and

Next, t a k i n g a cue from (8), d e f i n e

Er =

n ( - a ) xr.

By (12) and (81, f&+l D i f f e r e n t i a t i ng

(a),

g1 S c 1

(tr) f o r

l a r g e enough n.

So, consider

385

PERIOD-DOUBLING ONSET OF CHAOS

Accordingly , by (19),

an scales simply by powers 2(p) w h i l e by (12)

Thus, L(p) i s universal, w h i l e f o r l a r g e n, by (19), o f 2.

For s t a b l e p e r i o d i c behavior (p >

p), .$. =

L(p) = l n l p l < 0.

F i n a l l y , by (13) i f k(p) i s a v a i l a b l e , L(p) can be obtained as a f u n c t i o n o f . ,A Thus, L(p) i s a universal f u n c t i o n p e r i o d i c i n t h e v a r i a l b e log6(A ) ,A w i t h p e r i o d 1 f o r A < . A, For p = p * , L(p*) i s evaluated from go * and t h e i n ,cI v a r i a n t density a t p* from (20) and t h e ergodic theorem. ln(p*) achieves t h e A

-

-

values 2-nL(p*) a t An(p*),

so t h a t f o r A > ,A

a

l i e s w i t h i n a curve [5]

or

A t t h i s p o i n t we enlarge our scope from maps on an i n t e r v a l t o general m u l t i -

dimensional, d i s s i p a t i v e dynamical systems. For a wide class o f such systems i t turns o u t t h a t the mapping theory discussed above c a r r i e s over w i t h o u t modifications (see reference [9] f o r d e t a i l s and f u r t h e r general information).

How-

ever, beyond the usual 4.669 convergence r a t e , t h e dynamical consequences o f t h e theory are most n a t u r a l l y phrased i n the format o f a s c a l i n g theory. This scali n g formalism, w h i l e constructed from an "underlying" map, has t h e strong advantage o f being transparent t o any mention o f a map.

We proceed now i n a more

"physical" v e i n t o exposit t h i s formalism and determine i t s experimentally v e r i f i a b l e consequences. Simply put,

as a c o n t r o l parameter i s varied,

any dynamical v a r i a b l e demon-

s t r a t e s a systematic doubling o f i t s p e r i o d p r i o r t o t h e a p e r i o d i c l i m i t sign a l l i n g the onset o f chaos. However, since the p e r i o d doublings accumulate so q u i c k l y , gross features remain nearly constant, w h i l e the new temporal features c h a r a c t e r i z i n g t h e newly

306

M.J. FEIGENBAUM

doubled p e r i o d contain a small f r a c t i o n o f t h e energy.

Moreover, as period-

doubling i s p e r i o d i c i n terms o f l o g a r i t h m i c d e v i a t i o n s from the t r a n s i t i o n p o i n t , each new m o d i f i c a t i o n i s a constant f r a c t i o n o f i t s predecessor. T h i s e n t a i l s very d e f i n i t e consequences, whether viewed temporally o r through a Fourier transform, which we now explore. The basic t h e o r e t i c a l q u a n t i f i c a t i o n o f t h i s n o t i o n i s t h a t t h e d e v i a t i o n s from t h e o l d p e r i o d i c i t y t h a t determine t h e new and doubled p e r i o d s y s t e m a t i c a l l y (and u n i v e r s a l l y ) scale from one doub i n g t o t h e next.

where xn(t)

Forma ly, w r i t e

i s any coordinate when t h e parameter i s such t h a t t h e n - t h p e r i o d

doubling has occurred, f o r which t h e p e r i o d i s Tn = 2*Tn-1

.

(Actually, as t h e parameter v a r i e s Tn i s o n l y approximately constant. t h e parameter values accumulate, (22) i s a s y m p t o t i c a l l y c o r r e c t . ) shows dn t o be nonzero i f the p e r i o d o f xn i s r e a l l y Tn and

not

However,

Equation (21) Tn-l, and so dn

i s the d e v i a t i o n i n t h e t r a j e c t o r y associated w i t h t h e doubling period. t h e o r e t i c a l r e s u l t (which i s published elsewhere [4])

The

i s t h a t f o r l a r g e n,

so t h a t u has t h e symmetry

u(x +

5-)

= -u(x)

u(x + 1) = u(x)

.

Equation (23) i s a very strong p r e d i c t i o n : acquire t h e temporal data xn(t) and compute from i t the h a l f - p e r i o d d e v i a t i o n f u n c t i o n o f (21);

similarly obtain

dn+l and d i v i d e i t by dn; then f o r each n on an a p p r o p r i a t e l y scaled p l o t , t h e

same f u n c t i o n u(x) should be observed.

This i s already a s t r o n g s c a l i n g r e s u l t .

However, there i s an even stronger f e a t u r e being tested: known, a v a i l a b l e f u n c t i o n w i t h t h e approximate value

namely t h a t u(x) i s a

PERIODDOUBLING ONSET OF CHAOS

a-2 = 1/6.25

a(x) =

...

la-’ = 1/2.5029

A l l spectral

307

O < x < .25

...

.25 < x < .5

t e s t s f o r period-doubling u n i v e r s a l i t y a r e t e s t s o f (23) while, a t

present, the r e s u l t s f o l l o w i n g from t h e approximation (25) are as p r e c i s e as experiments can hope t o resolve. There are,

however,

d i f f i c u l t i e s i n employing (23)

as an experimental t e s t ,

which f o l l o w from t h e requirement o f making two sets o f measurements a t d i f f e r e n t parameter values.

The simpler problem i s the phasing o f the two time bases:

t o use (23) t h e time o r i g i n s must be t h e same. varying a delay, i n say dn(t), dent w i t h those o f dn+l(t).

This i s most e a s i l y handled by

u n t i l i t s zero crossings are most n e a r l y c o i n c i -

(There are indeed many n e a r l y c o i n c i d e n t crossings.

Also, the time scale o f dn(t) might r e q u i r e some adjustment so t h a t dn+l p r e c i s e l y t w i c e the period o f dn.)

has

More seriously, t h e parameter values must be This

chosen so t h a t the l i m i t cycles o f xn and x ~ + ~have , identical stability.

requires, f o r example, some convergence data which might be hard t o come by unless successive parameter values a t the b i f u r c a t i o n p o i n t s (when t h e p e r i o d doubles) are used.

I f t h i s d i f f i c u l t i e s are attended t o , however, (23) and (25)

then serve as a simple and very d i r e c t measurement o f t h e u n i v e r s a l i t y theory. (These same d i f f i c u l t i e s a f f l i c t the F o u r i e r measurements, except t h a t phasing i s obviously unimportant i f only amplitude spectra a r e determined.) I n f a c t , the universal f u n c t i o n u i s d i f f e r e n t f o r d i f f e r e n t s t a b i l i t i e s (e.g., a t superstable values as opposed t o b i f u r c a t i o n values o f the parameter).

In-

deed, f o r each s t a b i l i t y p , the appropriate u i s computed from t h e functions However, the differences show up o n l y i n the c o r r e c t i o n s t o t h e o f (8). gr,lJ approximation (25). Indeed there are o t h e r functions u agreeing t o t h e l e v e l o f (25) which determine d e v i a t i o n signals, d(t),

pertaining t o the y & parameter

value ( t y p i c a l l y t o be used a t t h e t r a n s i t i o n value) which then bypass t h e above d i f f i c u l t i e s .

We s h a l l explore these t e s t s a f t e r determining F o u r i e r

properties which s h a l l motivate them. Corresponding t o t h e d e v i a t i o n signal, the fundamental frequency o f xn i s 1 l / T n = 2(l/Tn-l), t h a t i s , a h a l f subharmonic o f t h e previous fundamental. Had dn(t) vanished, there would have been no components a t t h e odd m u l t i p l e s o f t h e new fundamental.

The even mu1t i p l e s represent t h e components a t t h e previous

fundamental ; since the parameter change becomes i n c r e a s i n g l y small, these components cannot s u f f e r s i g n i f i c a n t changes. Accordingly, t h e basic r e s u l t o f the doubled p e r i o d i s the s e t o f spectral l i n e s a t t h e odd m u l t i p l e s o f the new subharmonic fundamental.

These are determined n o t by

o f xn(t),

b u t o n l y by

M.J. FEIGENBAU M

388

i t s "subharmonic p a r t " dn(t).

Since these scale (by a), i t i s then c l e a r t h a t

these successive a d d i t i o n s t o t h e spectrum geometrically decrease i n a u n i v e r s a l way, as determined by the F o u r i e r a n a l y s i s o f u. vations q u a n t i t a t i v e .

L e t us now make these obser-

By d e f i n i t i o n ,

w i t h inverse,

Manipulating (26)

or

1

2(

(2p+l) =

Tn-l

J

d t dn(t) e2ni

n-1 0

n)

2p+l t 'n

w i t h inverse

dn(t) = 1 2

P (n)

(2p+l) e-2ni

t

Thus, t h e spectral components a t odd m u l t i p l e s o f t h e subharmonic fundamental are determined s o l e l y by the d e v i a t i o n signal dn(t). Since a l l the new subharmonic components a r i s e from t h e same time s i g n a l , they can be smoothly connected as an ensemble by i n t e r p o l a t i n g them.

The n a t u r a l

i n t e r p o l a t i o n t h a t r e f l e c t s t h e i r common source i s , then, t h e a n a l y t i c continua t i o n provided by (29):

389

PERIOD-DOUBLING ONSET OF CHAOS

(31) Tn-l I dt dn(t) eZniwt n-1 0

2n(w) = T 1

with

icn (-+I 2 + 1 - 2(n)(2P+l) n

Utilizing (301, this interpolation is 1

Tn-l

kn(w) = 1 2 (2p+l) T J n-1 0 P (n)

-2ni(Ep+l-uTn)t/Tn dt e

or niwT (1

an(w) =

"1

+

$ 1P

x^( ,)(ZP+l) 2p+l-uTn

(33)

*

We now ask how %n+l(w) is related to iCn(w).

gn+l(w) =

rn1 ITnO dt dn+l(t)

Be definition

Zniwt e

Employing the scaling formula for large n, we have

$,+l(u)

- 71 I dt dr)dn(t) t Tn n O

e2niwt

n

1 Tn

rn IO dt d

= t 2(,,)(2~+1)

P

= t2

P

(n)

(2p+l)

$- ;n-l

-2ni(2p+l-uTn)

~e )

-5

n

dtk(r)

n-1 0

-

n

niwtn e a(&-

n

+

-2ni(2p+l-wTn)t/T, ije

I f we now use the approximation o f (25), then

gn+l(w)

'

;(

-

niwT, e

1 t 2(,,)(2~+1) P

1 Tn-l -Zni(Zp+l-wT,)t/T, T I dt e n-1 0

M.J. FEIGENBAUM

390

by use o f the i n t e r p o l a t i o n formula (32). Equation (34) i s the basic r e s u l t o f t h i s paper regarding F o u r i e r analysis, and i s r i c h i n content [6].

The f i r s t conclusion t o be drawn i s t h a t r a t h e r than

s c a l i n g uniformly, d i f f e r e n t p a r t s o f t h e spectrum scale very d i f f e r e n t l y .

Thus

, the actual new spectral components c h a r a c t e r i z i n g the n + l - s t n+l l e v e l o f period doubling,

1.

At w =

or

That i s , the new spectral components a r e obtained by s c a l i n g t h e previous i n t e r p o l a t i o n a t these new p o s i t i o n s by a f a c t o r o f

or, dropped l o g a r i t h m i c a l l y by 10 loglO4.6

-"

6.6

(The approximation (35) i s t h e same as t h a t f o r the s c a l i n g o f successive RMS averages o f spectral l i n e s as obtained by Nauenberg [7].

2

, t h a t i s , a t those frequencies o f t h e n - t h l e v e l , one f i n d s n t h e n + l i n t e r p o l a t i o n scaled by

2.

At w =

391

PERIOD-DOUBLING ONSET OF CHAOS

That i s , i n the next generation o f doubling, the i n t e r p o l a t i o n scales by t h e anomalously small amount o f

or 10 1 0 g ~ ~ 3 .: 6 5 . 5 db

a t those frequencies t h a t had j u s t p r i o r l y come i n t o existence. 3.

2 At w = 9

, one has

T

'n

where

or 10 1 0 g ~ ~ 8 .: 3 9.2 db Thus,

.

i n the second and a l l f o l l o w i n g generations a f t e r a spectral l i n e has

appeared, the successive i n t e r p o l a t i o n s drop anomalously quickly.

(The approxi-

mation (37) i s e a s i l y seen t o be t h e same approximation o f Grossman [8] f o r t h e "leading" edge o f the spectrum). 4.

I n consequence o f 2. and 3.,

i f t h e n - t h i n t e r p o l a t i o n i s r a i s e d by x f o r

any 5.5 db < x < 9.2 db, these spectra w i l l a l l have regions o f overlap. In p a r t i c u l a r , the o r i g i n a l spectral p r e d i c t i o n [4] o f x = 8.2 db i s included i n t h i s range.

The present formula (34) i s the f u l l r e a l i z a t i o n o f the ideas o f

t h a t previous paper. 5.

Since d i f f e r e n t p a r t s o f the spectrum have d i f f e r e n t geometric scalings, a

geometric mean o f the n - t h l e v e l s p e c t r a l l i n e s i s a more s i g n i f i c a n t average than the mean o f the squares. Given (34), we can then compute t h e s c a l i n g o f the geometric mean, o r l o g a r i t h m i c a l l y , t h e average o f log-amplitudes:

M.J. FEIGENBAUM

392

Then,

However, t h e i n t e g r a l on t h e r i g h t i d e n t i c a l l y vanishes f o r A

Xn+l

-

.. Xn

l c r l > 1. Thus,

-2n(2cr)

or, the mean l o g amplitudes drop by 10 l 0 g ~ ~ 5 .=0 7.0 db

10 loglo(2a)

.

(This r e s u l t i s unchanged i f t h e averaging i s performed over

(39)

all t h e

n-th level

spectral l i n e s up t o any m u l t i p l e o f the o r i g i n a l fundamental, r a t h e r than j u s t the subharmonic p a r t . ) Equation (38) i s a new r e s u l t , and t h i s approximate value has been numerically v e r i f i e d t o be c o r r e c t t o t h e p r e c i s i o n o f (39).

A t t h i s p o i n t we want t o extend these r e s u l t s t o s p e c t r a l p r o p e r t i e s a t a f i x e d parameter value, r a t h e r than a t successive period-doubl i n g values. As prev i o u s l y demonstrated, as t h e parameter increases, t h e n - t h l e v e l l i n e s s l i g h t l y increase u n t i l they saturate a f t e r several f u r t h e r p e r i o d doublings.

This i n -

crease i s uniform f o r each l e v e l o f introduced s p e c t r a l l i n e s from which i t follows t h a t p r o p e r t i e s 1.-5.

a r e e q u a l l y v a l i d w i t h i n a spectrum a t f i x e d A f o r

a l l intermediate values o f n ( i . e . ,

a f t e r several p e r i o d doublings have occurred,

b u t n o t f o r t h e lowest l y i n g i n t e r p o l a t i o n s which have n o t y e t saturated.) Taking the fundamental p e r i o d t o be 1, UJ = 1 i s t h e o r i g i n a l fundamental, and the n - t h l e v e l o f spectral l i n e s are l o c a t e d a t t h e frequencies

w=2p+l 2n

.

(40)

393

PERIOD-DOUBLING ONSET OF CHAOS

Now, a l l the spectral p r o p e r t i e s are consequences o f (34) which i t s e l f i s a consequence o f (23).

Working backwards, i f we define, a t a f i x e d parameter value

An, d (t) as t h a t p a r t o f xn(t) constructed p u r e l y from t h e m-th l e v e l specn,m t r a l l i n e s , then (23) must again hold f o r a new s c a l i n g f u n c t i o n u which has t h e

same approximate value as (25).

That i s ,

where (41) i s v a l i d f o r a l l in such t h a t 1 << m << n and 6 has t h e approximate (t) i s specified, (41) i s now a d i r e c t time-domain t e s t o f value (25). Once d

n,m

the theory.

It i s probably the best t e s t t o perform since j u s t one time s e r i e s

i s required, and there are no phasing problems. fied i s d

n,m

A l l t h a t remains t o be speci-

(t):

where

n Xn(t)

so t h a t d,

= 1 d

Fl n,m

(t) +

< 2

2"l Z xn(t+rTo) r=O

(43)

i s j u s t t h a t p a r t o f xn which determines i t s m-th l e v e l spectrum.

For numerical experiments (41) i s very r e a d i l y v e r i f i e d ; t h e theory leading t o (41) w i l l be published elsewhere.

(41) can be i t e r a t e d so t h a t (41) and (43)

approximately determine xn(t) from t h a t p a r t o f i t w i t h p e r i o d 2T0.

This i s a

general s t r u c t u r e a r i s i n g from an underlying Cantor set, so t h a t t h i s type o f data processing might be more generally applicable t o dynamical systems w i t h a strange a t t r a c t o r present. The analogue t o (34) i s

so t h a t the log-amplitude spectrum i s b u i l t o u t o f sums o f determined p e r i o d i c

functions o f successively doubled periods.

M.J. FNGENBAUM

394

REFERENCES

[l] D. Ruelle, Comm. Math. Phys. [2] M. Misiurewicz, Studia Math.

55 67

[3] M. J. Feigenbaum, J . Stat. Phys.

[4] M. J. Feigenbaum, Phys. L e t t . (1980).

(1977). (1980).

11, 669 (1979). fi,375 (1979),

[5] B. A. Huberman, J . Rudnick, Phys. Rev. L e t t .

45,

Comm. Math. Phys.

11, 65

154 (1980).

[6] This formula f i r s t appeared as the s c a l i n g f o r the broadband p a r t o f a noisy spectrum i n A. Wolf, J . S w i f t . U n i v e r s i t y o f Texas a t Austin prep r i n t "Universal Power Spectra f o r t h e Reverse B i f u r c a t i o n Sequence" (1981).

[7] M. Nauenberg, J. Rudnick, " U n i v e r s a l i t y and the power spectrum a t the ons e t o f chaos." P r e p r i n t UCSC (1981). [8] S. Grossman, S. Thomae. U n i v e r s i t y o f Marburg, p r e p r i n t (1981). [9] P. C o l l e t and J.-P. Systems.

Eckmann, I t e r a t e d Maps on the I n t e r v a l as Dynamical (Birkhaeuser, Boston 1980).

Nonlinear Problems: Present and Future Campbell,B. Nicolaenko (eds.) 0 North-HollandPublishing Company,1982 A.R. Bishop,D.K.

395

TRICRITICAL POINTS AND BIFURCATIONS IN A QUARTIC MAP Shau-Jin Chang, Michael Wortis, Jon A. Wright Department of Physics University of Illinois at Urbana-Champaign 1110 West Green Street Urbana, IL 61801

We have studied the 1-dimensional iteration map sociated with Y This map the even quartic polynomial xn+l = 1 + a xf + b xn. allows a smooth transition from a single hump to a double hump. Bifurcations and higher-order transitions occur as we vary the parameters a, b. In addition to the usual universal bifurcation behavior discovered by Feigenbaum, we find a new universality class of bifurcations which is associated with a tricritical point. Tricritical points serve as natural boundaries to Feigenbaum critical lines. For the quartic map, the tricritical points which are the end-points of the original Feigenbaum line are (a,b) (0, -1.59490) and (-2.81403, 1.40701). Associated with each tricritical point, there are two unstable direct o s tlY = as well as t o independent exponents. The exponents are 6T 7.2851 and 6b) = 2.8571. At t 4 tricritical point, we can introduce a universal function fT(x) which obeys

-

-

with the scale factor a = -1.69031, The quartic map has a , T special duality transformation (a,b) (a',b'), such that the two mappings are intrinsically related. The tricritical points which are dual to the above pair of tricritical points are located at (-3.18980, 2.54371) and at (.95561, -1.14981) and are joined by a line which is the dual of the original Feigenbaum line. There are an infinite number of different tricritical points which form at least a Cantor set.

I. INTRODUCTION Recently, Feigenbaum has shown that the bifurcation sequence in a single hum iterative map xn+ = f( ) with a quadratic peak obeys a universal behavior.'s2 One can understand thisxhavior from a renormalization-group point of view. At the limiting point of a bifurcation sequence, the 2N itgraction of the map In the neighwith appropriate scaling approaches a universal function f (x). borhood of this universal function, it appears that there fs only one relevant direction, along which the eigenvalue 6 ( = 4.6692) is larger than one. We have recently studied a more general map described by a quartic polynomial x +1 = f(xn) = 1 + a x% + b xi. An important property of the quartic map is tgat f(x) can describe both a single-hump and a double-hump map. By changing the parameters a,b continuously, one can induce a smooth transition from a one-hump map to a two-hump map and vice versa. It is easy to see that a double-hump map admits many different kinds of stable cycles which do not exist in the iterations of a single-hump map. An important concept in a double-hump map i s that of doubly-stable cycles. These are cycles whose members pass through both the central peak and a side peak. In the vicinity of a doubly-stable cycle in the a,b parameter space, both peaks control the

S.J. CHANG, M. WORTIS, J.A. WRIGHT

396

s t a b i l i t y of t h e same cycle. Away from doubly-stable c y c l e s , however, t h e Thus, t h e peaks c o n t r o l d i f f e r e n t cycles which are u n r e l a t e d i n x-space. r e g i o n s c o n t r o l l e d by these d i f f e r e n t peaks become dynamically coupled at a doubly-stable cycle. This kind of dynamical coupling is a new phenomenon which cannot occur i n a single-hump map. I t s presence modifies t h e b i f u r c a t i o n processes. The l i m i t i n g p o i n t determined by a b i f u r c a t i o n sequence of 2N doubly-stable c y c l e s is a s s o c i a t e d w i t h a f i x e d point i n f u n c t i o n space with two r e l e v a n t d i r e c t i o n s i n analogy w i t h Feigenbaum's d i s c u s s i o n of t h e quad r a t i c case. This l i m i t i n g p o i n t a l s o serves as the end-point of a c r i t i c a l l i n e which d e s c r i b e s the usual Feigenbaum behavior. This typ of f i x e d p o i n t is known as a t r i c r i t i c a l point i n p h a s e - t r a n s i t i o n nguage. Tp$, t r i c r i t i c a l point discovered h e r e has two u n i v e r s a l exponents 6$"-7.2851, 6T =2.8571. A t t h e f i x e d p o i n t , t h e map is s e l f - s i m i l a r near the peaks a f t e r 2N i t e r a t i o n s . J u s t a s in*the Feigenbaum's c a s e , it is p o s s i b l e t o i n t r o d u c e a u n i v e r s a l f u n c t i o n f T ( x ) at a t r i c r i t i c a l point which obeys t h e same f u n c t i o n a l e q u a t i o n

'

a,,+i(f;(x/aT))

= ft(x)

Our expressed as b u t with a d i f f e r e n t scale f a c t o r 5 = 1.69031. a power a e r i e s in x W e have discovered an i d e n t i t y , r e l a t i n g one of t h e exponents t o the s c a l e f a c t o r . This i d e n t i f y is a member of a whole c l a s s of i d e n t i t i e s which e x i s t a t the f i x e d p o i n t of an i t e r a t i v e map.

.

11.

QUARTIC MAPS

(A)

S i n g l e vs multiple-hump maps

We s t u d y the one-dimensional i t e r a t i v e maps generated by an even q u a r t i c polynomial

x,,+~

-

f(xn) z 1

+a

x 2n

+b

4 xn

.

(2.1)

b

a

Ngure 1 I t e r a t i v e maps generated by t h e even q u a r t i c polynomial (2.1). The map i n each quadrant corresponds t o a f ( x ) with ( a , b ) i n t h a t quadrant.

TRICRITICAL POINTS AND BIFURCATIONS IN A OUARTIC MAP

Depending on the values of a and b, f ( x ) can have e i t h e r a s i n g l e hump or a double hump. See Fig. 1. For region defined by a > 0, b > 0, f ( x ) can a t most give rise t o a s t a b l e f i x e d point. For a < 0, b < 0, t h e f u n c t i o n f ( x ) has a s i n g l e q u a d r a t i c hump, and Feigenbaum u n i v e r s a l i t y holds. I n one parameter single-peaked maps, the limit point of i n f i n i t e l y b i f u r c a t e d ZN c y c l e s is a s i n g l e p o i n t . In two parameters single-peaked maps that point becomes a l i n e t h a t we r e f e r t o a s t h e Feigenbaum l i n e . For t h e regions a < 0, b > 0 and a > 0 , b < 0, f ( x ) has e i t h e r two peaks and one v a l l e y or v i c e versa. I n t h e following, we s h a l l refer t o the peak or v a l l e y a t x 0 as t h e c e n t r a l , peak, and t h e peaks or v a l l e y s at x # 0 a s t h e s i d e peaks. The e x i s t e n c e of both c e n t r a l and s i d e peaks implies t h a t f ( x ) may develop independent iterat i o n regions a s i n d i c a t e d i n regions I and I1 in Ng. 2.

-

Ngure 2 A map f o r a < 0, b > 0. Note t h a t regions I and I1 are independent i t e r a t i o n regions. I n region I, f ( x ) d e s c r i b e s a single-hump map with a q u a d r a t i c peak at x = 0. We s h a l l r e f e r t o region I a8 the region c o n t r o l l e d by the c e n t r a l peak. I n region 11, f ( x ) d e s c r i b e s an i n v e r t e d single-hump map with t h e peak a t x = m b . W e s h a l l r e f e r t o I1 as the r e g i o n c o n t r o l l e d by the s i d e peak. Since t h e two s i d e peaks have equal h e i g h t , they map i n t o t h e same x (=1Thus, we have a t a2/4b). Hence, they cannot both be members of an N-cycle. most one most s t a b l e region c o n t r o l l e d by the s i d e peaks. The e x i s t e n c e of two independent regions may a l s o appear i n the case a > 0, b < 0.

(B)

Duality transformation

For the study of cycles and t h e i r sequences, we only need t o consider t h e region c o n t r o l l e d by the c e n t r a l peak. The c y c l e s a s s o c i a t e d with t h e s i d e

397

398

S.-J. CHANG, M. WORTIS, J.A. WRIGHT

peaks can be obtained from those with t h e c e n t r a l peak by a d u a l i t y t r a n s f o r mation d e s c r i b e d below. Consider the q u a r t i c map (2.1).

W e can rewrite i t a s

~ ~ + ~ = l + a * + ,b 4b f O 2

= 1

-%+

b(%+

2 2 xn)

.

(2.2) Thus, we can f a c t o r the q u a r t i c map as the product of two q u a d r a t i c maps:

a2 + b 5,2 xn+1 a 1 W e can now v i s u a l i z e t h e o r i g i n a l mapping as

-

x1

+

c1

+

x2 +

c2

+ x3 +

,

* a * .

(2.5)

It is easy t o see t h a t i f the x-mapping has a s t a b l e N-cycle, so does t h e 6-mapping and v i c e versa. The mapping from to is simply

&+

-&+

.

En+1 = (1 b 58)2 After a r e s c a l i n g , we can put t h e S-mapping (2.6) = 1

+ a ' ST, + b'

(2.6) i n t o the s t a n d a r d form

F.4,

(2.7)

I t is easy t o v e r i f y t h a t the s i d e peaks of x a r e mapped i n t o t h e c e n t r a l peak of E , and t h e s i d e peaks of 5 a r e mapped i n t o the c e n t r a l peak of x. The d u a l i t y r e l a t i o n is r e c i p r o c a l : The d u a l i t y t r a n s f o r m a t i o n of ( a ' , b ' ) is t h e o r ig in a l (a,b). (C)

Most-stable c y c l e s and doubly-stable

cycles

For a q u a r t i c map, the most-stable N-cycles ( x l , x 2 . . . , ~ N )

a r e described by (2.10)

d N dx [f (x'],

- .

(2.11)

0

i

An N-cycle

is wst s t a b l e , i f e i t h e r t h e c e n t r a l peak (x = 0) o r one of t h e s i d e peaks ( x = * m b ) is a member of t h e c y c l e xi The polynomials a s s o c i a t e d with a s t a b l e N-cycle give rise t o a l i n e i n t h e a-b plane. In Fig. 3, the s o l i d l i n e s r e p r e s e n t th6 l o c i of most-stable 2-cycles. Fbnctions whose parameters s a t i s f y a + b + 1 0 have most-stable 2-cycles a s s o c i a t e d with the c e n t r a l peak. The curved l i n e s are the l o c i of p o i n t s of most-stable 2-cycle a s s o c i a t e d with the s i d e peaks. It is easy t o see t h a t , with the exception of t h e o r i g i n , the most s t a b l e N-cycle t r a j e c t o r i e s a s s o c i a t e d w i t h t h e c e n t r a l peak never i n t e r s e c t among themselves. Neither do t h e mst s t a b l e t r a j e c t o r i e s a s s o c i a t e d with t h e s i d e peaks. However, t h e t r a j e c t o r i e s assoc i a t e d w i t h t h e c e n t r a l peak do i n t e r s e c t those a s s o c i a t e d w i t h t h e s i d e peaks. Actually, they i n t e r s e c t i n two completely d i f f e r e n t ways. At i n t e r -

-

.

TRICRITICAL POINTS AND BIFURCATIONS IN A QUARTIC MAP

s e c t i o n s A and A' i n Mg. 3, the most s t a b l e 2-cycles pass through both t h e e c a l l t h i s kind of c y c l e a doublyc e n t r a l peak and one of the s i d e peaks. W s t a b l e cycle. Two t r a j e c t o r i e s which i n t e r s e c t a t a doubly-stable c y c l e a r e "dynamically c o u p l e d at t h e i n t e r s e c t i o n . A t i n t e r s e c t i o n s B and B ' , the 2c y c l e s a r e c o n t r o l l e d by d i f f e r e n t peaks which are not r e l a t e d : some regions of x a r e a t t r a c t e d t o one cycle and some t o the o t h e r . W e r e f e r t o t h i s kind of i n t e r s e c t i o n as "dynamically independent". Note t h a t doubly s t a b l e p o i n t s A and A' a r e dual t o each other. The concept of dynamical coupling is invari a n t under d u a l i t y transformation.

Figure 3 The s o l i d l i n e s r e p r e s e n t t h e most-stable 2 c y c l e s , t h e c r o s s e s denote t r i c r i t i c a l p o i n t s , and t h e dashed l i n e s d e s c r i b e t h e Feigenbaum l i n e s . 111. (A)

TRICRITICAL POINTS B i f u r c a t i o n along b-axis

I f we vary b ( n e g a t i v e ) , k e e p i n g 2 small and n e g a t i v e , we encounter t h e u s u a l b i f u r c a t i o n sequence, s t u d i e d by Feigenbaum. I f we vary b ( n e g a t i v e ) , keeping a small and p o s i t i v e , we f i n d t h a t the map goes through t h r e e d i f f e r e n t most These 2-cycles are cons t a b l e 2-cycles before b i f u r c a t i n g i n t o a 4-cycle. t r o l l e d by t h e t h r e e d i f f e r e n t peaks as shown i n f i g . 1. These phenomena The same phenomena r e p e a t themp e r s i s t no matter how small t h e value of s e l v e s at higher qN cycles. Thus, t h e r e are q u a l i t a t i v e d i f f e r e n c e s between b i f u r c a t i o n s on the tw d i f f e r e n t s i d e s of t h e b-axis. When a = 0 and b < 0 , t h e curve f ( x ) is single-humped with a q u a r t i c maximum. We f i n d a sequence of b i f u r c a t i o n s which always occur a t the doubly-stable 2N c y c l e s :

3Q9

400

S. J. CHANG, M. WORTIS, J.A. WRIGHT

bN

2N 2

-1

4

-1.503 93

8

-1.58225

16

-1.59316

32

-1.59466

...

...

The pN cycles have a l i m i t i n g point a t b, = -1.59490, i n c l o s e analogy t o t h e q u a d r a t i c map studied by Feigenbaum. However, because our map has a q u a r t i c ( r a t h e r than q u a d r a t i c ) maxim m, it is t o be expected t h a t t h e r e w i l l be d i f f e r e n t critical exponents,' as we now discuss. This l i m i t i n g point has s e v e r a l i n t e r e s t i n g p r o p e r t i e s which we s h a l l d i s c u s s below. (B)

C r i t i c a l exponents and the u n i v e r s a l f u n c t i o n

There a r e two relevant eigen-directions at the critical point (0 , -1.59490). Along a relevant eigen-direction and f o r l a r g e N, we have

where ST i s the c r i t i l a l eigenvalue g t h i s eigen-direction. One eigend i r e c t i o n is along the i s where 6T = 7.2851. The o t h e r eigen-direction is ( 1 , -1.2347) -where ' ; 6 2.8571. Note t h a t the kigenbaum f i x e d point has only one relevant d i r e c t i o n with 6 F = 4.6692. In phase-transition language, a c r i t i c a l point with two relevant d i r e c t i o n s which a l s o serves as the end-point of a c r i t i c a l l i n e is known as a " t r i c r i t i c a l " point. As we s h a l l see, our fixed point is indeed a t r i c r i t i c a l point. See Discussion Section below.

P

atPP

A t the c r i t i c a l point ( 0 , -1.59490),

t h e function is s e l f - s i m i l a r near x = 0 t o within a proper rescaling. The s c a l e f a c t o r here is aT = -1.6903. J u s t as 1%the Feigenbaum case, the t r i c r i t i c a l point has i t s own u n i v e r s a l f u n c t i o n f T ( x ) obeying the same f u n c t i o n a l equation pf:(2)(x/p)

= f%(x),

p

In the neighborhood of x = 0, f;(x) fT(x) * 1

-

.

-1.6903

can be represented approximately by

+ 0.01301 x8 x12 - 0.062035 x16 +

1.83413 x4

+ 0.31188 (C)

-

-***

.

(3.3)

An i d e n t i t y

F are

The second exponent 6 i 2 ) and t h e s c a l e f a c t o r

.

6$2) = a; This r e l a t i o n is exact2)as can be shown as follows: If vector belonging t o 6,

.

f ( x ) = f;(x> + Eh(x),

0

<

E

<<

1

,

then i t e r a t i o n under

-

f(2)(xf(l)) g(x) E f(l) leads by d e f i n i t i o n t o Since h(0) = 0 (because f ( 0 ) = l ) , we may take

r e l a t e d by (3.4) Let h(x) be the eigen-

TRICRITICAL POINTS AND BIFURCATIONS IN A QUARTIC MAP

h(x)

X2

+ C4X4 + C6X6 +

D i r e c t i t e r a t i o n of (3.6) gives

-

*.**

(5

*

l/fT(l))

<' &k2)h(x) = -h( l ) f f ( x ) + h(f*,(L)) +

*I

[h(l) x f

(")

T P

40 1

gpx 5

+ h(-)]

*I

fT

* (fT(k)) ,

(3.9)

which is a f u n c t i o n a l e q u a t i o n f o r h(x). I t - i s easy t o v e r i f y t h a t t h e r i g h t hand s i d e vanishes a t x = 0. Only t h e l a s t term c o n t r i b u t e s t o O(x2) and by equating both s i d e s of (3.9) one f i n d s 612) = f ; l ( l ) / a T

-

*I

.

(3.10)

One e v a l u a t e s f T ( 1 ) by d i f f e r e n t i a t i n g (3.2) w i t h r e s p e c t t o x and then setting x 0. The r e s u l t is *I 3 (3.11) fT (1) aT , w ich giv (3.4). The o t h e r eigenvalues belonging t o d i r e c t i o n s out of t h e xC, xe xfs.** subspace are i r r e l e v a n t . Eq. (3.4) is an example of a g e n e r a l c l a s s of i d e n t i t i e s which we d i s c u s s elsewhere.

-

,

(D)

The t r i c r i t i c a l p o i n t and doubly-stable c y c l e s

Our present t r i c r i t i c a l p o i n t ( 0 , -1.59490) is t h e b i f u r c a t i o n l i m i t of a sequence of doubly-stable 2N c y c l e s . This t u r n s out t o be a g e n e r a l p r o p e r t y of a b i f u r c a t i o n t r i c r i t i c a l point. W e have used t h i s property t o determine many o t h e r t r i c r i t i c a l p o i n t s which do not l i e on t h e b-axis. IV.

DISCUSSION

I n t h e a < 0 , b < 0 r e g i o n , we always have a single-hump f u n c t i o n f ( x ) . I f we i n c r e a s e t h e parameters -a, -b, we encounter an i n f i n i t e sequence of b i f u r c a t i o n s , as d e s c r i b e d by Ligenbaum. The l i m i t i n g p o i n t s of t h e s e i n f i n i t e b i f u r c a t i o n s now l i e on a l i n e i n t h e a-b plane which we r e f e r t o as a Feigenbaum l i n e . The t r i c r i t i c a l point ( a , b ) ( 0 , -1.59490) s e r v e s as a n a t u r a l boundary t o the o r i g i n a l Feigenbaum l i n e . When we extend t h e o r i g i n a l Feigenbaum l i n e i n t h e o t h e r d i r e c t i o n , we f i n d t h a t i t t e r m i n a t e s at a n o t h e r tric r i t i c a l p o i n t , l o c a t e d a t (-2.81402, 1.40701). One can check e a s i l y t h a t t h i s t r i c r i t i c a l p o i n t is a l s o t h e l i m i t of a sequen e of dou 1 - s t a b l e 2N c y c l e s and t h a t i t has t h e same c r i t i c a l exponents 6T1) and 6T 42v

-

c

.

Under t h e d u a l i t y t r a n s f o r m a t i o n , a tricritical p o i n t maps i n t o a t r i c r i t i c a l p o i n t and a Feigenbaum c r i t i c a l l i n e maps i n t o a Feigenbaum l i n e . The d u a l images of t h e previous t r i c r i t i c a l p o i n t s are l o c a t e d a t (-3.18980, 2.54371) and a t (0.95561, -1.14981). We have p l o t t e d t h e d u a l transformed Feigenbaum l i n e in Fig. 3. Since b i f u r c a t i o n occurs after each s t a b l e c y c l e , and s i n c e t h e r e a r e an i n f i n i t e number of d i f f e r e n t s t a b l e c y c l e s , t h e r e a r e an i n f i n i t e number of Feigenbaum l i n e s in t h e a-b plane. Hence, t h e r e aust also be an i n f i n i t e number of t r i c r i t i c a l p o i n t s i n t h e a-b plane. Indeed, we a r e a b l e t o show t h a t between t h e c r i t i c a l p o i n t s (0, -1.59490) and (.95561, -1.14981), t h e r e a r e an i n f i n i t e number of t r i c r i t i c a l p o i n t s , forming a Cantor set. W e shall d i s c u s s t h e geometrical meaning of t h e Cantor set elsewhere. ACKNOWLEDGMENTS This work was supported i n p a r t by t h e National Science Ebundation under Grant Nos. NSF PHY79-00272, NSF DMR78-21069, and by t h e O f f i c e of Naval Research under Contract No. NOOO14-80-C-0840.

SJ. CHANG, M. WORTIS, J.A. WRIGHT

402

REFERENCES

2, 25

21,

1.

Felgenbaum, M. J., J. S t a t . Phys. L e t t . *, 375 (1979).

2.

For reviews on I t e r a t i v e maps, see e.g. May, R. H., Nature 261, 459 (1976); C o l l e t , P. and Eckmann, J. P., I t e r a t e d Maps on t h e I n t e r v a l a s Dynamlcal Systems (Blrkhauser Co., Boston 1980).

3.

The term " t r l c r l t l c a l point'' was o r i g i n a l l y introduced by R. B. G r l f f i t h s : G r l f f l t h s , R. B., Phys. Rev. Lett. 24, 715 (1970). See a l s o , R l e d e l , E. K., Phys. Rev. Lett. 8, 675 (1972); G r l f f l t h s , R. B., Phys. Rev. g,545

(1978);

669 (1979), Phys.

(1 973).

4.

Derrida, B.,

Gervols, A. and Pomeau, Y.,

J. Phys.

g, 269

(1979).

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 8. Nicolaenko (eds.1 @ North-Holland Publishing Company, 1982

403

RECENT EXPERIMENTS ON CONVECTIVE TURBULENCE J e r r y P. Gollub P h y s i c s Department, H a v e r f o r d C o l l e g e , H a v e r f o r d , PA 19041 and P h y s i c s Department, U n i v e r s i t y o f P e n n s y l v a n i a , P h i l a d e l p h i a , PA 19174

We have used l a s e r D o p p l e r methods t o s t u d y t h e t r a n s i t i o n t o t u r b u l e n t c o n v e c t i o n i n f l u i d s o f m o d e r a t e P r a n d t l numb e r . T h i s work i s b r i e f l y summarized, w i t h emphasis on t h e m a j o r d i f f e r e n c e s between l a r g e and s m a l l ( r e l a t i v e t o t h e d e p t h ) f l u i d l a y e r s . I n s m a l l l a y e r s , we o b s e r v e phenomena t h a t a r e c h a r a c t e r i s t i c o f n o n l i n e a r models w i t h a f e w d e g r e e s o f freedom, e s p e c i a l l y q u a s i p e r i o d i c o s c i l l a t i o n s , phase l o c k i n g , and subharmonic g e n e r a t i o n . I n l a r g e l a y e r s , t h e o n s e t o f c h a o t i c m o t i o n i s due t o a s t r u c t u r a l i n s t a b i l i t y o f c o n v e c t i o n r o l l s , a n d l o w d i m e n s i o n a l models a r e n o t l i k e l y t o be u s e f u l .

INTRODUCTION The p r o b l e m o f u n d e r s t a n d i n g t h e o n s e t o f t u r b u l e n c e i n c l a s s i c a l f l u i d s i s a n o l d p r o b l e m i n n o n l i n e a r p h y s i c s , b u t o n e t h a t has a t t a i n e d new l i f e d u e t o r e c e n t advances i n b o t h m a t h e m a t i c s and e x p e r i m e n t . On t h e one hand, t h e h y p o t h e s i s ] t h a t random b e h a v i o r c a n a r i s e f r o m a s t r a n g e a t t r a c t o r i n a l o w dimens i o n a l d y n a m i c a l system has h e l d o u t a hope o f new p r o g r e s s . Dn t h e o t h e r hand, r e c e n t e x p e r i m e n t a l advances, such as l a s e r D o p p l e r v e l o c i m e t r y 2 a n d computer a u t o m a t i o n o f e x p e r i m e n t s , have made a q u a l i t a t i v e d i f f e r e n c e i n t h e s t u d y o f hydrodynamic i n s t a b i l i t i e s and t h e t r a n s i t i o n t o t u r b u l e n c e . The R a y l e i g h - B e n a r d i n s t a b i l i t y has f o r many y e a r s p l a y e d t h e r o l e o f t h e h y d r o g e n atom of n o n l i n e a r hydrodynamics, because o f i t s h i g h d e g r e e o f s i m p l i c i t y . Y e t even a system as s i m p l e as a f l u i d l a y e r w i t h a n imposed t e m p e r a t u r e g r a d i e n t shows a n amazing v a r i e t y o f i n t e r e s t i n g phenomena. D u r i n g t h e p a s t 5 y e a r s we have u n d e r t a k e n a d e t a i l e d s t u d y o f t h e p r o c e s s e s l e a d i n g t o t u r b u l e n t c o n v e c t i o n , w i t h p a r t i c u l a r emphasis on e v a l u a t i n g t h e r e l e v a n c e o f t h e s t r a n g e a t t r a c t o r c o n c e p t . S i n c e t h i s work has a p p e a r e d ( o r w i l l s o o n a p p e a r ) we w i l l summarize t h e m a i n r e s u l t s and r e f e r t h e r e a d e r t o t h e l i t e r a t u r e f o r d e t a i l s . We do n o t a t t e m p t t o r e v i e w t h e e x p e r i m e n t s o f o t h e r g r o u p s h e r e . Our e x p e r i m e n t s used w a t e r as t h e w o r k i n g f l u i d b e c a u s e i t s s t r o n g l y t e m p e r a t u r e dependent v i s c o s i t y p e r m i t t e d t h e P r a n d t l number t o b e v a r i e d o v e r a r e l a t i v e l y w i d e r a n g e s i m p l y b y v a r y i n g t h e mean w o r k i n g t e m p e r a t u r e . ( T h e P r a n d t l number i s t h e r a t i o of t h e kinematic v i s c o s i t y t o t h e thermal d i f f u s i v i t y . ) The r e c t a n g u l a r f l u i d c e l l s w e r e d e s i g n e d w i t h a n emphasis o n t e m p e r a t u r e s t a b i l i t y a n d h o r i z o n t a l u n i f o r m i t y , because e x t e r n a l p e r t u r b a t i o n s c a n o b s c u r e t h e o n s e t o f i n t r i n s i c dynamical n o i s e i n t h e f l u i d , and n o n u n i f o r m i t i e s can b l u r sharp transitions. The f l u i d v e l o c i t y a t a p o i n t i n t h e f l u i d was d e t e r m i n e d i n r e a l t ' m e f r o m t h e D o p p l e r s h i f t of s c a t t e r e d l a s e r l i g h t . V e l o c i t i e s as s m a l l as 10- cm/s c o u l d b e d e t e r m i n e d w i t h t h i s t e c h n i q u e . The f l u i d s y s t e m was mounted o n computer-

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c o n t r o l l e d s t e p p i n g motor d r i v e n t r a n s l a t i o n s t a g e s so t h a t t h e v e l o c i t y f i e l d c o u l d be mapped o u t i n a h o r i z o n t a l p l a n e , o r a l t e r n a t e l y s t u d i e d as a f u n c t i o n o f t i m e a t a f i x e d l o c a t i o n . A t o t a l o f a p p r o x i m a t e l y l o $ measurements were made i n the course o f t h i s study. The q u a l i t a t i v e n a t u r e o f t h e t r a n s i t i o n t o t u r b u l e n c e i s a s t r o n g f u n c t i o n of t h e a s p e c t r a t i o r , d e f i n e d as t h e r a t i o o f t h e l a r g e s t h o r i z o n t a l dimension t o t h e l a y e r d e p t h . The c e n t r a l r e s u l t i s t h a t s m a l l a s p e c t r a t i o f l u i d l a y e r s have many f e a t u r e s i n common w i t h low dimensional n o n l i n e a r models, w h i l e l a r g e a s p e c t r a t i o l a y e r s behave q u i t e d i f f e r e n t l y . SMALL ASPECT RATIO Small a s p e c t r a t i o l a y e r s ( r <- 5) t y p i c a l l y have a w i d e r e g i m e o f time-independent c o n v e c t i o n when t h e c r i t i c a l R a y l e i g h number Rc ( p r o p o r t i o n a l t o t h e tem e r a t u r e d i f f e r e n c e a c r o s s t h e l a y e r ) i s exceeded, Our Doppler mapping t e c h n i q u e $ f a c i l i t a t e d t h e d i s c o v e r y t h a t t h e r e a r e a number o f s t a b l e c o n v e c t i o n p a t t e r n s , depending on t h e p a s t h i s t o r y o r i n i t i a l c o n d i t i o n s o f t h e system. For example, c e r t a i n c e l l s c a n c o n t a i n e i t h e r two o r t h r e e c o n v e c t i v e r o l l s . Each p a t t e r n has a d i s t i n c t and somewhat d i f f e r e n t sequence o f i n s t a b i l i t i e s l e a d i n g t o t u r b u l e n c e as R i s v a r i e d , b u t each sequence i s r e p r o d u c i b l e as l o n g as t h e b a s i c c o n v e c t i o n p a t t e r n remains unchanged. By v a r y i n g t h e a s p e c t r a t i o , P r a n d t l number, and c o n v e c t i o n p a t t e r n , we found s e v e r a l qua1 i t a t i v e l y d i s t i n c t r o u t e s t o t u r b u l e n c e i n small r a t i o c o n v e c t i o n : ( a ) Q u a s i p e r i o d i c i t y and p h a ~ e - l o c k i n q : ~ ’ The ~ f l o w f i r s t becomes time-dependent by e n t e r i n g a s t r i c t l y p e r i o d i c regime a t R1. A t a h i g h e r R a y l e i g h number R , a second frequency begins t o grow smoothly as R i s i n c r e a s e d . These two o s c i l i a t i o n s i n t e r a c t w i t h each o t h e r , c a u s i n g t h e spectrum t o c o n s i s t o f s h a r p peaks a t t h e fundamental f r e q u e n c i e s f l and f 2 , and a l l o f t h e i r l o w o r d e r sums and d i f f e r e n c e s o f t h e form f = m l f l + mpf2. The presence o f t h e s e m i x i n g components i n t h e spectrum i n d i c a t e s t h a t t h e time-dependent processes a r e s t r o n g l y n o n l i n e a r . The r a t i o f 2 / f l decreases smoothly w i t h i n c r e a s i n g R, i n d i c a t i n g t h a t t h e two f r e quencies a r e g e n e r a l l y incommensurate, and hence t h a t t h e spectrum corresponds t o a q u a s i p e r i o d i c motion, which can be t h o u g h t o f as a t o r u s i n phase space. U s u a l l y , t h e two incommensurate o s c i l l a t i o n s phase-lock w i t h each o t h e r , s o t h a t t h e r a t i o f 2 / f l i s t h e r a t i o o f two s m a l l i n t e g e r s ( t y p i c a l l y 7 / 3 o r 9 / 4 ) . The spectrum t h e n c o n s i s t s o f an a r r a y o f s p i k e s a t a l l m u l t i p l e s o f t h e l o w e s t commensurate f r e q u e n c y o f f l and f 2 . The phase l o c k i n g p e r s i s t s o n l y o v e r a r e l a t i v e l y narrow r a n g e i n R. When t h e r a t i o f Z / f l began t o change again, t h e s p e c t r a l l i n e s became broadband, i n d i c a t i n g t h e o n s e t o f n o n p e r i o d i c i t y a t R t . A t h i g h R t h e l i n e s t r u c t u r e o f t h e spectrum i s no l o n g e r p r e s e n t , and i t s shape i s s i m i l a r t o t h a t found by A h l e r s and B e h r i n g e r g a t l a r g e r a s p e c t r a t i o , w i t h a Nearer t o t h e onset, t h e l i n e w i d t h o f t h e power l a w f a l l o f f as f-4.320.5. Phase l o c k i n g has a l s o been obs p e c t r a l peaks i s a l i n e a r f u n c t i o n o f (R-Rt). s e r v e d by L i b c h a b e r and Maurer.lO

bifurcation^:^'^

N o n p e r i o d i c i t y can be ( b ) Subharmonic ( p e r i o d d o u b l i n g ) o b t a i n e d a f t e r s e v e r a l s u c c e s s i v e subharmonic b i f u r c a t i o n s f o r c e r t a i n combin a t i o n s o f a s p e c t r a t i o and P r a n d t l number. T h i s b e h a v i o r i s a t l e a s t q u a l i t a t i v e l y s i m i l a r t o t h a t shown by n o n l i n e a r mappings o f t h e u n i t i n t e r v a l o f t h e form xn+! = f ( x n ) . The i n i t i a l t i m e dependence i s i n t h e f o r m of a p e r i o d i c o s c i l l a t i o n a t frequency f . However, a t a c r i t i c a l R a y l e i g h number R1 peaks appear a t +f and odd harmonics, and a t R 2 peaks a t %fand odd harmonics a l s o appear i n t h e spectrum. We have n o t r e s o l v e d h i g h e r o r d e r subharmonics, b u t c o u l d have missed them by i n c r e m e n t i n g R by t o o l a r g e a s t e p . The s p e c t r a l peaks become broadened and a broad background begins t o grow a t a w e l l d e f i n e d v a l u e This obserThe s t r e n g t h o f t h i s background n o i s e grows as (R-R,)1.43+0.2. .R, v a t i o n i s c o n s i s t e n t w i t h a r e c e n t p r e d i c t i o n by Huberman and Z i s o o k l l t h a t t h e However, i t i s p o s s i b l e n o i s e would grow as a power law w i t h exponent 1.5247

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RECENT EXPERIMENTS ON CONVECTIVE TURBULENCE

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t h a t t h e agreement i s f o r t u i t o u s s i n c e we have n o t observed an e x t e n s i v e sequence o f b i f u r c a t i o n s . Subharmonics have been s t u d i e d more e x t e n s i v e l y b y L i b c h a b e r and M a u r e r l o and r e c e n t l y by G i g l i o , Musazzi, and P e r i n i . I 2 ( c ) Three frequencies:Z q u a s i p e r i o d i c s t a t e s c h a r a c t e n z e d by t h r e e d i s t i n c t f r e q u e n c i e s were found i n s e v e r a l cases. I t i s n o n - t r i v i a l t o a c t u a l l y demons t r a t e t h a t t h i s i s t h e case. Our s p e c t r a t y p i c a l l y had over 20 s t a t i s t i c a l l y s i g n i f i c a n t peaks, b u t t h e y c o u l d a l l be f i t t e d t o l i n e a r combinations of t h r e e b a s i c f r e q u e n c i e s t o w i t h i n about one p a r t i n 104. Comparable f i t s c o u l d n o t be o b t a i n e d u s i n g two independent f r e q u e n c i e s , These f a c t s , combined w i t h t h e o b s e r v a t i o n t h a t t h e r a t i o s o f t h e t h r e e f r e q u e n c i e s v a r y smoothly w i t h R, p r o v i d e s t r o n g evidence t h a t t h e y a r e incommensurate w i t h each o t h e r . Thus, t h i s s t a t e can be r e p r e s e n t e d by a 3 - t o r u s i n phase space. N o n p e r i o d i c i t y e v o l v e s from t h i s s t a t e a t h i g h e r R. Summarizing t h e behavior observed a t s m a l l a s p e c t r a t i o , we f i n d a g r e a t v a r i e t y o f phenomena depending on t h e parameters o f t h e system and t h e i n i t i a l c o n d i t i o n s . However, t h e prevalence o f r e l a t i v e l y s i m p l e phenomena ( p e r i o d d o u b l i n g b i f u r c a t i o n s , q u a s i p e r i o d i c m o t i o n , and p h a s e - l o c k i n g ) i s r e m i n i s c e n t o f t h e b e h a v i o r shown by sim l e n o n l i n e a r models. For example, systems o f coupled e l e c t r o n i c o s c i l l a t o r s l ! a r e known t o e x h i b i t phase l o c k i n g . The f o l l o w i n g p i c t u r e may a s s i s t t h e r e a d e r i n o r g a n i z i n g t h e s e v a r i o u s o b s e r v a t i o n s . Imagine a phase space t h a t i s peppered w i t h l i m i t c y c l e s , t o r i , and s t r a n g e a t t r a c t o r s , each o b j e c t surrounded by i t s own b a s i n o f a t t r a c t i o n . As t h e R a y l e i g h number i s v a r i e d , some o f these o b j e c t s s h r i n k , w h i l e o t h e r s grow. A comprehensive quant i t a t i v e d e s c r i p t i o n o f t h e e n t i r e phase space w i l l p r o b a b l y be i m p o s s i b l e t o a t t a i n . On t h e o t h e r hand, t h e i n d i v i d u a l phenomena observed i n each b a s i n o f a t t r a c t i o n a r e r e a s o n a b l y w e l l understood i n p r i n c i p l e . LARGE ASPECT R A T I O When t h e l a r g e s t h o r i z o n t a l dimension i s much g r e a t e r t h a n t h e f l u i d depth, t h e b e h a v i o r o f t h e system i s q u a l i t a t i v e l y d i f f e r e n t . Experiments by A h l e r s and B e h r i n g e r g on c y l i n d r i c a l c e l l s o f c o n v e c t i n g l i q u i d h e l i u m showed t h a t t h e s t r i c t l y p e r i o d i c and q u a s i p e r i o d i c regimes a r e a b s e n t and t h a t t h e m o t i o n i s c h a o t i c whenever t h e v e l o c i t y f i e l d i s time-dependent. I n f a c t , t h e existence of regime o f s t r i c t l y time-independent c o n v e c t i o n was i n doubt i n t h e s e I t i s n o t p o s s i b l e t o d e t e r m i n e t h e cause o f t h e e a r l y t i m e depenexperiments. dence, because o n l y t h e t o t a l h e a t f l u x t h r o u g h t h e system c o u l d be measured. We have r e c e n t l y extended o u r l a s e r Doppler measurements t o a r e c t a n g u l a r c e l l o f aspect r a t i o 30. Our scanning t e c h n i q u e p e r m i t s t h e s t r u c t u r e o f t h e v e l o c i t y f i e l d t o be mapped i n t h e h o r i z o n t a l p l a n e . T h i s mapping c o u l d be done i n a t i m e short compared t o t h e c h a r a c t e r i s t i c t i m e s o f v e l o c i t y f l u c t u a t i o n s near t h e i r onset, so t h a t t h e space and t i m e s t r u c t u r e o f one component o f t h e v e l o c i t y f i e l d + v ( $ , t ) c o u l d be r e c o r d e d i n d i g i t a l form. We found t h a t t h e system i s c h a r a c t e r i z e d by v e r y l o n g t r a n s i e n t s , comparable t o t h e h o r i z o n t a l thermal d i f f u s i o n t i m e f r o m one edge o f t h e f l u i d l a y e r t o t h e other. This time i s o f the order o f 1 0 5 s . I n t h e i n t e r v a l Rc
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F i g . 1 . Doppler map showing a n example o f t h e s t r u c t u r e o f t h e v e l o c i t y f i e l d above t h e m i d p l a n e o f t h e 1 0 ~ 1 5 ~ 0 .cm5 c e l l i n t h e time-independent s t e a d y s t a t e . The d o t s mark t h e r o l l boundaries, where t h e f l o w i s e n t i r e l y v e r t i c a l ( o u t o f o r i n t o t h e page). The p a t t e r n c o n t a i n s d e f e c t s and t h e r o l l s a r e perpendicular t o the l a t e r a l boundaries. c h a r a c t e r i s t i c t i m e o f v e l o c i t y f l u c t u a t i o n s . T h i s t i m e becomes q u i t e long, about 7000 s a t 5Rc, t h u s p e r m i t t i n g us t o f o l l o w t h e space and t i m e s t r u c t u r e of t h e f l u c t u a t i o n s i n d e t a i l by r e p e t i t i v e l a s e r Doppler i m a g i n g . We scanned a p o r t i o n o f t h e c e n t e r o f t h e c e l l e v e r y 10 m i n u t e s f o r r u n s l a s t i n g a b o u t 1 0 hours, o b t a i n i n g 60 images per r u n i n d i g i t a l f o r m w h i c h c o u l d be l a t e r r e c o n s t r u c t e d . Several sequences o f t h e s e images a r e shown i n F i g . 2. Shaded and unshaded r e g i o n s a r e r o l l s o f o p p o s i t e v o r t i c i t y ( d i r e c t i o n o f c i r c u l a t i o n ) . A t t h e boundaries between t h e s e r e g i o n s , t h e f l u i d moves v e r t i c a l l y ( o u t o f o r i n t o t h e page). I n ( b ) t h e i n t r u s i o n o f a d e f e c t p i s v i s i b l e . I n ( c ) i t merges w i t h a r o l l q o f s i m i l a r v o r t i c i t y . T h i s process forms a d e f e c t o f o p p o s i t e p o l a r i t y (shaded) which i s t h e n e x p e l l e d from t h e f i e l d o f v i e w . A second example b e g i n s i n ( f ) where a r o l l l a b e l l e d r i s pinched o f f t o f o r m a d e f e c t , merges w i t h r e g i o n s i n ( h ) , and i s t h e n s e p a r a t e d once a g a i n i n ( i ) . These events i n d i c a t e t h a t t h e c o n v e c t i v e r o l l s a r e u n s t a b l e w i t h r e s p e c t t o deformations, and we b e l i e v e t h a t t h i s i n s t a b i l i t y i s r e s p o n s i b l e f o r t h e o n s e t o f t u r b u l e n c e . The observed o n s e t o f t h e i n s t a b i l i t y i s c o n s i s t e n t w i t h t h e p r e d i c t e d o n s e t o f t h e "skewed v a r i c o s e " i n s t a b i l i t y o f Busse and C 1 e ~ e r . l ~ While t h e l i n e a r s t a b i l i t y t h e o r y does n o t p r e d i c t a n y t i m e dependence, i t i s p o s s i b l e t h a t n o n l i n e a r e f f e c t s would cause t h i s i n s t a b i l i t y t o e v o l v e i n t o c h a o t i c t i m e dependence. Thus, n o n l i n e a r a n a l y s i s w i l l be c r u c i a l i n e x p l a i n i n g t h e o n s e t o f t u r b u l e n t c o n v e c t i o n a t l a r g e a s p e c t r a t i o . However, t h e s t r a n g e a t t r a c t o r concept i s n o t l i k e l y t o be u s e f u l i n t h i s case.

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40 min

(I) 30 min

F i g . 2. Two sequences o f Doppler images showing a p o r t i o n o f t h e v e l o c i t y f i e l d a t i n t e r v a l s o f 1 0 minutes, a t R =4.8Rc. Shaded and unshaded r e g i o n s have opp o s i t e c i r c u l a t i o n . The r o l l s a r e c l e a r l y u n s t a b l e . ACKNOWLEDGEMENT T h i s work was supported by a N a t i o n a l Science Foundation Grant (CME 79-12150) t o H a v e r f o r d C o l l e g e . The experiments were performed i n c o l l a b o r a t i o n w i t h J.F. Steinman and S.V. Benson. REFERENCES 1.

For a r e v i e w , see R u e l l e , D., Strange A t t r a c t o r s , La Recherche 108 ( F e v r i e r 1980).

2.

Gollub, J.P. and Benson, S . V . , Mech. 100 (1980) 449-470.

3.

Gollub, J.P. and Benson, S.V., C h a o t i c Response t o P e r i o d i c P e r t u r b a t i o n o f a Convecting F l u i d , Phys. Rev. L e t t . 41 (1978) 948-951.

4.

Gollub, J.P. and Benson, S . V . , Phase L o c k i n g i n t h e O s c i l l a t i o n s Leading t o Turbulence, i n : Haken, H. (ed.), P a t t e r n Formation by Dynamic Systems and P a t t e r n R e c o g n i t i o n ( S p r i n g e r - V e r l a g , B e r l i n , 1979).

5.

Gollub, J.P. and Steinman, J.F., E x t e r n a l Noise and t h e Onset o f T u r b u l e n t Convection, Phys. Rev. L e t t . 45 (1980) 551-554.

Many Routes t o T u r b u l e n t Convection, J. F l u i d

J. P. GO L LUB

408

6.

G o l l u b , J.P., The Onset o f Turbulence: Convection, S u r f a c e Waves, and O s c i l l a t o r s , i n : G a r r i d o , L. (ed.),Systems Far f r o m E q u i l i b r i u m , L e c t u r e Notes i n Physics V o l . 132 ( S p r i n g e r - V e r l a g , B e r l i n , 1 9 8 0 ) .

7.

G o l l u b , J.P., Benson, S.V. and Steinman, J.F., A Subharmonic Route t o T u r b u l e n t Convection, Ann. New York Acad. S c i . 357 (1980) 22-27.

8.

G o l l u b , J.P. and Steinman, J.F., Convection, t o appear.

9.

For example, see A h l e r s , G. and B e h r i n g e r , R.P., The R a y l e i g h Benard I n s t a b i l i t y and t h e E v o l u t i o n o f Turbulence, P r o g r . Theor. Phys. Suppl. NO. 64 (1978) 186-201.

Doppler Imaging o f t h e Onset o f T u r b u l e n t

10.

Libchaber, A. and Maurer, J . , Une Experience de Rayleigh-Benard de Geometrie Reduite: M u l t i p l i c a t i o n , Accrochage e t D e m u l t i p l i c a t i o n de Frequences, J . de Physique 41 (1980) C3-51 t o C3-53.

11.

Huberman, B.A. and Zisook, A.B., Rev. L e t t . 46 (1981) 626-628.

12.

G i g l i o , M., Musazzi, S . and P e r i n i , U., T r a n s i t i o n t o Chaos v i a a Well Ordered Sequence o f P e r i o d D o u b l i n g B i f u r c a t i o n s , p r e p r i n t .

13.

G o l l u b , J.P., Brunner, T.O. and Danly, B.G., P e r i o d i c i t y and Chaos i n Coupled N o n l i n e a r O s c i l l a t o r s , Science 200 (1978) 48-50.

14.

Busse, F.H. and C l e v e r , R.M., I n s t a b i l i t i e s o f C o n v e c t i o n R o l l s i n a F l u i d o f Moderate P r a n d t l Number, J. F l u i d Mech. 91 (1979) 319-335.

Power S p e c t r a o f S t r a n g e A t t r a c t o r s , Phys.

Nonlinear Problems: Present and Future

A.R. Bishop, D.K. Campbell, 6. Nicolaenko (eds.) @ North.Holland Publishing Company, 1982

409

EXPERIMENTAL OBSERVATIONS OF COMPLEX DYNAMICS

I N A CHEMICAL REACTION J.-C.

ROUX,* Jack S. Turner.,'

W.

0. McCormick, and H a r r y L. Swinney

Department o f P h y s i c s The U n i v e r s i t y o f Texas a t A u s t i n A u s t i n , Texas 78712 U.S.A.

I n experiments on a s t i r r e d f l o w chemical r e a c t o r we have observed a sequence o f p e r i o d i c and c h a o t i c regimes t h a t a l t e r n a t e as a f u n c t i o n o f t h e f l o w r a t e . The p e r i o d i c regimes a r e c h a r a c t e r i z e d by power s p e c t r a c o n s i s t i n g o f a s i n g l e fundamental frequency component and harmonics and by 1 i m i t c y c l e a t t r a c t o r s , w h i l e t h e c h a o t i c regimes a r e c h a r a c t e r i z e d by broadband power s p e c t r a and s t r a n g e a t t r a c t o r s . One o f t h e c h a o t i c regimes has been s t u d i e d i n d e t a i l and has been found t o correspond t o a smooth one-dimensional map t h a t has a s i n g l e maximum and a p o s i t i v e Lyapunov exponent.

INTRODUCTION O s c i l l a t i n g n o n e q u i l i b r i u m chemical systems were observed unambiguously e a r l y i n t h i s c e n t u r y , b u t t h e i r e x i s t e n c e remained l a r g e l y unknown u n t i l r e c e n t l y , even among chemists. Now n o n e q u i l i b r i u m chemical systems a r e b e i n g w i d e l y s t u d i e d as examples o f n o n l i n e a r systems which have i n t e r e s t i n g dynamics. Examples of n o n e q u i l i b r i u m systems t h a t o f t e n e x h i b i t p e r i o d i c o r more complex o s c i l l a t i o n s i n c l u d e l i v i n g systems and t h e chemical s t i r r e d f l o w r e a c t o r s used i n i n d u s t r y . By f a r t h e most e x t e n s i v e l y i n v e s t i g a t e d and b e s t understood o s c i l l a t i n g chemical system i s t h e B e l o u s o v - Z h a b o t i n s k i i ( B Z ) r e a c t i o n , which was d i s c o v e r e d [l] and s t u d i e d [2] i n t h e S o v i e t Union about 20 y e a r s ago. I n t h i s r e a c t i o n an a c i d i c bromate s o l u t i o n o x i d i z e s an o r g a n i c compound i n t h e presence o f a metal i o n c a t a l y s t . The mechanism o f t h e BZ r e a c t i o n , which i n v o l v e s more t h a n 20 r e a c t i o n s and as many chemical c o n s t i t u e n t s , has been e l u c i d a t e d by Noyes and coworkers i n a sequence o f more t h a n 30 papers [3-51. On t h e e x p e r i m e n t a l s i d e , work by s e v e r a l groups, p a r t i c u l a r l y Schmitz, Hudson. and coworkers [6-71 and t h e Bordeaux group [8-111, has r e v e a l e d s e v e r a l t y p e s o f dynamics i n t h e BZ r e a c t i o n , i n c l u d i n g m u l t i p l e steady s t a t e s , s i m p l e and complex o s c i l l a t i o n s , i n t e r m i t t e n c y , and c h a o t i c behavior.

We have conducted experiments on t h e BZ r e a c t i o n i n a s t i r r e d f l o w r e a c t o r and have observed a sequence o f d i s t i n c t dynamical regimes. The r e a c t i o n was s t u d i e d f o r d i f f e r e n t r e s i d e n c e times T (where T = r e a c t o r volume/flow r a t e ) w i t h a l l o t h e r v a r i a b l e s i n c l u d i n g t h e c o n c e n t r a t i o n s o f i n p u t chemicals h e l d f i x e d [12]. Note t h a t t h e i n v e r s e r e s i d e n c e time, T - ~ ,l i k e t h e Reynolds number i n a hydrodynamic system, i s a measure o f t h e d i s t a n c e away from thermodynamic e q u i l i b r i u m f o r a chemical r e a c t i o n m a i n t a i n e d i n a s t a t i o n a r y regime. P r e l i m i n a r y r e s u l t s o f t h i s work a r e p r e s e n t e d here; d e t a i l e d r e p o r t s , i n c l u d i n g a numerical s t u d y o f a model o f t h e r e a c t i o n , w i l l be p r e s e n t e d elsewhere [12-14).

410

J.C. ROUX et at.

EXPERIMENTAL SYSTEM The chemicals were i n j e c t e d c o n t i n u o u s l y w i t h a multi-channel p e r i s t a l t i c pump i n t o a v i g o r o u s l y s t i r r e d r e a c t o r t h a t was immersed i n a t e m p e r a t u r e - c o n t r o l l e d bath, The experimental parameters were as f o l l o w s : r e a c t o r volume, 31.0 c d ; b a t h temperature, 28.3'C; chemical c o n c e n t r a t i o n s i n t h e mixed feed--0.25 M malonic acid, 0.14 M potassium bromate, 0.00083 M cerous s u l f a t e octahydrate, and 0.2 M s u l f u r i c acid; residence time, 0.5 < T < 4 h r . The c o n c e n t r a t i o n o f one o f t h e r e a c t i o n products, bromide i o n , was measured w i t h a s p e c i f i c i o n probe, d i g i t i z e d , and recorded as a f u n c t i o n o f time i n a computer. The d i g i t a l t i m e s e r i e s records B ( t . ) (where ti = i A t , i = 1,...,16384, and A t = 0.88 s ) were F o u r i e r transformed 'to o b t a i n power spectra. In addition, the dynamical behavior o f t h e system was c h a r a c t e r i z e d by phase space p o r t r a i t s c o n s t r u c t e d f o l l o w i n g a procedure suggested by R u e l l e [15]: an n-dimensional (no) phase p o r t r a i t can be obtained by p l o t t i n g successive p o i n t s [B(ti+T1), B(ti+T2), ., B ( t i + T 1, where t h e T . ( j = l , n) a r e s u i t a b l y chosen constants. (A r e l a t e d procesure, i n which tht? nD p o r t r a i t i s o b t a i n e d from a t i m e s e r i e s and i t s f i r s t n-1 d e r i v a t i v e s , was proposed by Packard e t a l . [16] and t e s t e d i n experiments on t h e B Z r e a c t i o n by Roux ~t aJ-.- [lT.)-We r e p o r t here some 20 p o r t r a i t s obtained w i t h T =O and T2.8.8 s, and a P o i n c a e map o b t a i n e d f r o m a 3D p o r t r a i t w i t h T1=O, T2=8.81~, and T3=17.6 s .

..

....

EXPERIMENTAL RESULTS For T < 0.87 h r and T > 2.28 h r t h e r e a c t i o n e x h i b i t s simple p e r i o d i c o s c i l l a t i o n s . When T i s increased w i t h i n t h e range between these l i m i t s , t h e r e a c t i o n passes through an a l t e r n a t i n g sequence o f p e r i o d i c and c h a o t i c regimes; we c a l l these regimes P1 ( T < 0.87 h r ) , C1, P2, Ct, bCI , P ' , (T > 2.28 h r ) . The time r o p o r t i o n a j t o -log[Br-1) and t h e dependence o f t h e bromide i o n PO en 1 corresponding power spectra and 2D phase p o r t r a i t s are shown f o r t h e f i r s t s i x regimes i n F i g u r e s 1-6. (The bromide p o t e n t i a l i s p l o t t e d i n r e l a t i v e u n i t s w i t h a zero o f f s e t ; t h e p o t e n t i a l o s c i l l a t e s i n t h e range 155-200 mV.)

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As Figures 1, 3, and 5 i l l u s t r a t e , each regime Pk i s c h a r a c t e r i z e d by: ( 1 ) a p e r i o d i c time s e r i e s w i t h each p e r i o d c o n t a i n i n g one l a r g e amplitude r e l a x a t i o n o s c i l l a t i o n ( o f t h e t y p e i n regime P1) and k - 1 small amplitude n e a r l y s i n u s o i d a l o s c i l l a t i o n s ( o f t h e type i n regime P I ) , ( 2 ) a power spectrum c o n s i s t i n g o f a s i n g l e i n s t r u m e n t a l l y sharp fundamental \requency component and i t s harmonics, and (3) a phase p o r t r a i t t h a t i s a l i m i t c y c l e . As Figures 2, 4, and 6 i l l u s t r a t e , each regime c k i s c h a r a c t e r i z e d by: ( 1 ) a nonperiodic o s c i l l a t i n g time series, ( 2 ) a broadband power spectrum, and (3) a phase p o r t r a i t t h a t i s a strange a t t r a c t o r ( a s w i l l be discussed i n t h e f o l l o w i n g s e c t i o n ) . One way i n which we have d i s t i n g u i s h e d t h e p e r i o d i c and c h a o t i c regimes i s by comparing zero-frequency i n t e r c e p t s o f t h e background n o i s e l e v e l i n the power spectra f o r t h e d i f f e r e n t regimes. I n t h e p e r i o d i c regimes t h e broadband noise l e v e l , which i s presumably due t o i n s t r u m e n t a l n o i s e ( i n c l u d i n g f l u c t u a t i o n s i n f l o w r a t e , s t i r r i n g r a t e , e t c . ) , i s t y p i c a l l y two t o t h r e e orders o f magnitude lower than t h e broadband s p e c t r a l i n t e n s i t y a t low frequencies f o r t h e c h a o t i c regimes; t h i s i s i l l u s t r a t e d f o r P2 and C2 i n F i g u r e 7. The sequence o f dynamical regimes we have observed i s summarized i n F i g u r e 8, which shows t h e broadband s p e c t r a l noise l e v e l as a f u n c t i o n o f residence time. Each p e r i o d i c and c h a o t i c regime c l e a r l y e x i s t s over some range o f T , although t h e e x t e n t of t h e regimes, drawn as equal i n F i g u r e 8, has n o t been determined.

COMPLEX DYNAMICS IN A CHEMICAL REACTION

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J. C. ROUX et al.

414

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COMPLEX D YNAMICS IN A CHEMICAL REACTION

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COMPLEX DYNAMICS IN A CHEMICAL REACTION

417

DETERMINISTIC CHAOS We have c a l l e d t h e a t t r a c t o r s f o r t h e regimes ck " s t r a n g e a t t r a c t o r s , " w h i c h i m p l i e s t h a t t h e system i s d e t e r m i n i s t i c ( s e e e.g., L a n f o r d [18]). As S h o w a l t e r .__ et . a-l-. . [19] r i g h t l y emphasize, a l l e x p e r i m e n t a l systems a r e i n e v i t a b l y s u b j e c t t o e x t e r n a l p e r t u r b a t i o n s which r e s u l t i n n o n p e r i o d i c b e h a v i o r ; " t h e r e f o r e , f a i l u r e t o observe e x a c t r e p e t i t i o n i n a chemical s t i r r e d t a n k r e a c t o r cannot prove t h e system would be t r u l y c h a o t i c i n t h e complete absence o f p e r t u r b a t i o n " . We concur completely. No experiment w i l l ever be a b l e t o make an a b s o l u t e d i s t i n c t i o n between s t o c h a s t i c and d e t e r m i n i s t i c processes, and, i n f a c t , such a d i s t i n c t i o n cannot even be made i n p r i n c i p l e : as Kac has p o i n t e d o u t , a s t o c h a s t i c process can always i n p r i n c i p l e be d e v i s e d t o d e s c r i b e any n o i s y d a t a even when t h e d a t a can be simply and e l e g a n t l y d e s c r i b e d by a d e t e r m i n i s t i c model [20].

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410

Recognizing t h e s e d i f f i c u l t i e s , we now p r e s e n t t h r e e d a t a analyses t h a t p r o v i d e s u c c e s s i v e l y s t r o n g e r s u p p o r t f o r t h e i n t e r p r e t a t i o n o f regime C1 as c o r r e s p o n d i n g t o d e t e r m i n i s t i c chaos. F i r s t c o n s i d e r t h e 2D P o i n c a r e map shown i n F i g u r e 9. T h i s map was c o n s t r u c t e d f r o m a 3D R u e l l e - t y p e p o r t r a i t t h a t has, i n a d d i t i o n t o t h e two axes shown i n t h e phase p o r t r a i t i n F i g u r e 2 , a t h i r d a x i s ( p e r p e n d i c u l a r t o t h e f i r s t two axes) o b t a i n e d w i t h T -17.6 s. The P o i n c a r e map i n F i g u r e 9 was formed by t h e i n t e r s e c t i o n o f t h e 3D p&ise space o r b i t s w i t h a p l a n e t h a t passes (normal t o t h e paper) t h r o u g h t h e dashed l i n e i n F i g u r e 2. The o b s e r v a t i o n t h a t t h e P o i n c a r e map i s a smooth c u r v e ( F i g u r e 9) r a t h e r t h a n a s c a t t e r o f p o i n t s suggests t h a t t h e process i s d e t e r m i n i s t i c ; however, a s t o c h a s t i c process c o u l d be d e v i s e d t o g i v e t h e c u r v e i n F i g u r e 9. As a second t e s t o f t h e c h a r a c t e r of t h e dynamics, c o n s i d e r a r e t u r n ma d c o n s t r u c t e d f r o m t h e a t t r a c t o r i n F i g u r e 2. L e t X(n) b e t h e v a l u e o f t when an o r b i t i n t h e 20 p o r t r a i t crosses t h e dashed l i n e i n F i g u r e 2, and l e t X(nt1) b e t h e o r d i n a t e t h e n e x t t i m e t h e o r b i t c r o s s e s t h e dashed l i n e . A r e t u r n m p o b t a i n e d by p l o t t i n g t h e o r d e r e d p a i r s [ X ( n ) , X ( n t l ) ] f o r a l l n i s shown i n F i g u r e 10. Again t h e d a t a d e s c r i b e a smooth c u r v e t h a t suggests t h a t t h e system i s d e t e r m i n i s t i c r a t h e r t h a n s t o c h a s t i c : g i v e n any i n i t i a l X(n), a l l subsequent X ( n t k ) a r e determined by t h e map. The s t r o n g e s t evidence we have i n d i c a t i n g t h a t C1 corresponds t o d e t e r m i n i s t i c chaos i s g i v e n by t h e v a l u e of t h e Lyapunov exponent. Lyapunov exponents p r o v i d e a q u a n t i t a t i v e measure o f t h e degree o f chaos o r “ s e n s i t i v e dependence on i n i t i a l c o n d i t i o n s ” f o r n o n l i n e a r systems. A p o s i t i v e exponent means t h a t nearby o r b i t s w i l l e x p o n e n t i a l l y separate; t h e system i s t h e n h i g h l y s e n s i t i v e t o t h e i n i t i a l conditions. On t h e o t h e r hand, i f t h e exponent i s n e g a t i v e , nearby o r b i t s w i l l e x p o n e n t i a l l y converge; t h e system i s t h e n i n s e n s i t i v e t o i n i t i a l C o n d i t i o n s .

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RESIDENCE TIME (hr) F i g u r e 8. The a l t e r n a t i n g sequence o f p e r i o d i c and c h a o t i c regimes i s shown by t h i s graph o f t h e broadband s p e c t r a l n o i s e ( z e r o - f r e q u e n c y i n t e r c e p t s ) measured as a f u n c t i o n o f r e s i d e n c e time. I n t h e p e r i o d i c regimes Pk t h e n o i s e i s i n s t r u m e n t a l (see t h e a t t r a c t o r s i n F i g u r e s 1, 3, and 5 ) , w h i l e i n t h e c h a o t i c regimes c k t h e n o i s e i s s e v e r a l o r d e r s o f magnitude l a r g e r . The c u r v e i s drawn t o g u i d e t h e eye.

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For a 1D map t h e Lyapunov exponent has a maximum v a l u e o f I n 2 = 0.69. Robert Shaw has c a l c u l a t e d t h e Lyapunov exponent f o r t h e map shown i n F i g u r e 10 and has o b t a i n e d 0.520.1. The p o s i t i v e Lyapunov exponent g i v e s t h e r a t e o f d i v e r g e n c e o f o r b i t s i n t h e a t t r a c t i n g s h e e t shown i n c r o s s s e c t i o n i n t h e P o i n c a r e map ( F i g u r e 9). I n c o n t r a s t , o r b i t s o f f o f t h e sheet converge e x p o n e n t i a l l y f a s t t o t h e sheet ( t h e Lyapunov exponent i n d i r e c t i o n s normal t o t h e s h e e t i s n e g a t i v e ) . T h e r e f o r e , t h e i n e v i t a b l e s c a t t e r i n o u r e x p e r i m e n t a l d a t a due t o f l u c t u a t i o n s i n t h e environment i s much l e s s n o t i c e a b l e i n t h e P o i n c a r e map (where t h e s c a t t e r i s a l o n g t h e c u r v e i n F i g u r e 9) t h a n i n t h e r e t u r n map ( F i g u r e 10).

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/" B ( t +T) F i g u r e 9. A P o i n c a r e map ( w i t h T = 8.8 s) f o r t h e s t r a n g e a t t r a c t o r i n regime C shown i n F i g u r e 2. The map was formed by t h e i n t e r s e c t i o n o f t h e 3D o r b i t s wit!! t h e p l a n e (normal t o t h e paper) p a s s i n g t h r o u g h t h e dashed l i n e i n F i g u r e 2.

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MODEL

STUDIES

Numerical s t u d i e s o f a mathematical model o f t h e BZ r e a c t i o n have p l a y e d an i m p o r t a n t r o l e i n t h e d e s i g n and guidance o f o u r e x p e r i m e n t a l program. I n these s t u d i e s a f o u r - v a r i a b l e model based on t h e Oregonator o f F i e l d and Noyes [21,22] has been found t o e x h i b i t a v a r i e t y o f phenomena r e l e v a n t t o e x p e r i m e n t a l s t u d i e s i n a s t i r r e d f l o w reactor. These i n c l u d e m u l t i p l e s t a t i o n a r y and o s c i l l a t o r y s t a t e s , h y s t e r e s i s , b u r s t s o f o s c i l l a t i o n , complex p e r i o d i c and n o n p e r i o d i c s t a t e s , and two d i s t i n c t sequences o f t r a n s i t i o n s i n v o l v i n g mixed p e r i o d i c and c h a o t i c s t a t e s . D e t a i l s of t h e model and i t s p r e d i c t i o n s a r e p r e s e n t e d elsewhere [12,14],

F i g u r e 10. A r e t u r n map f o r t h e s t r a n g e a t t r a c t o r i n regime C shown i n F i g u r e 2. The map was formed by p l o t t i n g as o r d e r e d p a i r s [X(n),X(n+l)] t h e successive values X(n) o f t h e o r d i n a t e o f phase space o r b i t s when t h e y c r o s s t h e dashed l i n e i n F i g u r e 2.

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O f p a r t i c u l a r i n t e r e s t h e r e a r e t h e sequences o f t r a n s i t i o n s which occur as t h e i n v e r s e r e s i d e n c e t i m e i s v a r i e d . The o s c i l l a t i o n s which appear f i r s t i n t h e model as ~ - 1 i n c r e a s e s a r e n e a r l y s i n u s o i d a l . Further increase i n T - ~ reveals a p e r i o d - d o u b l i n g sequence o f t r a n s i t i o n s o f t h e t y p e discussed i n t h i s volume by Feigenbaum [see a l s o Ref. 231. For s t i l l l a r g e r 1 - 1 . a second t y p e o f o s c i l l a t i o n appears i n t h e model s t u d i e s , h a v i n g l a r g e r a m p l i t u d e and l o w e r frequency t h a n t h o s e p r e v i o u s l y found. It i s i n t h i s r e g i o n o f T-1 t h a t t h e p e r i o d i c - c h a o t i c sequence observed i n o u r experiments i s found i n t h e model. The model p r e d i c t i o n s o f t h e p e r i o d i c - c h a o t i c sequence, which were o b t a i n e d p r i o r t o t h e conduct o f t h e experiments, a r e i n e x c e l l e n t q u a l i t a t i v e agreement w i t h t h e experiments [12]. We expect t h e q u a n t i t a t i v e agreement t o improve w i t h parameter v a r i a t i o n and model r e f i n e m e n t now i n progress.

DISCUSS I O N

A s i m i l a r sequence o f a l t e r n a t i n g p e r i o d i c and c h a o t i c regimes has been observed i n experiments by Hudson e t al.[7] and by t h e Bordeaux group [10,24]. However. t h e sequences o f regimes observed i n our experiments [12] and i n p r e v i o u s experiments [7,10,24] have t h e o o s i t e dependence on r e s i d e n c e t i m e T . The e x p e r i m e n t a l c o n d i t i o n s a r e s i m i k c e p t f o r t h e range i n r e s i d e n c e t i m e s s t u d i e d : 0.08 < T < 0.2 h r f o r p r e v i o u s experiments and 0.5 < T <4 h r f o r o u r experiments. I n a d d i t i o n t o t h e numerical s t u d i e s by Turner [12,14], the periodic-chaotic sequence has been found by Tomita and Tsuda [25] i n a dynamical systems model (map) developed t o d e s c r i b e t h e o b s e r v a t i o n s o f Hudson. Thus t h e models and t h e experiments suggest t h a t t h e p e r i o d i c - c h a o t i c sequence, l i k e t h e p e r i o d - d o u b l i n g sequence, may o c c u r w i d e l y i n n o n l i n e a r systems. ACKNOWLEDGMENTS We acknowledge h e l p f u l d i s c u s s i o n s w i t h Robert Shaw and Jack Hudson. We thank Jack Hudson f o r a p r e p r i n t [26], which we r e c e i v e d a f t e r t h e p r e s e n t r e s u l t s were presented a t t h e Los Alamos conference; t h e p r e p r i n t d e s c r i b e s an a n a l y s i s o f h i s experiments u s i n g phase p o r t r a i t s and 10 maps s i m i l a r t o t h o s e which we have presented here. T h i s research i s supported by N a t i o n a l Science F o u n d a t i o n Grant CHE79-23627 and The Robert A. Welch Foundation Grant F-805. *Permanent address: Centre de Recherche Paul Pascal, U n i v e r s i t e de Bordeaux Oomaine U n i v e r s i t a i r e , 33405 Talence Cedex, France. 'Also, Center f o r S t u d i e s i n S t a t i s t i c a l Mechanics, U n i v e r s i t y o f Texas, Aust Texas 78712. REFERENCES

[l] Belousov, B. P., Ref. Radiats. Med. 1958 (1959) 145. Ookl. Akad. Nauk. SSSR 157 (1964) 392; B i o f i z i k a 9 [2] Z h a b o t i n s k i i , A. M., (1964) 306. [3] F i e l d , R. J., Ktlrtls, E . , and Noyes, R. M., J. Am. Chem. Soc. 94 (1972) 8649. [4] F i e l d , R. J. and Noyes, R. M., Accounts Chem. Res. 1 0 (1977) 214. [5] Noyes, R . M. and F i e l d , R. J., Accounts Chem. Res. 1 0 (1977) 273. [6] Schmitz, R. A., G r a z i a n i , K. R., and Hudson, J. L.. J. Chem. Phys. 67 (1977) 3040. [ 7 ] Hudson, J. L., Hart, M., and Marinko, O., J. Chem. Phys. 7 1 (1979) 1601. [8] De Kepper, P., Rossi, A,, and P a c a u l t , A., C. R. Acad. Sci. P a r i s 283C (1976) 371. and Rossi, A., J. Pm. Chem. SOC. 102 (1980) 1241. [9] V i d a l , C., Roux,J.-C., [ l o ] V i d a l , C., Roux, J.-C., B a c h e l a r t , S., and R o s s i , A., N.Y. k a d . Sci. 357 (1980) 377.

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Y., Roux, J.-C., Rossi A., and B a c h e l a r t , S. and V i d a l C., submitted t o J. Phys. L e t t . ( P a r i s ) . Turner, J. S., Roux, J.-C., McCormick, W. D., and Swinney, H. L., submitted t o Phys. L e t t . Swinney, H. L., Roux, J.-C., and King, G. P.. i n : M e i j e r , P. H. E., Mountain, R. D., and Soulen, R. J . (eds.), Noise i n P h y s i c a l Systems (Proceedings o f t h e Sixth International Conference) ( N a t i o n a l Bureau o f Standards Special P u b l i c a t i o n , U.S. Govt. P r i n t i n g O f f i c e , 1981). Turner, J. S., Discussion Meeting, K i n e t i c s o f Physicochemical O s c i l l a t i o n s , Aachen, September, 1979, and t o be published. Ruelle, D., p r i v a t e communication. Packard, N. H., Crutchfield, J . P., Farmer, J. D., and Shaw, R. S., Phys. Rev. L e t t . 45 (1980) 712. Roux, J.-C., Rossi, A., Bachelart, and V i d a l , C., Phys. L e t t . 77A (1980) 391; Physica D ( i n press). Lanford, 0. E., Strange a t t r a c t o r s and turbulence, i n : Swinney, H. L. and Gollub, J. P. (eds.), Hydrodynamic I n s t a b i l i t i e s and t h e T r a n s i t i o n t o Turbulence, Topics i n A p p l i e d Physics Vol. 45 ( S p r i n g e r , B e r l i n , Heidelberg, New York ,1981),p. 7. Showalter, K. ., Noyes, R. M., and B a r - E l i , K., J. Chem. Phys. 69 (1978) 2514. Kac, M., p r i v a t e communication. F i e l d , R. J. and Noyes, R. M.,J. Chem. Phys. 60 (1974) 1877. F i e l d , R. J . , J . Chem. Phys. 63 (1975) 2289. Feigenbaum, M. J., J. Stat. Phys. 19 (1978) 25. V i d a l , C., i n : Haken, H. (ed.), I n t e r n a t i o n a l Symposium i n Synergetics (Springer, B e r l i n , Heidelberg, New York, 1981). Tomita, K. and Tsuda, I . , Prog. Theor. Phys. 64 (1980) 1138. Hudson, J. L. and Mankin, J. C., J. Chem. Phys. (June, 1981).

[11] Pomeau, [12] [13]

[14]

[15] [16]

[17] [18]

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Nonlinear Plasma Dynamics Below the Cyclotron Frequency John M. Greene Princeton Plasma Physics Laboratory Princeton, New Jersey, 08544

Plasma i s a soup o f electromagnetic f i e l d s , ions, and electrons. The essential aspect o f t h i s state o f matter i s t h a t the charged p a r t i c l e components i n t e r a c t strongly w i t h the electromagnetic f i e l d . There are a v a r i e t y o f plasma regimes. Here I am thinking o f a hot, completely ionized gas so t h a t there are no neutral atoms i n the soup. This s o r t o f plasma can be found i n a number o f places, such as thermonuclear fusion devices, the magnetosphere, and a v a r i e t y o f astrophysical locations. A good way t o know a plasma i s through i t s waves. Each o f the independent components t h a t compose i t has i t s own modes o f o s c i l l a t i o n , and these are generally coupled together. A f u l l description o f a plasma t h i s way i s extremely complicated. However, i f one r e s t r i c t s oneself t o the study o f modes w i t h frequencies below the i o n cyclotron frequency the s i t u a t i o n i s considerably simplified. Since I have spent most o f my career working i n t h i s frequency range, t h i s t a l k w i l l be l i m i t e d appropriately. On the other hand, t h i s t a l k would seem t o be a good opportunity t o speculate, so I w i l l wander beyond the bounds o f calculations t h a t I have completed. The electrons are very l i g h t , so t h e i r v i b r a t i o n frequencies are high. Thus i n the low frequency regime they closely f o l l o w the ions. Furthermore, the ions a l l tend t o move together over the course o f these slow time scales. The appropriate model f o r these circumstances i s one t h a t t r e a t s the plasma as a single f l u i d . Only i t s mass density, momentum density, and stress need be considered. Further, since the l i g h t electrons react r a p i d l y t o screen o u t e l e c t r i c f i e l d s , only the magnetic p a r t o f the electromagnetic f i e l d need be treated completely. These considerations lead on t o the ideal MHD equations,

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The f i r s t o f these i s conservation o f mass density, p. Next i s the conservation of momentum, w i t h y t h e f l u i d v e l o c i t y . The f l u i d s t r e s s has been taken t o be a scalar, the pressure p, and I i s t h e u n i t tensor. The equation f o r t h e evolut i o n o f the magnetic f i e l d ,B i s the i n d u c t i o n equation w i t h an e l e c t r i c f i e l d t h a t vanishes i n the frame moving w i t h t h e f l u i d v e l o c i t y ,

Consistent w i t h t h e neglect o f v i s c o s i t y i n t h e momentum equation and r e s i s t i v i t y i n the i n d u c t i o n equation, the energy equation has been reduced t o the conservation o f entropy, S. T h i s i s then r e l a t e d back t o t h e pressure through t h e i n t e r n a l energy, U. The f i r s t t h i n g t o n o t i c e about these equations i s t h a t the pyy and @ terms i n the momentum f l u x have a s i m i l a r form. As a r e s u l t t h e r e a r e u s e f u l analogies between f l u i d dynamics, w i t h B=O, and i d e a l MHD. I n another close analogy, w i t h i n the i d e a l MHD model t h c magnetic f l u x and magnetic l i n e s o f f o r c e are convected w i t h the f l u i d j u s t as v o r t i c i t y i s convected i n i n v i s c i d f l u i d flow. The next t h i n g t o n o t i c e i s t h a t these equations a r e a g e n e r a l i z a t i o n o f the Euler equations o f f l u i d dynamics. Analogously, one can expect t h a t nonlinear flows develop s i n g u l a r i t i e s i n f i n i t e time, leading t o shocks. That i s , t h e sources o f entropy have been assumed t o be s u f f i c e n t l y weak i n r e l a t i o n t o t h e wavelengths o f t h e problem under i n v e s t i g a t i o n , whatever i t may be, t h a t d i s s i p a t i o n i s p r i m a r i l y confined t o t h i n layers. Shocks are indeed a f e a t u r e o f t h i s overview, b u t t h e r e are some complications a r i s i n g from t h e plasma s t a t e and plasma waves t h a t w i l l be discussed below. An idea o f the waves e x i s t i n g i n t h i s model can b e s t be obtained by studying an i n f i n i t e homogeneous medium. The d i s p e r s i o n r e l a t i o n can be displayed schemati c a l l y as i n Fig. 1.

Fig. 1. The phase v e l o c i t y o f plane waves i n a plasma as a f u n c t i o n o f t h e angle between t h e normal t o t h e plane wave and t h e magnetic f i e l d d i r e c t i o n . S i s the sound speed and a i s t h e speed o f the Alfven modes. Figure 1 i l l u s t r a t e s t h e phase v e l o c i t i e s o f plane waves as a f u n c t i o n o f the angle between t h e magnetic f i e l d and t h e normal t o t h e constant phase surfaces, where t h e h o r i z o n t a l a x i s i s t h e d i r e c t i o n o f B. Since an e v o l u t i o n equation f o r t h e magnetic f i e l d i s included i n the setlabove, t h e r e are transverse as

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w e l l as l o n g i t u d i n a l components t o these waves. The c o u p l i n g o f t h e two t r a n s verse and one l o n g i t u d i n a l p o l a r i z a t i o n y i e l d t h e t h r e e modes i l l u s t r a t e d i n Fig. 1. I n t h i s f i g u r e t h e c h a r a c t e r i s t i c speed o f t h e l o n g i t u d i n a l ("sound") modes, s2=ap/ap, has been taken $0 ;Pe smaller than t h e c h a r a c t e r i s t i c speed o f t h e transverse ( A l f v e n modes, a =B /p. The o u t e r curve describes a wave t h a t spreads more o r l e s s u n i f o r m l y i n a l l d i r e c t i o n s , i n t h e manner o f an o r d i n a r y sound wave, b u t w i t h approximately t h e A l f v e n speed. I t i s known as t h e f a s t magnetosonic mode. The middle curve i s p u r e l y transverse, and w i l l be c a l l e d the shear A l f v e n mode. The i n n e r curve i s t h e slow magnetosonic branch o f t h e d i s p e r s i o n r e l a t i o n . The important f e a t u r e o f t h i s sketch f o r t h e remainder o f . t h i s review i s t h e vanishing o f t h e phase speed f o r t h e two slowest modes propagating perpendicular t o t h e magnetic f i e l d . To o b t a i n a deeper understanding o f these equations, i t i s u s e f u l t o consider a s p e c i f i c c o n f i g u r a t i o n . The primary example used here w i l l be a tokamak. T h i s i s an axisymmetric c o n f i g u r a t i o n w i t h magnetic f i e l d l i n e s winding around t o r o i dal surfaces, and w i t h a h o t plasma c o n f i n e d t o t h e i n n e r t o r o i d s . A cross s e c t i o n o f t h e magnetic surfaces t h a t c o n t a i n t h e f i e l d l i n e s can be sketched as i n Fig. 2.

C

Fig. 2.

3

A cross s e c t i o n o f magnetic f i e l d

ines i n a tokamak.

Our method o f a p p l y i n g t h e MHD equations t o t h i s c o n f i g u r a t i o n has been t o f i r s t c a l c u l a t e an e q u i l i b r i u m , next t o t r e a t i t s l i n e a r s t a b i l i t y , and f i n a l l y t o consider t h e n o n l i n e a r meaning o f any l i n e a r i n s t a b i l i t y . Much o f t h e e f f o r t has gone i n t o t h e l i n e a r problem. I n c i d e n t l y , t h e c o n f i g u r a t i o n s we c a l l , " e q u i l i b r i u m " are e q u i v a l e n t t o t h e c o n f i g u r a t i o n s t h a t Orszag c a l l e d "quasie q u i l i b r i u m " i n another paper i n t h i s c o l l e c t i o n [l]. They a r e i n e q u i l i b r i u m w i t h respect t o t h e generation o f waves through t h e E u l e r equations, b u t may be changing s l o w l y due t o t h e d i f f u s i o n o f mass, momentum, energy, and magnetic f l u x . Our e q u i l i b r i a are two-dimensional as were those o f Orszag. The l i n e a r s t a b i l i t y problem i s c h a r a c t e r i z e d by s i n g u l a r i t i e s t h a t a r e very s i m i l a r t o those o f t h e i n v i s c i d l i m i t o f t h e Orr-Sommerfeld equation. One o f t h e f i r s t people t o - u t i l i z e these s i n g u l a r i t i e s i n a s t a b i l i t y c a l c u l a t i o n was The s i n g u l a r i t i e s o f t h e i n v i s c i d Orr-Sommerfeld Suydam a t Los Alamos [Z]. system a r i s e where t h e f l u i d streaming v e l o c i t y i s equal t o t h e phase v e l o c i t y o f t h e p e r t u r b a t i o n . That i s a l s o t h e case here, w i t h t h e v a r i a t i o n t h a t t h i s v e l o c i t y vanishes. I t occurs along closed f i e l d l i n e s f o r p e r t u r b a t i o n s t h a t

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have constant phase along these l i n e s . Thus these p e r t u r b a t i o n s represent waves propagating perpendicular t o t h e magnetic f i e l d a t t h a t p o i n t . Remember t h a t two modes have vanishing phase v e l o c i t y i n t h i s d i r e c t i o n . Since these f i e l d l i n e s wind around t h e t o r o i d a l magnetic surfaces, t h e s i n g u l a r p e r t u r b a t i o n s are three-dimensional. Closer examination o f these s i n g u l a r i t i e s from a n o n l i n e a r v i e w p o i n t shows t h a t they a r i s e from an attempt t o change t h e topology o f t h e magnetic surfaces, as i l l u s t r a t e d i n Fig. 3.

Fig. 3. A schematic sketch o f cross s e c t i o n s o f surfaces c o n t a i n i n g magnetic f i e l d l i n e s showing " i s l a n d s t r u c t u r e " and " x - p o i n t s . " Again, these l i n e s a r e t h e cross s e c t i o n o f surfaces t h a t c o n t a i n t h e magnetic f i e l d l i n e s . What w i l l be c a l l e d an i s l a n d s t r u c t u r e has opened o u t o f a t o r o i d a l surface o f i n t e r m e d i a t e radius. I t s associated tube o f f l u x winds h e l i c a l l y around t h e tokamak. Thus t h e l i n e o f f o r c e a t t h e c e n t e r o f t h e f l u x c e l l , and a l s o t h e l i n e a t t h e j u n c t i o n o f t h e c e l l s , c l o s e a f t e r t r a v e r s i n g t h e tokamak f o u r times, f o r t h e case shown here. T h i s change i n topology i s i n c o n s i s t e n t w i t h convection o f magnetic f i e l d l i n e s w i t h t h e f l u i d . The s t r i n g s o f f l u i d elements connected by f i e l d l i n e s a r e rearranged as t h e magnetic f l u x and f l u i d f l o w s i n t o and f a t t e n s o u t t h e f o u r f o l d i s l a n d s t r u c t u r e i l l u s t r a t e d i n t h e sketch above. There i s reconnection o f these s t r i n g s a t t h e p o i n t s where t h e i s l a n d s j o i n . Since these p o i n t s a r e i n t e r s e c t i o n s o f t h e magnetic surfaces t h a t separate t h e d i f f e r e n t f l u x regions, t h e y a r e c a l l e d x - p o i n t s . Thus an e s s e n t i a l aspect o f n o n l i n e a r plasma dynamics below t h e c y c l o t r o n frequency i s t h e study o f reconnection a t x-points. The f i r s t problem t h a t a r i s e s when s t u d y i n g x - p o i n t s i s t o d e s c r i b e what i s meant by t h e term. The f i r s t step i n much o f x - p o i n t t h e o r y i s t o draw an x on t h e blackboard ( o r t h e page, i n t h e p r e s e n t case), as i n Fig. 4.

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4.

427

A schematic i l l u s t r a t i o n o f magnetic f i e l d l i n e s near an x - p o i n t .

and study problems w i t h a t r a n s l a t i o n a l i n v a r i a n c e perpendicular t o t h e page. This i s a u s e f u l approximation i n many respects, b u t i t does n o t c o n t r i b u t e t o t h e i d e n t i f i c a t i o n o f x - p o i n t s i n complicated c o n f i g u r a t i o n s . T h i s page repres e n t a t i o n o f an x - p o i n t t r e a t s a plane orthogonal t o a chosen l i n e o f force. The magnetic f i e l d i n t h i s plane vanishes a t t h e x - p o i n t , since t h a t i s how t h e r e p r e s e n t a t i o n was chosen. However, t h e r e i s no reason f o r t h e t o t a l magnetic f i e l d t o vanish there. Indeed, i f t h e magnetic f i e l d vanishes along a l i n e i n some c o n f i g u r a t i o n , almost any small p e r t u r b a t i o n w i l l y i e l d a f i n i t e magnetic f i e l d everywhere except a t a d i s c r e t e s e t o f p o i n t s . The conclusion from t h i s l i n e o f thought i s t h a t t h e r e i s no way t o i d e n t i f y an x - p o i n t from l o c a l considerations. An x - p o i n t i s t h e consequence o f g l o b a l r e 1a t i ons. Returning t o the tokamak c o n f i g u r a t i o n , t h e s i n g u l a r i t y t h a t produced t h e xp o i n t i s associated w i t h a closed f i e l d l i n e . I n Fig. 3 t h e x - p o i n t s are t h e closed f i e l d l i n e s t h a t l i e a t t h e j u n c t u r e o f d i f f e r e n t f l u x c e l l s . T h i s i s t h e general d e s c r i p t i o n o f a tokamak x - p o i n t . Cowley [3] has discussed c o n f i g u r a t i o n s i n which reconnection and x - p o i n t beh a v i o r occurs w i t h o u t a closed f i e l d l i n e o f t h e tokamak type. From h i s work i t i s c l e a r t h a t t h e r e i s another k i n d o f x - p o i n t . This c o n f i g u r a t i o n has n o t been thoroughly e x p l o i t e d , even though i t i s probably very common i n t h e magnetosphere, t h e sun, and elsewhere. B r i e f l y , these x - p o i n t s are t h e magnetic l i n e s t h a t connect p o i n t s a t which t h e magnetic f i e l d vanishes. This w i l l be d i s cussed i n more d e t a i l i n t h e next few paragraphs. I f the magnetic f i e l d p o i n t s i n every d i r e c t i o n i n t h e neighborhood o f a p o i n t a t which t h e f i e l d vanishes, a p e r t u r b i n g f i e l d can move t h e n u l l p o i n t , b u t cannot destroy it. Consider the l o c a l behavior around t h e n u l l p o i n t . I f t h e coordinates a r e chosen so t h a t t h i s p o i n t l i e s a t t h e o r i g i n , t h e l o c a l magnetic can be given i n terms o f a constant m a t r i x , 6B,

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The c o n d i t i o n t h a t tJ be divergence f r e e becomes

T r (6B) = 0 From t h i s i t f o l l o w s t h a t two eigenvalues o f 6B have one s i g n and t h e t h i r d has t h e o t h e r sign. Thus t h e magnetic n u l l s can be d i v i d e d i n t o two classes, those f o r which t h e odd eigenvalue i s p o s i t i v e , and those f o r which i t i s negative. F o l l o w i n g Cowley, we w i l l c a l l t h e f i r s t c l a s s t y p e A n u l l s , and t h e second c l a s s t y p e 6. Next consider t h e l i n e s t h a t a s y m p t o t i c a l l y approach o r d i v e r g e from t h e n u l l p o i n t , The eigenvectors o f 6B d e f i n e t h r e e such l i n e s . F u r t h e r , consider t h e plane d e f i n e d by the two eigenvectors whose corresponding eigenvalues have t h e same sign. Then a l l magnetic l i n e s t h a t a s y m p t o t i c a l l y l i e i n t h i s p l a n e e i t h e r approach type A n u l l p o i n t s o r d i v e r g e from t y p e B n u l l p o i n t s . Extending t h i s plane by f o l l o w i n g l i n e s o f f o r c e then d e f i n e s a two-dimensional m a n i f o l d o f l i n e s t h a t are asymptotic t o a given n u l l . We w i l l c a l l t h i s m a n i f o l d t h e sheet associated w i t h the given n u l l . Now consider t h e i n t e r s e c t i o n o f two sheets. Since each sheet i s composed o f l i n e s o f f o r c e , t h e magnetic f i e l d cannot have a component p e r p e n d i c u l a r t o e i t h e r sheet. Thus t h e l i n e o f i n t e r s e c t i o n o f two sheets must be a magnetic f i e l d l i n e . Since each l i n e i n one sheet must d i v e r g e from a 6 n u l l , and each l i n e i n t h e o t h e r sheet must converge toward an A n u l l , t h i s l i n e o f i n t e r s e c t i o n must be a n u l l l i n e .

A sketch o f t h e near i n t e r s e c t i o n o f two sheets, such as t h i s ,

I F i g . 5. A sketch o f t h e near i n t e r s e c t i o n o f two sheets, i l l u s t r a t i n g t h a t n u l l t o n u l l l i n e s a r e formed i n p a i r s . can e a s i l y convince one t h a t n u l l t o n u l l l i n e s a r e c r e a t e d o r destroyed i n pairs.

To show t h a t n u l l t o n u l l l i n e s l i e a t t h e j u n c t u r e o f d i f f e r e n t f l u x regions, and thus q u a l i f y as x-points, r e q u i r e s some complex three-dimensional v i s u a l i z ation. Cowley worked o u t a p a r t i c u l a r example, a u n i f o r m f i e l d orthogonal t o t h e a x i s o f a p o i n t d i p o l e . They can be c a l l e d t h e s o l a r wind and e a r t h f i e l d s , respectively. A sketch o f a cross s e c t i o n o f t h i s c o n f i g u r a t i o n through t h e n u l l p o i n t s i s shown i n Fig. 6.

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Fig. 6. A schematic i l l u s t r a t i o n o f t h e n u l l l i n e s r e s u l t i n g u n i f o r m f i e l d orthogonal t o the a x i s o f a p o i n t d i p o l e . The n u l l t o n u l l l i n e s l i e on a c i r c l e whose plane i s orthogonal t o t h e paper. C l e a r l y , t h e sheet o f l i n e s t h a t go i n t o t h e n u l l p o i n t A cannot decide whether t o d i v e i n t o the e a r t h , o r wander o u t i n t o space, and thus wind up a t A. Those t h a t diverge f r o m 6 have the same problem w i t h t h e opposite o r i e n t a t i o n . Thus these sheets d i v i d e f l u x bundles t h a t a r e d e f i n e d by t h e asymptotic behavior o f t h e f i e l d l i n e s t h a t compose t h e bundle. The n u l l t o n u l l l i n e s a r e common t o b o t h sheets, and thus a r e a t t h e j u n c t u r e o f a l l t h e f l u x bundles. I n this sense, they should be q u i t e acceptable x - p o i n t s . The program o f i n v e s t i g a t i n g x - p o i n t behavior on n u l l t o n u l l l i n e s remains t o be c a r r i e d o u t i n d e t a i l . No doubt a thorough i n v e s t i g a t i o n would modify t h e p i c t u r e suggested here. S t i l l , the idea t h a t n u l l t o n u l l l i n e s are x-points would seem t o be based on s o l i d i n t u i t i o n . The events t h a t happen when two n u l l t o n u l l l i n e s are t h r u s t i n t o existence by t h e i n t e r s e c t i o n o f two sheets m i g h t be p a r t i c u l a r l y i n t e r e s t i n g .

A c e n t r a l problem o f x - p o i n t theory i s t o e x p l a i n how f a s t t h e i s l a n d s o f Fig. 3 can f a t t e n up. To what e x t e n t does t h e x - p o i n t c o n f i g u r a t i o n impose narrow d i s s i p a t i o n l a y e r s o r shocks t h a t promote r a p i d b r e a k i n g o f t h e t o p o l o g i c a l c o n s t r a i n t s o f t h e E u l e r equations? T h i s i s a l a r g e and c o n t r o v e r s i a l subject. I t i s impossible t o do j u s t i c e t o i t here, b u t i t cannot be ignored i n t h i s overview e i t h e r . A f a i r l y r e c e n t review paper has been p u b l i s h e d by Vasyliunas

C41. A c e n t r a l f e a t u r e o f an x - p o i n t t h a t i s r e l a t e d t o t h i s q u e s t i o n i s t h e angle between the two bars t h a t compose t h e x. The component o f ,B i n t h e plane o f t h e page can be w r i t t e n i n terms o f a stream f u n c t i o n ,

The stream f u n c t i o n can be decomposed i n t o a p a r t t h a t i s imposed from outside, and p a r t t h a t i s due t o c u r r e n t f l o w i n g a t t h e x - p o i n t ,

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Fig. 7. x-point.

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A schematic i l l u s t r a t i o n o f the tendency t o form a c u r r e n t sheet o f an

where I' i s t h e magnitude o f t h e imposed quadrapole f i e l d and j i s the c u r r e n t density a t the x-point. Thus i f t h e x - p o i n t s i t s i n a vacuum f i e l d t h e bars meet a t r i g h t angles. Larger c u r r e n t d e n s i t i e s can be accommodated a t an xp o i n t i f they l i e on a bar, so t h a t t h e o u t e r p a r t s c o n t r i b u t e t o t h e imposed quadrapole f i e l d a t t h e x-point, Fig. 7. Thus any movement o f f l u i d toward and through an x-point, c a r r y i n g magnetic f i e l d l i n e s w i t h it, i s going t o a c t t o close t h e angle o f t h e x, and create a c u r r e n t sheet. Consequently, t h e r e w i l l be l a r g e forces on the f l u i d i n t h i s region. Here we have a region o f nonlinear flow. It seems reasonable t h a t t h e r e should be shock waves around. This was suggested by Petschek [5] i n a famous paper some years ago. He drew a p i c t u r e as i n Fig. 8, where t h e dotted l i n e s denote shocks.

Fig. 8. point.

A schematic i l l u s t r a t i o n o f the shock wave f r o n t s formed near an x-

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The best recent work on t h i s c o n f i g u r a t i o n t h a t I know o f has been done by Hamieri [6]. I am n o t going t o get i n t o t h i s , b u t o n l y deal w i t h one s l i g h t l y tangential aspect. Dealing w i t h a f l u i d l i k e a i r , one i s comfortable considering a shock as a l i n e . Shock widths are o f the order o f a mean-free path, which i s smaller than any length o f i n t e r e s t . For problems dealing w i t h f l o w through x-points, however, t h a t i s n o t so clear. The shock s t r u c t u r e may engulf and transform a s i g n i f i c a n t volume, p a r t i c u l a r l y near t h e x-point where several shocks converge. S i m i l a r problems have a r i s e n i n plasma physics before. Crude estimates o f t h e thickness o f shocks o f t h e f a s t magnetosonic wave mode y i e l d e d unreasonable shock dimensions. Much t h e o r e t i c a l and experimental e f f o r t has been invested i n a thorough study o f the actual shock s t r u c t u r e associated w i t h t h i s mode. T h i s r e s u l t e d i n t h e discovery o f the " c o l l i s i o n l e s s f a s t magnetosonic shock." A s i m i l a r e f f o r t applied t o the slow magnetosonic mode might produce very i n t e r -

e s t i n g r e s u l t s . Slow magnetosonic shocks are no doubt very d i f f e r e n t from f a s t magnetosonic shocks, since the waves themselves a r e very d i f f e r e n t . A major d i f f i c u l t y w i t h the Petschek mechanism f o r f l o w through an x - p o i n t i s t h a t we do n o t know w i t h c e r t a i n t y what i s meant by t h e shock l i n e s drawn i n t h i s p i c t u r e . There i s one other aspect o f i s l a n d formation i n a tokamak t h a t w i l l be t r e a t e d i n t h i s overview. That i s the question o f wandering f i e l d l i n e s . I n drawing Fig. 3, i t was assumed t h a t magnetic f i e l d l i n e s l a y on surfaces. That i s n o t always true. There are configurations i n which a f i e l d l i n e , on successive i n t e r a c t i o n s o f the cross section shown i n Fig. 3, w i l l wander e r r a t i c a l l y over the page. This i s bad news f o r tokamaks. F l u i d and heat f l o w more o r l e s s unimpeded along f i e l d l i n e s , so a tokamak w i t h such a c o n f i g u r a t i o n i s n o t a good containment device. Following a magnetic f i e l d l i n e around a tokamak from one i n t e r s e c t i o n o f a cross s e c t i o n t o another defines a mapping o f the cross s e c t i o n onto i t s e l f . This map conserves f l u x , since a given area i s mapped i n t o another w i t h t h e same flux. Conservation o f f l u x d i f f e r s o n l y t r i v i a l l y from t h e conservation o f area, and thus i s equivalent t o Poincare's i n v a r i a n t f o r Hamiltonian systems. Hence the t r a c i n g o f f i e l d l i n e t r a j e c t o r i e s i s c l o s e l y r e l a t e d t o a wide var i e t y o f mechanical and c e l e s t i a l problems, and r e s u l t s from one f i e l d can be applied t o another. As a r e s u l t , one can make r a t h e r general statements about t h e behavior o f magn e t i c f i e l d l i n e s t h a t are t r u e o f n e a r l y a l l c o n f i g u r a t i o n s s u b j e c t t o t h e c o n s t r a i n t t h a t v-_B=O. Consider, f o r example, an i d e a l tokamak as i l l u s t r a t e d i n Fig. 2. Here t h e f i e l d l i n e s loop around t o r o i d a l surfaces w i t h a winding number t h a t v a r i e s continuously from surface t o surface. When such a configurat i o n i s perturbed w i t h an a r b i t r a r y magnetic f i e l d , each surface on which t h e winding number i s r a t i o n a l sprouts an i s l a n d s t r u c t u r e such as i l l u s t r a t e d i n Fig. 3. Thus there are dense sets o f islands o f a l l m u l t i p l i c i t i e s . However, i f the magnetic f i e l d i s f u r t h e r r e s t r i c t e d by t h e c o n d i t i o n t h a t the magnetic stress of Eq. (1) be reasonable, these i s l a n d s do n o t appear. That i s , modes t h a t produce islands o f h i g h m u l t i p l i c i t i e s are s t a b l e according t o MHD theory. O f course, t h e r e i s s t i l l the p o s s i b i l i t y o f e x c i t a t i o n o f these modes by t h e r mal f l u c t u a t i o n s , and t h a t may be a very i n t e r e s t i n g t o p i c o f i n v e s t i g a t i o n , b u t b a s i c a l l y t h e tokamak c o n f i g u r a t i o n appears t o be p r e f e r r e d experimentally. The exception t o t h i s r u l e are islands w i t h low m u l t i p l i c i t i e s , two, three, o r four, as shown i n Fig. 3. These observed and can grow t o q u i t e l a r g e amplitude. I n the remainder o f t h i s paper I w i l l speculate on t h e i r e f f e c t on t h e s t a b i l i t y o f very nearby high m u l t i p l i c i t y islands.

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Since t h e winding numbers o f t h e magnetic f i e l d l i n e s i n t h e i d e a l tokamak are continuous, t h e f i e l d l i n e t h a t closes a f t e r 25 r o t a t i o n s around t h e c r o s s s e c t i o n shown i n Fig. 3 w h i l e making 1 0 1 loops i s h i g h l y c o n t o r t e d by t h e growth o f t h e f o u r - f o l d i s l a n d . This should have a considerable e f f e c t on i t s s t a b i l i t y properties. That i s , t h e s e l f - c o n s i s t e n t c o n s t r a i n t t h a t t h e magnetic s t r e s s be acceptable i s probably l e s s c o n s t r i c t i n g i n t h i s circumstance. Thus, t h e r e should be a 1 0 1 - f o l d i s l a n d and a l l o t h e r s w i t h a nearby winding number, c l u s t e r e d around t h e f o u r f o l d i s l a n d .

A t t h e c e n t e r o f each o f these i s l a n d s i s a c l o s e d f i e l d l i n e . The i s l a n d f l u x tube winds around t h i s l i n e , something i n t h e nature o f a small, t h i n , h e l i c a l tokamak. The r a t e o f r o t a t i o n w i t h i n t h i s f l u x tube i s a q u a n t i t y o f i n t e r e s t . As i n c r e a s i n g f l u x and c u r r e n t a r e drawn i n t o t h e i s l a n d t h i s s u b s i d i a r y r o t a t i o n increases. A c r i t i c a l p o i n t i s reached when t h e c e n t e r o f t h e f l u x tube r o t a t e s a t h a l f t h e r a t e a t which t h e c e n t e r l i n e winds around t h e tokamak. Then, i n almost every case, t h e c e n t e r o f t h e f l u x tube takes t h e form shown i n Fig. -9.

Fig. I l l u s t r a t i o n o f t h e formation o f an x - p o i n t , and subsequer doubl.,,,d b i f u r c a t i o n , i n an i s l a n d f l u x tube i n a tokamak.

period

That i s , t h e c e n t r a l magnetic l i n e becomes an x - p o i n t , and a new c e n t r a l l i n e o f t w i c e t h e l e n g t h i s created. This i s known as a p e r i o d d o u b l i n g b i f u r c a t i o n . F o l l o w i n g t h i s argument on, i f t h e d r i v ' i n g f o u r f o l d i s l a n d f u r t h e r increases i n size, more f l u i d and f l u x w i l l be f o r c e d i n t o t h e 1 0 1 - f o l d i s l a n d , and i n t o t h e It t o o can undergo a p e r i o d d o u b l i n g b i f u r c a t i o n . i n t e r i o r 202-fold island. As one might expect here a t Los Alamos, i n general systems, w i t h o u t t h e magneti c s t r e s s c o n s t r a i n t , an i n f i n i t e sequence o f d o u b l i n g b i f u r c a t i o n s a r e produced by a f i n i t e d r i v i n g p e r t u r b a t i o n , and these b i f u r c a t i o n s have a u n i v e r s a l behavi o r . I t remains t o be seen i f t h e s t r e s s c o n s t r a i n t e f f e c t s t h i s u n i v e r s a l i t y . I would guess t h a t i t won't.

NONLINEAR PLASMA DYNAMICS BELOW THE CYCLOTRON FREQUENCY

433

A t each stage o f t h e b i f u r c a t i o n new x-points, l i n e s from which a l l o t h e r l i n e s diverge, are created. I n the l i m i t o f i n f i n i t e b i f u r c a t i o n s , then, a l l l i n e s diverge from each other. This i s t h e e s s e n t i a l c h a r a c t e r o f wandering f i e l d l i n e s . Thus as the f o u r f o l d i s l a n d grows i t w i l l be surrounded by an expanding sea o f wandering f i e l d l i n e s . When t h i s sea laps c l e a r o u t t o t h e edge o f t h e tokamak containment region, t h e r e i s a catastrophe. C e r t a i n l y t h e sudden spreadi n g o f contained plasma i n a " d i s r u p t i v e i n s t a b i l i t y ' ' i s a major c o n s t r a i n t on tokamak operation. I f my i n t e r p r e t a t i o n i s c o r r e c t , and t h e observed d i s r u p t i v e spreading i s due t o sudden onset o f wandering f i e l d s , u n i v e r s a l p e r i o d d o u b l i n g b i f u r c a t i o n s are an important f e a t u r e o f tokamaks.

REFERENCES

1.

A. T. Patera and S. A. Orszag, t h i s volume.

2.

El. R. Suydam, Proc. 2nd I n t . Conf. Peaceful Uses o f Atomic Energy, Geneva, Vol. 31, p. 157 (1950).

3.

S. W. H. Cowley, Radio Science

4.

V.

5.

H. E. Petschek, NASA Spec. Publ. SP 50, 425 (1964).

6.

E. Hamieri, J. Plasma Phys.

8,

903 (1973).

M. Vasyliunas, Rev. Geophys. Space Phys.

21, 245 (1979).

13, 303

(1975).

This Page Intentionally Left Blank

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North.Holland Publishing Company, 1982

435

Turbulence and Self-consistent F i e l d s i n Plasmas

D. Pesme and D. DuBois Theoretical D i v i s i o n University o f California Los Alamos National Laboratory Los Alamos, NM 87545 USA This paper i s concerned w i t h the r o l e o f self-consistency o f the e l e c t r i c f i e l d i n 1-0 plasma turbulence. We f i r s t show t h a t i n the non-self consistent e l e c t r i c f i e l d problem excell e n t agreement i s found between numerical experiments and quasi1 inear theory whenever t h e imposed e l e c t r i c f i e l d Fourier components have random phase. A discrepancy i s e x h i b i t e d between q u a s i l i n e a r p r e d i c t i o n and numerical simulations i n t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d case. This discrepancy i s explained by t h e c r e a t i o n o f a long c o r r e l a t i o n time o f the e l e c t r i c f i e l d r e s u l t i n g from a strong wave-particle i n t e r a c t i o n . A comparison i s made between q u a s i l i n e a r and renorma1i z e d propagator theories, and the Dupree "Clump" theory. These three theories are found t o be s e l f - c o n t r a d i c t o r y i n the regime o f strong wave-particle i n t e r a c t i o n because they make an a p r i o r i quasi-gaussian assumption f o r t h e e l e c t r i c f i e l d . The D i r e c t I n t e r a c t i o n Approximation i s then shown t o be a closure t a k i n g i n t o account i n a s e l f - c o n s i s t e n t way nongaussian e f f e c t s . INTRODUCTION

This paper i s concerned w i t h the r o l e o f s e l f - c o n s i s t e n t electromagnetic f i e l d s i n the theory o f t u r b u l e n t plasmas. I n recent years dramatic progress has occurred i n our understanding o f the motion o f charged p a r t i c l e s i n prescribed electromagnetic f i e l d s : We have learned t h a t t h e r e are thresholds f o r the onset of chaotic motion which are governed by conditions such as resonance ( o r i s l a n d ) overlap; the chaotic motion i s characterized by d i f f u s i o n o f p a r t i c l e s i n veloci t y and by t h e exponential divergence o f i n i t i a l l y nearby t r a j e c t o r i e s . Above the stochastic threshold p a r t i c l e s can wander a l l over and indeed t h e i r t r a j e c t o r i e s appear t o completely f i l l c e r t a i n regions o f phase space. The d e t a i l e d studies o f i n t r i n s i c s t o c h a s t i c i t y have concentrated on t h e prescribed f i e l d problem. I n t h i s paper we w i l l t r y t o make a p r e l i m i n a r y assessment o f t h e importance o f the self-consistency o f the e l e c t r i c f i e l d s i n the t u r b u l e n t o r c h a o t i c behavior o f plasmas. We w i l l f i r s t review i n Section 2 some numerical studies o f presc r i b e d f i e l d models i n parameter regimes w e l l above t h e stochastic threshold. I n c e r t a i n cases i t i s known t h a t , even f o r a s i n g l e r e a l i z a t i o n o f t h e imposed f i e l d , a s t a t i s t i c a l theory on the l e v e l o f t h e f a m i l i a r q u a s i l i n e a r theory provides an accurate d e s c r i p t i o n o f the d i f f u s i o n i n v e l o c i t y space when compared t o numerical solutions. I t i s l e s s well-known t h a t t h e exponential separation o f o r b i t s i s l i k e w i s e accurately p r e d i c t e d by a s t a t i s t i c a l theory o f t h e twop o i n t c o r r e l a t i o n function. I n Section 3 we review t h e r e s u l t s o f s t a t i s t i c a l theories which involve the ensemble average over many r e a l i z a t i o n s o f t h e imposed f i e l d s ; i n p a r t i c u l a r the imposed e l e c t r i c a l f i e l d s a r e considered t o be quasi-

D. PESME. D. F. DUEOlS

436

gaussian random v a r i a b l e s , i n a sense which w i l l be d e f i n e d i n Section 2. Usua l l y t h i s p r o p e r t y i s r e a l i z e d by assuming random phases f o r each mode. We exami n e the c o n d i t i o n s under which t h e p r e d i c t i o n s o f such a s t a t i s t i c a l t h e o r y are a b l e t o describe t h e p r o p e r t i e s o f a s i n g l e r e a l i z a t i o n such as a numerical simul a t i o n run. I n Section 4 we w i l l review some aspects o f t h e s e l f - c o n s i s t e n t f i e l d case. S e l f - c o n s i s t e n t numerical s i m u l a t i o n s i n t h e q u a s i l i n e a r parameter regime a r e n o t i n agreement w i t h t h e imposed f i e l d s t a t i s t i c a l t h e o r i e s . The reason i s t h e f o l l o w i n g : t h e s e l f - c o n s i s t e n t Maxwell equations a r e n o n l i n e a r and impose phase c o r r e l a t i o n s between modes. T h i s i n v a l i d a t e s t h e random phase o r quasigaussian f l u c t u a t i o n assumption and introduces new c o r r e l a t i o n times i n t o t h e problem; t h i s i s shown i n an i t e r a t i v e c a l c u l a t i o n i n Section 5. The imposed f i e l d s t a t i s t i c a l t h e o r i e s cannot be made s e l f - c o n s i s t e n t a p o s t e r i o r i because t h e y n e g l e c t important nongaussian c o r r e l a t i o n s . I n a sense ( t o be discussed) t h e assumption t h a t E i s a gaussian random v a r i a b l e i s an assumption o f maximum chaos. Since s e l f - c o n s i s t e n c y i n v a l i d a t e s t h i s assumption then a s e l f - c o n s i s t e n t system must have l e s s than maximum chaos. Self-consistency i m p l i e s a k i n d o f s e l f - o r g a n i z a t i o n o f t h e chaos. The f o r m u l a t i o n o f a s e l f - c o n s i s t e n t , s t a t i s t i c a l d e s c r i p t i o n o f a t u r b u l e n t plasma i s being pursued w i t h renewed v i g o r a t t h e p r e s e n t time. An approach t h a t shows considerable promise i n v o l v e s t h e a p p l i c a t i o n o f t h e s t r o n g turbulence approximation known as t h e d i r e c t i n t e r a c t i o n approximation ( D I A ) t o Vlasov turbulence. T h i s approximation i n c o r p o r a t e s c e r t a i n nongaussian c o r r e l a t i o n s i n t o a s t a t i s t i c a l t h e o r y t h a t r e t a i n s t h e b a s i c conservation laws and o t h e r r e a l i z a b i l i t y p r o p e r t i e s . I n Section 5 we w i l l examine c e r t a i n f e a t u r e s o f t h i s theory and make some comparisons w i t h t h e imposed f i e l d t h e o r i e s .

REVIEW OF INTRINSIC STOCHASTICITY STUDIES I N PRESCRIBED FIELDS The imposed f i e l d s t u d i e s seek t o e s t a b l i s h t h e p r o p e r t i e s o f s i n g l e part i c l e t r a j e c t o r i e s i n p r e s c r i b e d e l e c t r i c and magnetic f i e l d s . Here we w i l l r e s t r i c t our c o n s i d e r a t i o n s t o one-dimensional e l e c t r o s t a t i c models w i t h e l e c t r i c f i e l d s o f t h e form

E(x,t) =

IEklexp(iek) expi[kx-wn(k)t] n=l

(2.1)

k

We consider a f i n i t e system o f l e n g t h L so t h a t t h e k modes a r e d i s c r e t e and we a l l o w t h e p o s s i b i l i t y o f a s e t o f frequencies wn(k) associated w i t h each mode, where

1 5 n 5 Nm.

For s i m p l i c i t y we w i l l w r i t e most o f our equations e x p l i -

c i t l y f o r t h e case Nm = 1 and wl(k)

= w(k).

Newton's equations f o r t h e motion

o f a p a r t i c l e o f charge q and mass m i n t h i s f i e l d a r e o f course: x = v

=

E(x,t)

=

q

(2.2)

I Ekl

expi [ k ~ - w ( k ) t + 8 ~ ]

and a r e h i g h l y n o n l i n e a r . We f i r s t focus on a s i n g l e resonance associated w i t h a g i v e n mode ko; Newton's equation f o r a p a r t i c l e i n t e r a c t i n g w i t h t h i s s i n g l e mode can be w r i t t e n

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

x = v

431

(2.3)

= 9 E

sin[kox-u(ko)t] ko

I t i s convenient t o transform t o t h e coordinates X,V

i n t h e wave frame:

x = v (ko)t + X

4

(2.4) v = v (k ) + V $

0

where v ( k ) = w(ko)/ko i s the wave phase-velocity. $

0

I n t h i s frame the equations

o f motion are x = v

(2.5)

\j = 9 E

s i n koX ko

*

These are the well-known equations f o r t h e nonlinear pendulum whose s o l u t i o n i s e x a c t l y known. The phase space i s d i v i d e d i n t o t r a p p i n g and passing regions as shown i n Fig. 1 by t h e s e p a r a t r i x whose equation i s

v=*

[

2 q+]I::'

cos

y2

"T

Fig. 1

Fig. 1. S e p a r a t r i x i n phase-space f o r a s i n g l e resonance.

The " t r a p p i n g h a l f width" o f the i s l a n d s o f t h e s e p a r a t r i x i s (2.7)

I n the case o f many modes we can l a b e l each resonance by i t s C..x.e v e l o c i t y and t r a p p i n g h a l f - w i d t h as i f i t were an i s o l a t e d resonance. The d i s t a n c e i n v between t h e phase v e l o c i t i e s o f the i s o l a t e d resonances i s 6v = Vg(kn+l) =

& ve(k)

-v ( k 1 $ n x 2n

D. PESME, D. F. DUBOIS

430

where k = 2 d L . For very small amplitudes t h e resonances, a t a f i r s t approximation,"can be considered as i s o l a t e d and we have the p i c t u r e o f superimposed resonances shown i n Fig. 2.

VA

Fig. 2.

Approximate resonance domains i n phase space f o r several resonances such t h a t kS << 1.

0

The work o f Chirikov parameter kS:

x

1 showed t h a t i t was useful t o d e f i n e t h e s t o c h a s t i c i t y

He showed t h a t f o r k << 1 the motion i s r e g u l a r (except i n small disconnected regions o f phase spa&e) b u t f o r ks > 1 i t i s c h a o t i c i n the phase space domain f o r which the v e l o c i t y belongs t o the i n t e r v a l o f phase v e l o c i t i e s o f t h e resonances. As an example we d i s p l a y the r e s u l t s o f Escande and Dovei12 f o r the case o f two we1 1-separated resonances w i t h t h e equations o f motion

= M1sin x + M p i n k(x-t) where v (k) = 1 and v (0) = 0.

4

4

They computed t h e o r b i t s x ( t ) , v ( t ) as a func-

t i o n o f t from some i n i t i a l conditions xo, vo a t t = 0.

The r e s u l t s are summar-

i z e d i n Fig. 3 f o r the case kS << 1. The primary resonances are d i s t o r t e d as evidenced by t h e d e v i a t i o n o f t h e separatices from the sinusoidal shape. In addl t i o n , new secondary resonances appear. The t r a j e c t o r i e s near t h e separat i c e s become fuzzy forming a s t o c h a s t i c layer. The important r e s u l t i s t h a t there s t i l l e x i s t s d e f i n i t e passing t r a j e c t o r i e s and these i s o l a t e t h e phase space i n the sense t h a t a p a r t i c l e cannot t r a v e l from one s e t o f t r a p p i n g i s l a n d s t o another. Thus t h e motion i s s t a b l e and I v ( t ) v I i s bounded so a p a r t i c l e As ks i s increased c a n ' t wander f a r i n v e l o c i t y space; thus t h e r e i s no'diffusion.

-

(but ks

2 1) t h e width o f the s t o c h a s t i c l a y e r grows as does t h e s i z e o f the

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

secondary resonances.

439

There e x i s t s a value o f kS (near u n i t y ) where a l l t h e

s t o c h a s t i c l a y e r s j o i n together a t which p o i n t t h e system undergoes a gross s t o c h a s t i c i n s t a b i l i t y i n which p a r t i c l e s can wander over t h e phase space between the resonances. C h i r i k o v o r i g i n a l l y p o s t u l a t e d t h a t t h e s t o c h a s t i c t h r e s h o l d occurred a t ks = 1. More r e c e n t work3”

has shown t h a t t h e p r e c i s e

value o f t h e t h r e s h o l d i s somewhat l e s s t h e u n i t y . I n t h e f o l l o w i n g we w i l l assume kS >> 1, w e l l above the c h a o t i c threshold. We w i l l n o t attempt a p r e c i s e d e f i n i t i o n o f chaos here; s u f f i c e i t t o say f o r our purposes, t h e c h a o t i c regime i s c h a r a c t e r i z e d by d i f f u s i o n o f p a r t i c l e s i n v e l o c i t y and exponential d i v e r gence o f nearby o r b i t s . The former i s a very p r a c t i c a l consequence o f chaos t h e l a t t e r i s a more t h e o r e t i c a l consequence r e l a t e d t o t h e r a t e a t which informat i o n i s l o s t by t h e system as c h a r a c t e r i z e d by the s o - c a l l e d Kolmogorov entropy.

Fig. 3.

D i s t o r t i o n o f primary resonances and appearance o f secondary resonances f o r kS = 0.4 (from (Escande e t a l . , Ref. 2).

Fig. 3 There have been many numerical studies o f these q u a n t i t i e s f o r v a r i o u s models o f t h e imposed f i e l d o f t h e form i n equation (2.2) which appear t o f a l l i n t o t h r e e classes:

i) The f i r s t c l a s s considers o n l y a s i n g l e wavenumber b u t associates an i n f i n i t e numer o f frequencies (harmonics) w i t h t h i s wave number ( i . e . , wn(k) = nwo and N, = =. This c l a s s includes t h e famous C h i r i k o v - T a y l o r 4 5 mapping and has been i n v e s t i g a t e d by Rechester and White and others. ii) A second c l a s s o f models has an i n f i n i t e ( o r l a r g e ) number o f modes k each w i t h an i n f i n i t e number o f harmonics o f a fundamental frequency ( i . e . , wn(k) = nw(k) and Nm = =). Such models have been i n v e s t i g a t e d by Rechester, Rosenbluth and 8

Molvig

e t al.

iii) The class o f models c l o s e s t t o our i n t e r e s t s have an i n f i n i t e ( o r l a r g e ) spectrum o f k values each w i t h a d e f i n i t e frequence w(k) ( i . e . , N, =

D. PESME, D.F. DUEOIS

440

= w(k)).

1 and wl(k)

Flynn9 and D o v e i l . l 0

Simulations o f t h i s case have been c a r r i e d o u t by I n these numerical c a l c u l a t i o n s values o f l E k l f o r

Eq. 2.2 were assigned and t h e phases Ok were randomly chosen between 0 and 2n w i t h no c o r r e l a t i o n between d i f f e r e n t values o f k. T h i s i s u s u a l l y This procedure generates a c a l l e d t h e Random Phase Approximation (RPA). s t o c h a s t i c process which i s a s y m p t o t i c a l l y e q u i v a l e n t t o a gaussian process 12 i n t h e l i m i t o f a l a r g e number o f modes." S. P. Gary and D. Montgomery,

K. N. Graham and J. A. Fejer13 a l s o d i d s i m i l a r c a l c u l a t i o n s , though they used somewhat d i f f e r e n t processes than random phases i n order t o generate a quasigaussian e l e c t r i c f i e l d . I n t h e f o l l o w i n g , f o r s i m p l i c i t y , we w i l l consider i n t h e imposed e l e c t r i c f i e l d case o n l y t h e models w i t h random phase F o u r i e r components. The s i n g l e p a r t i c l e t r a j e c t o r i e s x(xo, vo, t) and v(xo, vo, t) were computed n u m e r i c a l l y from i n i t i a l c o n d i t i o n s xo, vo a t t = 0. V e l o c i t y d i f f u s i o n was s t u d i e d by computing t h e s p a t i a l average over t h e i n i t i a l p o s i t i o n s o f t h e squared v e l o c i t y d e v i a t i o n X

2 Av (t)

(V(XO,VO,t)

-

vo)

2O

(2.10)

where we d e f i n e the s p a t i a l average o f any f u n c t i o n +(x) by

Results f r o m these numerical c a l c u l a t i o n s showed t h a t : i) 3 was p r o p o r t i o n a l t o t (except f o r very s h o r t times). The observed v e l o c i t y d i f f u s i o n c o e f f i c i e n t i n t h e numerical experiments can thus be d e f i n e d as Dexp ( d i f ) h 2ti

qTj

(2.11)

ii) The observed d i f f u s i o n c o e f f i c i e n t was i n good agreement w i t h t h e pred i c t i o n o f quasilinear theory (2.12) where

DQaL = m

'c k

IEkI2 n

(t

dr exp i [wn(k)

-

kv]r

(2.13)

0

The q u a s i l i n e a r theory i s a s t a t i s t i c a l theory, averaged over many r e a l i z a t i o n s o f the phases, 0 , which are assumed t o be random. We w i l l discuss s t a t i s t i c a l t h e o r i e s i n t h e k e x t section. S i m i l a r good agreement between t h e observed d i f f u s i o n c o e f f i c i e n t and t h e pred i c t i o n s o f q u a s i l i n e a r theory was found i n t h e work o f Rechester, Rosenbluth and White

7 as i t can be seen i n Fig. 4.

For t h e i r model t h e q u a s i l i n e a r pre-

d i c t i o n i s DQL = c2M/4 where c i s t h e amplitude o f t h e F o u r i e r components and M i s t h e number o f harmonics (Nm i n o u r n o t a t i o n ) .

44 1

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

f

Fig. 4.

Comparison between numerical experiments (dots 0) and the quasi 1i n e a r p r e d i c t i o n ( s o l i d l i n e ) f o r the value o f t h e d i f f u s i o n c o e f f i c i e n t (from Rechester e t a l . , Ref. 2).

I

,

,

,

,

,

,

Nuatrirol Rinla

D

Id'

,

* O I

ai

.01

a

10

1.0

Fig. 4 Rechester, Rosenbl u t h and White a1 so computed t h e s o - c a l l e d "Kolmogorov entropy" d e f i n e d as h =

lim

1

t-

t

- . .

l i m In do+O

11

(2.14)

do

where d i s t h e "distance" between two nearby o r b i t s .

We can take t h i s t o be

(2.15)

where x1 = x ( t , xo,l

vo),l

v1 = v ( t , x,ol

vo)l

and x2 = x ( t , xO2, vO2), v2 =

x ( t , xo2, vo2) corresponding t o d i f f e r e n t b u t nearby i n i t i a l c o n d i t i o n s a t t =

0; do = d ( t = 0).

The n o r m a l i z a t i o n constants i n t h i s expression a r e v+ = %(vl

v ) and k+ = k(v+); k(v) i s t h e s o l u t i o n o f t h e i m p l i c i t equation w(k(v)) = 2 k(v)v. P o s i t i v e values o f h are i n t e r p r e t e d t o be a s i g n a t u r e o f c h a o t i c behav i o r . Roughly speaking, i n p h y s i c a l terms, t h i s d e f i n i t i o n i m p l i e s t h a t f o r s h o r t times d ( t ) 2 d exp h t and t h i s i m p l i e s a loss o f memory o f i n i t i a l condit i o n s and a s u f f i c i e g t c o n d i t i o n f o r m i x i n g and e r g o d i c i t y . +

The r e s u l t s o f the numerical experiments o f Rechester e t a l . ,6'7 a r e shown i n f i g u r e 5. They showed t h a t h e x i s t s and i s independent o f i n i t i a l condiexP t i o n s i f do i s s u f f i c i e n t l y small and obtaiped e x c e l l e n t agreement w i t h t h e p r e d i c t i o n h Q e Lobtained from a s t a t i s t i c a l theory assuming a quasigaussian e l e c t r i c f i e l d . This theory, which i s discussed i n t h e n e x t s e c t i o n provides an e x p l i c i t r e l a t i o n o f h Q a L *t o t h e q u a s i l i n e a r d i f f u s i o n c o e f f i c i e n t h Q e L *= 0.36

(

Q*L3

k+D

13'

(2.16)

442

D. PESME, D. F. DUBOlS

h

F i g . 5.

Comparison between numeric a l experiments (dots 4) and t h e q u a s i l i n e a r p r e d i c t i o n ( s o l i d l i n e ) f o r the value o f the Kolmogorov entropy ( f r o m Rechester e t a l . , Ref. 6, 7).

Numoical hints

Fig. 5 The numerical experiments showed t h a t t o e x c e l l e n t accuracy hQ' L' = 0.36 (k+D 2 Q. L.) 13'

(2.17)

hexp = STATISTICAL TURBULENCE THEORIES FOR PRESCRIBED RANDOM FIELDS

We next g e n e r a l i z e our considerations t o a system o f many-charged p a r t i c l e s I f t h e f i e l d s a r e p r e s c r i b e d , we have a system moving i n e l e c t r o s t a t i c f i e l d s . o f independent p a r t i c l e s whose t r a j e c t o r i e s have t h e p r o p e r t i e s discussed i n Section 2. The many p a r t i c l e s i t u a t i o n i s c o n v e n i e n t l y described by t h e one p a r t i c l e phase space d i s t r i b u t i o n f u n c t i o n f ( x , v , t ) which evolves a f t e r a t i m e t from some i n i t i a l d i s t r i b u t i o n fo(x,v) a t t = 0 according t o t h e formula f(x,v,t)

[

= I d x o I d v o 6 x-x(t,xo,vo)]

where x(t,xo,vo) Section 2.

[at +

and v(t,xo,vo)

6 [v-v(tlxolvo)

1

f o (x,v)

(3.1)

are j u s t t h e p o s i t i o n and v e l o c i t y discussed i n

The phase-space d e n s i t y evolves according t o t h e L i o u v i l l e equation: v

L

ax

+

$ E(x,t)a,,lf(x,v,t)

= 0

(3.2)

J

which expresses t h e conservation o f d e n s i t y along phase-space t r a j e c t o r i e s . I n t h e s e l f - c o n s i s t e n t case we can s t i l l use Eq. 3.2 i n which E(x,t) i s t h e s e l f 14 c o n s i s t e n t f i e l d determined by Poisson's equation 4.1. I n t h e p r e s c r i b e d f i e l d problem we assume E(x,t) i s a random v a r i a b l e over an ensemble each r e a l i z a t i o n o f which i s given by an expression o f t h e form (2.1). This i s t h e s o - c a l l e d s t o c h a s t i c a c c e l e r a t i o n problem based on t h e l i n e a r stoThis problem has r e c e i v e d much attention.'' We g i v e c h a s t i c equation (3.2). here t h e r e s u l t s o f t h i s a n a l y s i s i n t h e q u a s i l i n e a r regime which i n v o l v e s t h e f o l l o w i n g assumptions

i) kS >> 1

TURBULENCE AND SELF-CONSISTENTFIELDS IN PLASMAS

443

iii) quasigaussian approximation f o r e l e c t r i c f i e l d f l u c t u a t i o n s , e.g., phase Bk random and uncorrelated. Condition i) simply assumes t h a t t h e system i s w e l l above t h e s t o c h a s t i c t h r e s hold. I n c o n d i t i o n iii),t h e quasigaussian approximation i s d e f i n e d as t h e case i n which t h e c o r r e l a t i o n f u n c t i o n s o f any order a r e c h a r a c t e r i z e d by a unique c o r r e l a t i o n length. More p r e c i s e l y , throughout a l l t h i s a r t i c l e , we name quasigaussian any random process f o r which an i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n o f any order i s n o t zero o n l y i f a l l the arguments l i e i n s i d e a same i n t e r v a l o f l e n g t h Q , where Q i s t h e c o r r e l a t i o n l e n g t h o f t h e two p o i n t c o r r e l a t i o n function: This c o f i d i t i o n can be r e a l i z e d by t h e random phase approximation. I n c o n d i t i o n ii)we have d e f i n e d a c h a r a c t e r i s t i c time o f the Kolmogorov entropy;

th

o f i n i t i a l conditions i s l o s t .

T~

as t h e i n v e r s e

i s a measure o f t h e t i m e over which i n f o r m a t i o n This time, rh, must be much l e s s than t h e

time o f e v o l u t i o n o f t h e mean d i s t r i b u t i o n f u n c t i o n , T.<~>. l a t t e r as

We can estimate t h e

(3.4)

where Avf i s a measure o f the w i d t h i n v e l o c i t y space f o r which waves i n t e r a c t resonantly w i t h p a r t i c l e s and DQ’L’ 2.13.

Note t h a t DQ’L’

(q/m)‘

t h e q u a s i l i n e a r d i f f u s i o n c o e f f i c i e n t o f Eq.

<E 2>tCwhere r c = [Ak(v-V,)]-l

i s the correlation

time o f the e l e c t r i c f i e l d f l u c t u a t i o n s as seen by a t y p i c a l resonant p a r t i c l e ; We can then w r i t e V i s the group v e l o c i t y o f the wave w i t h wave number k(v). cl<E2> % ~ < ~ >=/ (t I ~J J ~ ~where ~ ) - ~wb = i s o f t e n c a l l e d t h e e f f e c t i v e bounce frequency; i . e . ,

t h e frequency o f a p a r t i -

c l e trapped i n a monochromatic wave o f amplitude <E2>% and wave number i . these d e f i n i t i o n s i t i s found t h a t c o n d i t i o n ii)can a l s o be w r i t t e n as

With (3.5)

Under these c o n d i t i o n s i t i s known16 t h a t t h e equation o f e v o l u t i o n o f t h e ensemble averaged o r mean d i s t r i b u t i o n < f > i s

a t c f > = a V D Q - ~ . ( V ) av

cf>

(3.6)

where DQeL. is given i n 2.13. The ensemble average i s over many r e a l i z a t i o n s o f t h e e l e c t r i c f i e l d s . We assume s t a t i s t i c a l homogeneity, i . e . <E(x,t)> = 0 and fo(X,V) = fo(v). An equation f o r t h e two-phase p o i n t , one time c o r r e l a t i o n f u n c t i o n
bf(x:v:t’)>

, Gf(x,v,t)

can a l s o be d e r i v e d i n t h i s l i m i t (Dupree

= f 17

).

-

,

,

I n t h e l i m i t o f small separation

i n v e l o c i t y space I k ( v + ) v - ) << sup [ ( k DQ’L’)1/3, [(k2(v+)DQ‘L’ )-1/3,.y,1]

y,],

and Ix-/v+l

t h i s equation can be w r i t t e n as:

<< i n f

0.PESME, 0.F. DUBOIS

444

(3.7) where

D12

i s the r e l a t i v e d i f f u s i o n c o e f f i c i e n t

These equations are expressed i n r e l a t i v e coordinates: v- = v - v ' , x- = x - x ' and - v+v' c e n t r i x coordinates: v+ - - assuming s p a t i a l homogeneity. The propagator G k , r o ( ~ ) t o be used i n 3.7 i s t h e usual renormalized propagator given by: (3.9) Dupree17 was t h e f i r s t t o show t h a t equation (3.7) p r e d i c t s t h a t i n i t i a l l y nearby t r a j e c t o r i e s w i l l diverge e x p o n e n t i a l l y i n time. He considered t h e source f r e e s o l u t i o n t o eqn. (3.7),

G2(x-,v-;v+,t),

and generated a closed s e t

JdxJdv x_? vlk;,. The s o l u t i o n o f 2 2t h e r e s u l t i n g equations showed t h a t <x-(t) >, increased from t h e i r

o f equations f o r the moments <x-(t)nv-(t)m>

t ) . p r e c i s e c o n t a c t between t h e i n i t i a l values as exp ([4k+ DQ * L * ] 1 /A3more

dynamical d e f i n i t i o n has been made by Rechester, Rosenbluth and White.6 Since t h e q u a n t i t y h as d e f i n e d by Eq. 2.14 i s n u m e r i c a l l y observed t o be independent o f i n i t i a l c o n d i t i o n s , these authors argue t h a t i t i s a s t a t i s t i c a l q u a n t i t y . More e x p l i c i t l y they invoke ergodic p r o p e r t i e s which p e r m i t them t o r e p l a c e t h e t i m e average In 0

]

d(t') [do

by the s t a t i s t i c a l average

dt' =

1 I n d(t) t

a <

In

do

>.

By w r i t i n g t h e q u a s i l i n e a r equation

corresponding t o t h e two-point, one-time c o r r e l a t i o n f u n c t i o n they f i n d :

(3.

lo)

These authors observed an e x c e l l e n t agreement between t h e s t a t i s t i c a l p r e d i c t i o n hQ'L' and t h e experimental value h obtained from numerical i n t e g r a t i o n o f t h e exP o r b i t s (Fig. 5). Thus t h e r e i s e x c e l l e n t agreement between t h e p r e d i c t i o n s o f t h e q u a s i l i n e a r s t a t i s t i c a l theories f o r velocity diffusion (i.e.,

DexpY D Q ' L * ) and t h e corres-

N hQ'L.) i n t h e case h ex? o f p r e s c r i b e d quasigaussian random f i e l d s when t h e c o n d i t i o n s (3.3) f o r t h e I n Section 4 we w i l l show t h e r e v a l i d i t y o f q u a s i l i n e a r theory are s a t i s f i e d . i s no corresponding agreement between standard s t a t i s t i c a l t h e o r y and t h e r e s u l t s o f c e r t a i n s e l f - c o n s i s t e n t computer s i m u l a t i o n s .

ponding p r e d i c t i o n s f o r t h e Kolmogorov entropy ( i . e . ,

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

445

We complete this section by pointing out the following question: why the results of an ensemble averaged statistical theory can accurately describe the results of a single numerical experiment; i.e., a single realization of the ensemble. The answer is really not obvious as it is sometimes considered to be; the statistical quantities DQ'L* and hQ'L* clearly do not depend on the phases, but the and h of one-numerical experiment with a given assignment of results D exP exP the phases could in principle depend on the phases. The simulations described above are essentially spatially homogeneous and explicitly calculated only spatial-averages. In particular, the spatially averaged phase space distribution f(v,t) is defined as

Pesme and Brisset18 have considered the following situation: for each realization the mode amplitudes lEkl form a smooth function of k and the phases Ok are randomly distributed and-independent of each other. For such a given realization they _have shown that f obeys the quasilinear equation (3.5) with replaced by f the conditions ( 3 . 3 ) are satisfied. More precisely they have shown that the corrections to this equation are statistically small under these conditions. The condition k >> 1 is easily shown to be equivalent to the following condition on the lsngth of the system:

L/v >> h-l or L >> !Lh

E

v

th

(3.11)

In this case during the transit time of a particle o f velocity v through the system it sees many independent partitions of the system which are equivalent to independent realizations. This is because on one hand, after a time h-l the particle has lost all memory of its initial conditions. On the other hand, the correlation length of the electric field fluctuations is ac- l/A k and when the conditions 3 . 3 are satisfied, we have Ilc << a, so that the electric field in each partition is independent. Therefore spatial averaging is then equivalent to ensemble averaging. COMPARISON OF SELF-CONSISTENT NUMERICAL SIMULATIONS WITH STANDARD QUASILINEAR THEORY Here we will review the simulations of the bump on tail instability in a onedimensional, one-component plasma which were carried out and analyzed by Adam, Lava1 and Pesme." The conditions of these simulations were those commonly In fact, the thought to be in the regime of validity of quasilinear theory.*' simulations did not follow the quasilinear predictions and this failure is now attributed to the breakdown of the random phase assumption in a self-consistent plasma. The fact that self-consistency results in qualitatively and quantitatively different behavior than in the imposed field case has also been verified in a rather different physical situation by the simulations of Bezzerides, Gitomer and Forslund." These authors investigated the production of hot electrons in resonance absorption of light. Detailed consideration of this case i s not possible here and we refer the reader to this work. In the self-consistent problem the electric field is, of course, determined by the charge density of all the particles in the system. Poisson's equation is written as:

0. PESME. 0.F. DUBOIS

446

V

*

E(x,t) = 4nq Jdvf(x,v,t)

-

4nqN

(4.1)

where f i s a smooth function14 and N i s t h e i o n d e n s i t y . The Vlasov-Poisson system o f equations thus becomes a n o n l i n e a r f i e l d theory. I n t h e s e l f - c o n s i s t e n t case t h e f i e l d E(x,t) cannot be s p e c i f i e d , b u t evolves dynamically from t h e i n i t i a l c o n d i t i o n s (fo(xlv) = f(x,v,t=O)). The bump on t a i l i n s t a b i l i t y develops from t h e i n i t i a l c o n d i t i o n s shown i n Fig. 6a. Here i s shown t h e i n i t i a l s p a t i a l averaged d i s t r i b u t i o n f(v,t=O) = The phases o f fk(v,t=O) a r e randomly chosen from one mode t o t h e fk=O(v,t=O). next. The f o u r i e r amplitudes Ek(t=O) f o r k # 0 are computed from Poisson's equation, i . e . ,

Ek(t=O)

amplitudes I E k ( t =

Fig. 6.

= (4nq/ik) Jdvfk(v,t=O)

b u t a r e n o t shown.

The i n i t i a l

0)l are shown s c h e m a t i c a l l y i n Fig. 6.a.

Time e v o l u t i o n o f t h e spaceaveraged d i s t r i b u t i o n f u n c t i o n f ( v . t ) and o f t h e e l e c t r i c f i e l d made amplitude IEkI(t),

4%

6.b

a. i n i t i a l c o n d i t i o n s , b. e a r l i e r stage: growth o f lEkl and no m o d i f i c a t i o n o f f , d. stationary f i n a l state.

/ii

6.L

0

Fig. 6

I t i s well-known t h a t t h e l i n e a r d i s p e r s i o n r e l a t i o n d e r i v e d from EO(k,w) = 0

where E ~ ( J U U ) i s t h e l i n e a r d i e l e c t r i c f u n c t i o n p r e d i c t s t h e f a m i l i a r Landau growth r a t e f o r Langmuir waves p r o v i d e d y:'L'/wp

<< 1:

P a r t i c l e s w i t h v e l o c i t i e s v near t h e phase v e l o c i t y v (k) = w(k)/k o f t h e wave

4

s e e - e s s e n t i a l l y a s t a t i c f i e l d and i n t e r a c t s t r o n g l y w i t h t h e wave. I f (avf)v=v > 0 then t h e waves grow because t h e r e i s energy t r a n s f e r from 4J

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

p a r t i c l e s t o the wave.

For the i n i t i a l c o n d i t i o n on

447

chosen by Adam e t a l . ,

t h e r e i s a band o f wave numbers whose growth r a t e s ( Y Q ~ . >~ '0) a r e very c l o s e i n value and f o r s h o r t times a broad spectrum o f waves grows up as shown i n Fig. 6b. A t l a t e r times (Fig. 6c) t h e spectrum becomes n o i s y and uneven b u t s t i l l broad and-begins t o cause d i f f u s i o n o f p a r t i c l e s i n v e l o c i t y which tends t o f l a t t e n o u t f i n t h e unstable r e g i o n o f v e l o c i t i e s . A t longer times (Fig. 6d) f becomes e s s e n t i a l l y f l a t i n t h i s region, t h e waves cease t o grow and a broad b u t noisy q u a s i s t a t i o n a r y spectrum o f waves r e s u l t s . Except f o r v e r y s h o r t times the spectrum has grown t o a l e v e l f o r which kS >> 1 and t h e c o n d i t i o n f o r s t o c h a s t i c i t y expected f o r an imposed f i e l d i s exceeded. s i m u l a t i o n t h e c o n d i t i o n f o r q u a s i l i n e a r theory h - l = s a t is f ied.

Also, throughout t h e

<<

T~

17 i s

well

The comparison between t h e numerical r e s u l t s and t h e q u a s i l i n e a r p r e d i c t i o n f o r t h e growth r a t e f o r a t y p i c a l unstable mode i s shown i n Fig. 7. The numeric a l r e s u l t s f o r the ensemble average over t e n s i m u l a t i o n runs ( s o l i d l i n e ) l i e s i g n i f i c a n t l y above t h e p r e d i c t e d q u a s i l i n e a r growth r a t e (dashed l i n e ) .

Fig. 7.

Growth r a t e s versus phase v e l o c i t y : t h e dashed l i n e i s t h e quasi 1i n e a r p r e d i c t i o n , t h e d o t t e d curve i s obtained by numerical time d e r i v a t i o n o f t h e ensemble average o f t h e mode energy and t h e s o l i d curve i s a l e a s t square f i t o f t h e preceding one (from Adam e t a l . , Ref. 19).

3

4

6

6

7

8

9

10

v/vc. Fig. 7 Now r e c a l l t h e usual q u a s i l i n e a r equation f o r t h e t i m e development o f t h e mean i n t e n s i t y i n mode k: (4.3) I n t h e usual q u a s i l i n e a r theory t h e source term Sk which a r i s e s from mode-mode c o u p l i n g e f f e c t s i s argued t o be n e g l i g i b l e when t h e c o n d i t i o n s (3.3) a r e f u l f i l l e d . The coupled equations (4.3) w i t h Sk = 0 and (3.5) c o n s t i t u t e t h e usual q u a s i l i n e a r theory. Together they conserve p a r t i c l e number, momentum, and energy. The simulations appear t o show t h a t t h e growth r a t e which should appear i n (4.3) i s g r e a t e r than yQkSL-;i n t h i s case equation (3.5)

must a l s o be a l t e r e d

i n order t o m a i n t a i n conservation o f energy. !e.g., D # i n f e r from t h e simulations t h a t

Therefore, we

448

D. PESME, 0.F. DUBOIS

Since o f t h e c o n d i t i o n s (3.3) f o r t h e v a l i d i t y o f q u a s i l i n e a r theory b o t h

i)( k s >> 1) and i f ) ( h - l << T < ~ > )are s a t i s f i e d d u r i n g t h e s i m u l a t i o n we must conclude t h a t iii) t h e quasigaussian f l u c t u a t i o n assumption i s n o t v a l i d i n t h e sel f - c o n s i s t e n t f i e l d case. This statement i s n o t s u r p r i s i n g i n i t s e l f since we know t h a t t h e s e l f - c o n s i s t e n t Poisson equation i s nonlinear. As a r e s u l t nongaussian e l e c t r i c f i e l d f l u c t u a t i o n s w i l l evolve from i n i t i a l l y gaussian e l e c t r i c f i e l d f l u c t u a t i o n s . SELF-CONSISTENT STATISTICAL THEORIES OF PLASMA TURBULENCE Introduction The understanding o f t h e s e l f - c o n s i s t e n t s t a t i s t i c a l d e s c r i p t i o n o f Vlasov t u r bulence i s i n a new stage o f development today and t h e study o f t h i s problem w i l l c e r t a i n l y be t h e s u b j e c t o f f u t u r e research i n t h e n o n l i n e a r physics o f plasmas. I n t h e p r e s c r i b e d e l e c t r i c f i e l d problem we were concerned w i t h t h e motion o f p a r t i c l e s i n random forces. The p r o p e r t y f o r t h e e l e c t r i c f i e l d o f b e i n g p r e s c r i b e d d i d n o t appear e x p l i c i t l y i n t h e d e r i v a t i o n o f t h e equations o f e v o l u t i o n f o r < f > and <€if €if’>. These equations (eq. 3.6 and 3.7) a c t u a l l y reduce t o two Fokker-Planck equations and t h e y can be e a s i l y d e r i v e d by computing t h e f r i c t i o n and d i f f u s i o n c o e f f i c i e n t s corresponding t o t h e motion o f a p a r t i c l e i n t h e random f i e l d . T h i s i s t h e reason why i t has been thought f o r a long t i m e t h a t t h e s t a t i s t i c a l d e s c r i p t i o n o f t h e p a r t i c l e motion i n s e l f - c o n s i s t e n t Vlasov t u r b u lence was t h e same as i n the p r e s c r i b e d e l e c t r i c f i e l d case. We have, however, t o keep i n mind t h a t a basic v a l i d i t y c o n d i t i o n f o r q u a s i l i n e a r t h e o r y as w e l l as f o r t h e Fokker-Planck equation i s t h e quasigaussian p r o p e r t y o f t h e e l e c t r i c f i e l d . I n t h e s t o c h a s t i c a c c e l e r a t i o n problem t h i s p r o p e r t y was always imposed, e.g., through random phases, b u t i n t h e Vlasov-Poisson case i t i s necessary t o check t h e v a l i d i t y o f t h i s hypothesis.

-

-

We consider e x p l i c i t l y a s e l f - c o n s i s t e n t Vlasov t u r b u l e n c e s i t u a t i o n f o r which t h e c o n d i t i o n s 3.31 and 3 . 3 i i a r e s a t i s f i e d . We assume a l s o t h e usual weak turbulence hypothesis t h a t assigns N, we1 1-defined frequencies wn(k) t o a wave number k. These frequencies have t o be computed s e l f - c o n s i s t e n t l y through a n o n l i n e a r d i s p e r s i o n r e l a t i o n . For s i m p l i c i t y we w i l l t a k e N, = 1. A l l t h e q u a n t i t i e s such as k,

wb, t c are now computed through t h e instantaneous values

o f t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d . I n e q u a l i t y 3.3i) i s e a s i l y s a t i s f i e d i n unstable s i t u a t i o n s . I n e q u a l i t y 3 . 3 i i ) i s u s u a l l y considered as t h e v a l i d i t y c o n d i t i o n o f q u a s i l i n e a r t h e o r y and i s f u l f i l l e d , f o r instance, i n t h e warm beam-plasma i n s t a b i l i t y . We d e f i n e as quasigaussian t h e o r i e s t h e s t a t i s t i c a l d e s c r i p t i o n s o f Vlasov t u r bulence which assume a p r i o r i t h a t t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d has quasigaussian p r o p e r t i e s . I n t h i s s e c t i o n we w i l l f i r s t p r e s e n t t h e usual argument given f o r j u s t i f y i n g t h i s assumption. We w i l l then show t h a t t h e quasigaussian t h e o r i e s are s e l f - c o n t r a d i c t o r y i n t h e regimes where t h e n o n l i n e a r i t y p l a y s a r o l e . The reason f o r discrepancy w i l l be e x h i b i t e d through a p e r t u r b a t i v e c a l culation: the s t a t i s t i c a l properties o f the self-consistent e l e c t r i c f i e l d

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

are not characterized by a unique c o r r e l a t i o n l e n g t h stochastic d e c o r r e l a t i o n l e n g t h Rh = vth.

449

= l / A k b u t a l s o by the

We close t h i s Section by describing some o f the features o f the d i r e c t i n t e r a c t i o n approximation ( D I A ) which i s a renormalized theory which takes i n t o account c e r t a i n nongaussian e f f e c t s . The quasi gaussi an hypothesis I n the e a r l i e r d e r i v a t i o n s o f quasi1 i n e a r theory the random-phase assumption was usually i m p l i c i t l y made. z z 9 z 3 The underlying reason has been made e x p l i c i t by OlNei124 as follows: Let us assume on physical grounds t h a t w e l l above t h e stochastic threshold the e l e c t r i c f i e l d i s i t s e l f t u r b u l e n t and i s characterized by an a u t o c o r r e l a t i o n I n a f i n i t e box o f length L, the F o u r i e r components o f the e l e c t r i c length Qeff. f i e l d ar& defined as:

1 L

Ek(t) =

dx E(x,t) exp-ikx

I f we now d i v i d e t h e i n t e r v a l [O,L]

w i t h k = n2n

eff i n t o segments o f l e n g t h go where gc

<< go << L, t h e e l e c t r i c f i e l d i n each segment i s independent o f t h e f i e l d i n other segments.

Ek =

Therefore Ek can a l s o be w r i t t e n as

dx E(x,t) gcN j=l (j-l)Qo

exp-ikx

w i t h N = L/Qo

.

Thus Ek appears t o be the sum o f many independent v a r i a b l e s and by t h e c e n t r a l

l i m i t theorem Ek behaves as a Gaussian v a r i a b l e i n t h e l i m i t L/go very large. Let us emphasize t h a t the o n l y hypothesis i n t h i s demonstration i s t h e existence o f an a u t o c o r r e l a t i o n l e n g t h gFff and t h a t seems q u i t e reasonable i n a turbulent situation. However, O'Neil observedz4 t h a t the argument o u t l i n e d above cannot be used t o j u s t i f y t h e quasigaussian assumption (which he c a l l s the random-phase approximation).

This assumption would say, f o r example, t h a t the t<Ek-qEqEp-kE-p> l2>. The neglected terms are24 i n d i v i d u a l l y can be replaced by 2
D. PESME, D. F. DUBOlS

450

On t h e b a s i s o f t h e previous s e c t i o n , t h e quasigaussian t h e o r i e s assume a p r i o r i t h a t t h e l a s t c o n d i t i o n iii)o f 3.3 i s s a t i s f i e d i.e., the e l e c t r i c f i e l d behaves as a quasigaussian process.

-

The scheme o f these t h e o r i e s i s t h e f o l l o w i n g :

1) Two Fokker-Planck equations describe t h e e v o l u t i o n o f < f > and < 6 f 6 f ' > , namely t h e q u a s i l i n e a r equation 3.6 and t h e Dupree equation 3.7. The f r i c t i o n and d i f f u s i o n c o e f f i c i e n t s i n v o l v e t h e s p e c t r a l d e n s i t y < l E k l 2>. 2 2 ) The spectrum < I E k l > i s then computed by w r i t i n g Poisson's equation as: k2 = (471)2q2 Jdvdv' c 6 f 6 f ' > k

(5.1)

We thus o b t a i n a closed s e t o f t h r e e equations f o r t h e t h r e e unknowns < f > , < 6 f

2

6 f ' > , and .

However, i t i s necessary t o go beyond t h i s l e v e l i n o r d e r t o

compute t h e i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n s o f t h e e l e c t r i c f i e l d o f h i g h e r order than 2. This scheme would be c o n s i s t e n t o n l y i f these c o r r e l a t i o n funct i o n s s a t i s f y t h e d e f i n i t i o n o f a quasigaussian process given i n 52. An i n d i r e c t demonstration o f t h e i n c o n s i s t e n c y o f t h i s scheme i n t h e regime (k2D)1/3 >> y f a L * has been given by Adam, Lava1 and P e ~ m e . They ~ ~ solved t h e Dupree equation 3.7 and by i n s e r t i n g i t s s o l u t i o n i n t o Poisson's equation 5.1; they found a growth r a t e d i f f e r e n t from t h e q u a s i l i n e a r p r e d i c t i o n . This r e s u l t , combined w i t h q u a s i l i n e a r equation 3.6 does n o t a l l o w energy conservation. These authors conclude t h a t t h e q u a s i l i n e a r theory i s i n c o n s i s t e n t i n one dimension and i n t h e regime ( k DQ'L')1/3

>> y f e L * and showed t h a t t h e c o n t r i b u t i o n o f t h e 4-

point irreducible correlation function i s not negligible, i n contradiction t o t h e quasigaussian hypothesis. Even though t h e quasigaussian scheme appears t o be i n v a l i d , t h e s o l u t i o n o f t h e Dupree equation 3.7 e x h i b i t s c e r t a i n f e a t u r e s worth b e i n g mentioned. I n t h e

l i m i t o f i n e q u a l i t i e s 3 . 3 i ) and ii).and f o r Ik(v+) v-I (k2DQ*L.)1/3),

and Ix-/v+I

<< i n f [(k(v+) DQ.L)-1/3,

y;]',

<< sup (Y!'~., equation 3.7

becomes:

[at + v-ax - - 2 D Q ' L * ( ~ + )

< 6f 6fl>

=

2 2 DQ'L*(v+) cos (k(v+)x-)

kv;f>]

The l a s t term o f t h e R.H.S. o f Eq. 5.2 i s c a l l e d t h e mode-coupling term s i n c e i t couples t o g e t h e r t h e k and k-k(v+), k + k(v+) F o u r i e r components o f < 6 f 6 f ' > . The r a t i o R = (k2DQ*L')1/3/yf*L* t u r n s o u t t o be a measure o f t h e n o n l i n e a r i t y and can be considered as a Reynolds number. For R << 1 t h e mode-coupling term i s n e g l i g i b l e and t h e q u a s i l i n e a r r e s u l t s a r e recovered. For R >> 1 t h e mode c o u p l i n g terms can be shown t o be o f t h e same order o f magnitude as t h e o t h e r and generate terms. The mode c o u p l i n g terms g i v e a c o n t r i b u t i o n f o r v - v ' = w k / k

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

451

harmonics coupled together through t h e resonant p a r t i c l e s . As a r e s u l t , t h e c o r r e l a t i o n f u n c t i o n c 6 f 6 f ' > i s found t o have t h e f o l l o w i n g convenient harmonic expansion <6f(x,v,t)

Bf

(XI , V ' , t ' ) >

= cgn(v-)

exp

n k(v+)(x-x')

] [-inw exp

Ik(v+)l (t-t')]

(5.3)

I n t h e regime R >> 1, where t h e n o n l i n e a r i t y p l a y s a r o l e , t h e c o r r e l a t i o n funct i o n i s thus the sum o f an i n f i n i t e number o f harmonics coupled together. When i n s e r t e d i n t o Poisson equations t h e c o n t r i b u t i o n coming from t h i s harmonic coupl i n g appears as a source term r a d i a t i n g coherently. We can, t h e r e f o r e , say t h a t t h e n o n l i n e a r i t y r e i n f o r c e s t h e coherence o f t h e s o l u t i o n . As a consequence, Adam, Laval, and Pesme indeed have found an increase o f t h e growth r a t e compared t o the q u a s i l i n e a r r e s u l t . This f e a t u r e i s generalized i n a s e l f - c o n s i s t e n t way i n the D I A . The Dupree "clump theory" l i k e w i s e p r e d i c t s an enhanced r a d i a t i o n due t o macroscopic g r a n u l a t i o n o f t h e phase space d e n s i t y a s o r t o f enhanced discreteness noise due o n l y t o n o n l i n e a r processes i n t h e continous Vlasov approximation. The Dupree t h e o r y a p p l i e s t o a "strong" turbulence l i m i t i n which no d e f i n i t e normal mode frequency e x i s t s f o r a given k. Therefore k(v) = wk(,,)/v i s not defined i n t h i s l i m i t .

-

The r e s u l t s o f Adam, Laval and Pesme apply i n t h e l i m i t o f q u a s i l i n e a r o r d e r i n g (eq. 3 . 3 i and 3 . 3 i i ) when yk/wk << 1 so t h a t a d e f i n i t e frequency i s associated w i t h a d e f i n i t e k. P a r t i c l e s o f v e l o c i t y v i n t e r a c t s t r o n g l y w i t h waves i n t h e spectrum o f wave number k(v) such t h a t p a r t i c l e s tend t o be p e r i o d i c a l l y c o r r e l a t e d w i t h t h e separation 2n/k(v). P h y s i c a l l y , t h e p a r t i c l e s are more l i k e l y t o be found near t h e troughs o f these resonant waves. The quasigaussian hypothesis r e v i s i t e d I n t h i s subsection we w i l l show i n a p e r t u r b a t i v e way how t h e mode-coupling terms generate a nongaussian e l e c t r i c f i e l d . We w i l l f o l l o w e s s e n t i a l l y t h e 26 arguments o f Laval and Pesme. Let io(v,t)

t h e space-averaged d i s t r i b u t i o n f u n c t i o n and d e f i n e 6 f as 6 f =

Following Kadomtsev, t h e equation f o r 6 f can be w r i t t e n i n f(x,v,t)-fo(v). F o u r i e r space as:

i n which

01

provides a contour p r e s c r i p t i o n f o r v - i n t e g r a t i o n .

Equation 5.5 can be solved i t e r a t i v e l y beginning w i t h i n a f u n c t i o n a l s e r i e s i n powers o f E. equation ikEkw = 4nq J6fb(v)dv

= g k [Ek fOlw T h i s s e r i e s i s then i n s e r t e d i n t o Poisson's

and y i e l d s a n o n l i n e a r equation f o r Ekw o f t h e

f o l l o w i n g form: &(k,w)Eb

= u2(k,w)E

E + p3 (k,w)EEE +

...

(5.6)

D. PESME, D.F. DUBOlS

452

(5.7)

(5.8a)

with Vku,,k'w' =

Ig-

h0

0

0

ku, g k - k ' ,

w-w'

(5.9)

f~dv

Equation 5.7 can be solved i t e r a t i v e l y by s e t t i n g

E = E(l)

+

E(2)

+

E(3)

...

(5.10)

The lowest order gives z E ( l ) = 0. T h i s i s t h e approximation o f q u a s i l i n e a r theory i n which the imaginary p a r t o f t h e r o o t o f E(k,w) determines t h e quasif o r s i m p l i c i t y i n t h e f o l l o w i n g we s e t y f a L ' = yk.

l i n e a r growth r a t e

= Ek(0) exp[-iwkt + y k t ] .

We have E(kl)(t) field.

We assume Ek(0) as a random phased

The mode-coupling terms w i l l generate nongaussian c o r r e l a t i o n s :

E(2) = I-r2 E ( l ) E ( l )

(5. l l a )

We now consider t h e 4 - p o i n t c o r r e l a t i o n f u n c t i o n <Ek +k +k E-k E-k2 E-k3 > 1 2 3 1 On s u b s t i t u t i o n o f t h e expansion (5.10) f o r each Ek we f i n d f i r s t o f a l l t h e usual terms <E19 E-kl (l)

E-$ ( l ) E-k3 ( l ) > which, because o f t h e gaussian assumption

f o r E ( l ) can be c a s t i n t o a sum o f <E ( l ) ~ ( l ) > < E ( ~ ) E ( ~ ) .> L e t us now compute t h e nongaussian c o n t r i b u t i o n s .

A t t h e lowest order we o b t a i n t h e sum o f f o u r

terms l i k e <E(3)E(1)E(1) E ( l ) > w i t h t k m = 0. The terms E(2) a r e indeed s m a l l e r kj k kR s i n c e E(u,k) i n Eq. 5.11a has t o be taken a t frequency w = 0 by a f a c t o r y /w k Pe o r f 20 and cannot be made small. Pe From Eq. 5.11b we f i n d

ki

453

TURBULENCE AND SELF-CONSISTENT FIELDS IN PLASMAS

where Qk = wk + i y k and 2Q = R k l + Rkll + Rk-kl-kll expression i n t o <E(3)E(1)E(1)E(1)>

.

After substitution o f t h i s

and use o f t h e gaussian assumption f o r E (1) ,

we o b t a i n t h e f o l l o w i n g expression:

exp- i w

( tl- t2) exp- i w k 2 ( tl- t 3) exp- iwk3( tl- t4) kl

W

W

(2kl+k2+k3)

3 -2iu

+ permutations (kl,k2,

k3)

, with u = y

(5.13)

kl The Here t h e m a t r i x element V has been e x p l i c i t l y evaluated u s i n g 5.8b. d i e l e c t r i c f u n c t i o n has been taken t o be on resonance, i , e . , wkl + wk2 + wk3 which i m p l i e s t h a t frequencies and wave numbers have n o t a l l t h e same +k +k kl2 3 sign. We have a l s o assumed t h a t t h e d i s p e r s i o n o f t h e r o o t s wk i s weak so

2:w

t h a t e(m)

N

(E)

.

2 i ~ t . ~ .

The expression 5.13 enables us t o compute t h e

contribution o f the irreducible 4-point correlation function i n the evaluation o f the mode amplitudes. A t the lowest order we have

which leads t o t h e r e s u l t : (5.15)

D. PESME, D.F. DUBOlS

454

The m o d i f i c a t i o n o f t h e spectrum as g i v e n by t h e expression i n s i d e t h e b r a c k e t of Eq. 5.15 shows again t h a t t h e mode c o u p l i n g terms l e a d t o an increase o f t h e energy t r a n s f e r from resonant p a r t i c l e s towards waves. T h i s r e s u l t i s s t r i c t l y v a l i d o n l y f o r y!'L'>>

( k 2 D ) , l l 3 b u t i t c l e a r l y shows t h a t t h e i r r e d u c i b l e

4 - p o i n t c o r r e l a t i o n f u n c t i o n g i v e s a n o n - n e g l i g i b l e c o n t r i b u t i o n as soon as (k2D)1'3

2

y i ' L. which i n v a l i d a t e s t h e quasigaussian assumption.

By F o u r i e r transform o f Eq. 5.13 we o b t a i n t h e expression i n r e a l space o f t h e 4-point correlation function <E(3)(x,,tl)

E(l)(x2,t2)

E(')(x3,t3)

E(')(x4,t4)>

=

(5.16) expi kl(xl-x2)

expi k2(x1-x3)

expi k3(x1-x4)

I f the expression i n t h e c u r l y brackets o f t h e R.H.S. o f Eq. 5.13 was a smooth f u n c t i o n o f kl, kz and k3, then t h i s c o r r e l a t i o n f u n c t i o n would be e s s e n t i a l l y a product o f t h r e e two-point c o r r e l a t i o n f u n c t i o n s o f t h e form

$([,r) = 2 exp-i[wkr-k[l For c o r r e l a t i o n s along t h e unperturbed o r b i t s l a t i o n time

T~

5

= v t , t h i s has t h e usual c o r r e -

= I A k ( v - V g ) l - l discussed p r e v i o u s l y .

On t h e c o n t r a r y , however, the expression i n c u r l y brackets o f t h e R.H.S. o f Eq. 5.13 i s a peaked f u n c t i o n f o r vlkl+kzl

5 u.

2

I n t h e regime R = ( k 0)

1/3 /yk

>> 1, Eq. 5.15 shows t h a t t h g s e r i e s expansion i n power o f E ( l ) i s n o t convergFnt. 9

The f r e e propagator g

may then be replaced by t h e renormalized propagator [(k21f*L')~]3

m

gw,k

= (-q/m$

drexp

i(w-kv)r.

exp

-

3 av

(5.17)

0

I n the e s t i m a t i o n o f order o f magnitudes i t i s s u f f i c i e n t t o r e p l a c e u = y i m L . by v = ( k DQ'L*)1/3.

The nongaussian c o r r e l a t i o n f u n c t i o n i s , t h e r e f o r e , charac-

t e r i z e d by t h e time (k2DQ*L*)-1'3

which i s o f t h e o r d e r o f t h e i n v e r s e o f Kolmo-

gorov entropy hQ. L' = 0.36( k2DQ*L'

)ll3 encountered e a r l i e r .

Since ( k2D)-1'3

>>

tC

we see t h a t s e l f - c o n s i stency i n t r o d u c e s a tendency toward s e l f - o r g a n i z a t i o n o f t h e t u r b u l e n t sys tem.

A d e t a i l e d diagrammatic expansion leads t o t h e conclusion t h a t i n t h e regime (k2D)1'3

>> y k a l l t h e i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n s c o n t r i b u t e a t t h e same

order o f magnitude as the q u a s i l i n e a r term. T h i s r e s u l t i n v a l i d a t e s t h e t h e o r i e s which c o n s i s t o f a simple r e n o r m a l i z a t i o n o f t h e propagator such as developed by

TURBULENCE AND SELF-CONSISTENT FIELDS IN PLASMAS

Rudakov and Tsytovich, Choi and H o r t ~ n . ~In~ the regime defined by ii) they lead to the quasilinear result since they don't contain the ling term of Eq. 5.2. It can be shownz8 that a partial summation of irreducible correlation functions leads to the Dupree equation 3.7. we may summarize the different levels of renormalization: 1.

455

3.3i) and mode coupthe neglected Therefore,

WEAK TURBULENCE THEORY - This makes the usual quasilinear prediction but suffers from the defect of secular divergences arising in the neglected irreducible correlation functions. These secularities are removed by renormalization of the propagator which leads to:

2.

SIMPLY RENORMALIZED THEORIES - These again make the usual quasilinear prediction but still suffer from non-negligible contributions from neglected irreducible correlation functions. A partial summation of the irreducible correlation functions leads to:

3.

DUPREE'S THEORY - This theory predicts a zero order modification of the quasilinear result coming from mode coupling terms. However, this theory considers only part of t h e irreducible correlation functions and is not properly self-consistent.

SELF-CONSISTENT STATISTICAL DESCRIPTIONS AND THE DIA It appears from the considerations above that most familiar statistical theories of Vlasov turbulence until recently, have been quasigaussian theories in the sense that they explicitly or implicitly make the assumption of quasigaussian electric field correlations. A self-consistent statistical description of Vlasov turbulence must take into account the relationship between perturbations in the averaged phase space distribution = and perturbations in the mean electric field <El> = <E(xltl)> as well as the relationship between the These fluctuations in these quantities: 6fl = fl- and 6E1 = EI-<E1>. relationships are, of course, imposed by Poisson's eqn. (4.1). For the discussion here we follow closely the results of reference 29. From the Vlasov eqn. (32) the well-known equations for the mean distribution function is easily found: r

(5.18)

This couples to the fluctuation correlation function <6E 6f>. The averaged Poisson equation is (5.19) V1 * <E1> = 4nq J dvl - 4nqN We have added an artificial phase space source density q1 = q(xlxltl) which we can consider as a probe of the system. Suppose for r) = 0 the system of equations has a well-defined solution , <El>; we then add a local phase space source perturbation 6r) which changes to + 6 and <E1> to <E1> + &<El>. The increments 6 are functional differentials such as those used in the calculus of variations of continuous fields. A relationship between the increments 6, 6<E> and 6<rp is readily found from eq. (5.18) to be of the form

0. PESME, 0.F. DUBOIS

456

G;:

6

a

+

r126<E2> = 6ql v1

(5.20)

By t a k i n g t h e f u n c t i o n a l d i f f e r e n t i a l o f eqn. (5.18) c o e f f i c i e n t s as

i t i s easy t o i d e n t i f y t h e

(5.21)

(5.22) To s i m p l i f y w r i t i n g t h e equations we a r e u s i n g a summation convention where, f o r example, :;G

6

Jdv2Jdx2Jdt2 G;:

6, etc. and 6(1-2)

6(vl-v2)

6(x1-x2)6 (tl-t2). The l a s t terms on t h e r i g h t hand s i d e o f eqns. (5.21) and (5.22) c o n t a i n f u n c t i o n a l d e r i v a t i v e s o f t h e c o r r e l a t i o n f u n c t i o n <6E16fl> w i t h respect t o changes i n t h e mean values and <E>, r e s p e c t i v e l y . These funct i o n a l d e r i v a t i v e s completely describe t h e renormal i z a t i o n o f t h e response funct i o n s G12 and rI2. The response f u n c t i o n G12

i s c a l l e d t h e " q u a s i p a r t i c l e propagator"; w i t h o u t

r e n o r m a l i z a t i o n i t i s c l e a r from eqn. (5.21)

t h a t G12

simply describes p a r t i c l e

propagation along unperturbed o r b i t s ( i n t h e mean f i e l d

<El> which i s zero i n

a homogeneous system.) The r e n o r m a l i z a t i o n term takes i n t o account t h e s e l f c o n s i s t e n t response o f the f l u c t u a t i n g t u r b u l e n t "background" through which t h e p a r t i c l e propagates. I n an imposed f l u c t u a t i o n l i m i t t h i s propagator reduces t o t h e f a m i l i a r Dupree renormalized propagator g i v e n i n eqn. 3.9. The response f u n c t i o n

aVlrl2

i s c a l l e d t h e " q u a s i p a r t i c l e p o t e n t i a l source."

I t takes i n t o account t h e e f f e c t i v e source d e n s i t y 6q which must a r i s e when t h e r e i s a change i n t h e mean e l e c t r o s t a t i c f i e l d 6<E> i n o r d e r t o keep < f > constant. Without r e n o r m a l i z a t i o n , t h i s term i s simply p r o p o r t i o n a l t o a . v1 The r e n o r m a l i z a t i o n o f t h i s q u a n t i t y i s n o t f a m i l i a r b u t was a c t u a l l y considered

by K a d o m ~ t e vi ~n ~Section I11 o f h i s 1965 book (equation 21a). (Kadomstevls renormalized theory has some elements i n common w i t h t h e D I A , b u t d i f f e r s i n several important respects r e l a t e d t o s e l f - c o n s i s t e n c y . ) We next consider t h e f l u c t u a t i o n s . By s u b t r a c t i n g eq. (5.18) from eqn. (3.2) and n o t i n g t h e d e f i n i t i o n s (5.21) and (5.22) we can d e r i v e t h e exact equation f o r bf,

6fl

=

6f;nCA

+ ;q/m)

G12 8,

[6f26E2-<6f26E2>

2

-

(5.23) 6<6f26E2> 6f3

6

-

6<6f26E2> <6E3>

D. PESME. D.F. DUBOIS

We have i n d i c a t e d t h a t 6fl

457

can be separated i n t o a coherent p a r t p r o p o r t i o n a l t o

t h e e l e c t r o s t a t i c f l u c t u a t i o n 6E and an incoherent p a r t .

I n a s p a t i a l l y homo-

geneous plasma i t i s easy t o see t h a t t h e F o u r i e r component 6 f p h i s p r o p o r t i o n a l t o 6Ek.

For t h e incoherent p a r t terms l i k e 6f16E1

such as

i n v o l v e F o u r i e r convolutions

6fk-k16Ekl ( w i t h 6Ekz0=0) and thus i n v o l v e s t h e c o u p l i n g o f d i f f e r e n t

F o u r i e r modes. I f t h e expression (5.23) i s i n s e r t e d i n t o Poisson's equation Vl* GE1=4nqJdvl

[ 6 f i o h + 6 f i n c ] one can rearrange t h e coherent c o n t r i b u t i o n t o w r i t e

V1 where

-

E~~

cI2

6L2 = 4nqJdv16fl

inc

(5.24)

i s t h e d i e l e c t r i c operator 29

The coherent p a r t has produced t h e s u s c e p t i b i l i t y f u n c t i o n i n t h e d i e l e c t r i c operator. For a homogeneous s t a t i o n a r y system, t h e F o u r i e r c o e f f i c i e n t o f E can be w r i t t e n

2

2

E(k,w) = 1 + (4nq /mk ) Jdv3 Jdv

i k * a r (k,w) v4 v4

(k,w)

Gv

4

3 4

We see from eqn. (5.24) t h a t t h e charge d e n s i t y associated w i t h 6finC a c t s as an incoherent source f o r t h e e l e c t r o s t a t i c f l u c t u a t i o n . For example, t h e twop o i n t e l e c t r o s t a t i c c o r r e l a t i o n f u n c t i o n can be w r i t t e n as

i n terms o f t h e inverse d i e l e c t r i c operator.

Or

n

(5.26b) f o r t h e s p e c t r a l i n t e n s i t y i n a homogeneous, s t a t i o n a r y case. The two-point c o r r e l a t i o n f u n c t i o n f o r t h e phase space d i s t r i b u t i o n can l i k e w i s e be w r i t t e n as c6f16fll

> = (q 2 /m 2 1 G 1 2 ~ v 2 ~ 2 3 G 1 1 2 1 ~ v ; ~ 2 ~ 3 ~ ~ ~ E 3 ~ E 3 ~ >

where P12

6(1-2)P1,2,-P126~l'-21)

1

<6f2i n c6f2' inc,

(5.27)

i s a " p o l a r i z a t i o n " operator

2 a v3r 34(v2& 1-l '12 = (4nq lm) '13 4 42

(5.28)

The f i r s t term on t h e r i g h t side i s t h e c o r r e l a t i o n <6fyh6fi:h> while the f i r s t inchf inc, 1l The second and t h i r d term i n term i n square brackets a r i s e s from <6fl

.

458

D. PESME, D.F. DUBOlS

square brackets a r i s e r e s p e c t i v e l y from <6finc6f::h>

coh inc, and <6fl 6fll

.

In

w r i t i n g these terms i n terms o f the operator P we have used Poisson's equation i n the form o f eq. (5.24) t o e l i m i n a t e 6E i n terms o f 6finc.

By doing t h i s we

have expressed c o r r e l a t i o n f u n c t i o n s o f t h e form <6E6E6f> i n terms o f < 6 f i n c 6finc 1l > which can be seen f r o m t h e d e f i n i t i o n o f 6finc, eq. 5.23, t o i n v o l v e 4 - p o i n t and higher c o r r e l a t i o n f u n c t i o n s . The vanishing o f t h e cross c o r r e l a t i o n terms i s made e x p l i c i t l y i n t h e theory o f Dupree ( r e f . 2 eqn. (21)). The equations w r i t t e n i n t h i s s e c t i o n up t o t h i s p o i n t a r e f o r m a l l y exact. We inc6f inc, see t h a t the system i s n o t closed since t h e c o r r e l a t i o n f u n c t i o n <6fl i n v o l v e s t h r e e - p o i n t and f o u r - p o i n t c o r r e l a t i o n f u n c t i o n s . An approximation which closes these equations c a l l e d t h e " D i r e c t I n t e r a c t i o n Approximation," D I A , has a t t r a c t e d renewed i n t e r e s t . This t y p e o f theory, f i r s t a p p l i e d by Kraichnan t o Navier-Stokes t u r b ~ l e n c e ,t r~e~a t s c e r t a i n nongaussian c o r r e l a t i o n s i n a n o n p e r t u r b a t i v e way. This approach was reasonably successful f o r moderate Reynold's numbers b u t f a i l e d t o a c c u r a t e l y d e s c r i b e t h e i n e r t i a l range f o r h i g h Reynold's numbers. The f a i l u r e was due t o t h e i n a c c u r a t e treatment o f t h e convection o f s h o r t scale turbulence by t h e l o n g s c a l e motion. 32 The D I A was w r i t t e n down f o r t h e Vlasov-Poisson system by Orzsag and Kraichnan i n 1967. This work r e c e i v e d very l i t t l e a t t e n t i o n u n t i l r e c e n t l y when DuBois and E ~ p e d a showed l~~ how t o " f a c t o r " t h e t h e o r y i n t o s i n g l e (quasi) p a r t i c l e and c o l l e c t i v e responses. This exposed t h e existence i n t h e D I A o f new s e l f - c o n s i s t e n t f l u c t u a t i o n ( o r n o n l i n e a r p o l a r i z a t i o n ) c o n t r i b u t i o n s which were n o t found i n t h e o l d e r quasigaussian imposed f i e l d t h e o r i e s . The new terms a r e proport i o n a l t o t h e d i e l e c t r i c response E - ~such as t h e terms i n t h e p o l a r i z a t i o n operator i n eqn. (5.27); various estimates have shown these terms t o be o f t h e same order as t h e imposed f i e l d terms. The D I A based theory was a l s o i n v e s t i gated by Krommes and coworkers34 who a l s o considered a p p l i c a t i o n t o t h e g u i d i n g 35

center model and t o d r i f t wave turbulence.

Here we w i l l l i s t the approximations t o t h e s e t o f exact equations l i s t e d above which a r e e q u i v a l e n t t o t h e D I A :

1.

y

The incoherent c o r r e l a t i o n f u n c t i o n <6fl inchf i > i s computed t o terms o f

q u a d r a t i c order i n t h e c o r r e l a t i o n s u s i n g a quasigaussian approximation f o r t h e mixed f o u r - p o i n t c o r r e l a t i o n f u n c t i o n

q2/m 2 G12av2G1121av21[<6E26E21><6f26f21>+ <6E26f2'> <6f26E21>]

(5.29)

Note t h a t t h i s i s n o t a t a l l j u s t t h e quasigaussian approximation f o r t h e c o r r e l a t i o n f u n c t i o n o f f o u r e l e c t r i c f i e l d s . The f l u c t u a t i o n 6 f c o n t a i n s , as a consequence o f t h e incoherent c o n t r i b u t i o n , 6finc orders i n t h e e l e c t r i c f i e l d f l u c t u a t i o n 6E.

, i n eqn. (5.23), terms o f a l l (These terms can be obtained from

eqn. (5.23) by i t e r a t i o n s t a r t i n g w i t h 6 f y ) = 6 f E o h . )

Thus t h e D I A c o n t a i n s

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

459

e l e c t r c f i e l d c o r r e l a t i o n s o f a l l orders; i n p a r t i c u l a r i t produces i n t h i r d 2 order n (6E ) t h e nongaussian c o r r e l a t i o n s c a l c u l a t e d by Lava1 and Pesme. n

The inverse propagator G - l and t h e response f u n c t i o n lowest (linear)order i n the c o r r e l a t i o n f u n c t i o n s .

L.

r

a r e computed t o

To c a r r y o u t the l a s t step t h e f u n c t i o n a l d e r i v a t i v e s o f <6E6f> i n eqs. (5.21) and (5.22) must be evaluated t o t h i s order. This can be accomplished by u s i n g an exact r e l a t i o n connecting the f l u c t u a t i o n s which i s e a s i l y d e r i v e d from eq. (5.23): <6f16ElI>

=

{ G12av;23GE36E11>

+

4nq a l(V~lcl121 )-1,6finc1 6f2inc,

(5.3ij

x1 This equation can a l s o be used f o r t h e r i g h t - h a n d s i d e o f eq. (5.18) f o r t h e mean d i s t r i b u t i o n . I n computing t h e r e q u i r e d f u n c t i o n a l d e r i v a t i v e s o f <6f6E> f o r s e l f - c o n s i s t e n t f l u c t u a t i o n s i t i s necessary t o consider t h e f u n c t i o n a l dependence on < f > o r <E> o f t h e d i e l e c t r i c response, c-', which e n t e r s e x p l i c i t l y i n the l a s t t e r m o f eq. (5.31) and a l s o i n <6E6E> as shown e x p l i c i t l y i n eq. (5.26). We can g i v e here o n l y the r e s u l t o f t h i s c a l c u l a t i o n f o r :G; homogeneous, s t a t i o n a r y system:

-1 Gvvl (k,u) = -i(w-k.v)b(v-v')

+

42m2 av. Jdk'Jdw'

I

for a

<6E6E>klul

~ v v I ( k - k ' , u - w ' ) + Jdy P v ( k - k ' , w - w ' ) G ~ v l ( k - k ' , w - w l )

1

(5.32)

The l a s t two terms i n c u r l y brackets a r e s e l f - c o n s i s t e n t f l u c t u a t i o n terms which a r i s e from t h e f u n c t i o n a l dependence discussed above and do n o t occur i n t h e quasigaussian f i e l d t h e o r i e s . These s e l f - c o n s i s t e n t f l u c t u a t i o n c o n t r i b u t i o n s have been estimated i n various cases t o be as important as t h e remaining term which appears i n t h e quasigaussian f i e l d t h e o r i e s and produces t h e renormal i z e d I n the q u a s i l i n e a r o r d e r i n g o f ~ ~ < < t and < ~ kS>> > 1f o r propagator i n eq. 5.1. a one-dimensional one-component plasma we have been a b l e t o show t h a t t h e s e l f c o n s i s t e n t f l u c t u a t i o n terms cannot be neglected. A s i m i l a r equation r e s u l t s f o r the renormalized Tv(k,u) which we w i l l n o t discuss here. F u r t h e r r e s u l t s a r e given i n reference 29. I f the D I A approximation o f eqn. (5.30) f o r <6finc6finc> i s i n s e r t e d i n t o eq. (5.27) we o b t a i n t h e f o l l o w i n g equation f o r t h e two-point c o r r e l a t i o n f u n c t i o n

460

D. PESME, D.F. DUBOIS

r

s . c. A

= (q2/m2) G12~v;23G1121~vh~2131<6E36E31>

(5.34)

When the terms labelled S.C. are dropped in this expression and only the imposed field renormalization terms in G are retained, this equation can be shown to reduce to the Dupree equation for <6f(x,v,t)6f(x'v1t)> given in equation (3.7). The latter approximation is valid only for quasigaussian imposed electric field fluctuations; we can see explicitly in this case that the remaining contribution from <6finc 6fivc> in eqn. (5.34); i.e., (q 2/m 2)G12G11216 v 2 6 v 2 1 ~ 6 f 2 6 f 2 1 ~ ~ 6 E 2 6 E 2 1 ~ reduces to the mutual diffusion (or mode coupling) term of equations (3.7) or (5.2). It is this term which produces the finite Kolmogorov entropy, h, discussed in Section 3. Thus there is a direct connection in this case between a finite level for the incoherent noise function <6f~"6f~vc> (sometimes called the nonlinear noise) and the intrinsic stochastic behavior of the system. The reduction of these DIA equations to a calculable form is very difficult even in the limit of quasilinear ordering. The harmonic expansion of eq. (5.3) still appears to be useful in the DIA. Using this technique we have been able to show that n ~ n eof the self-consistent terms can be dropped even in quasilinear ordering when R >> 1. Only the equations for the harmonic components n = 0, ? 1 and 2 are significantly changed by the self-consistent terms. Work is still in progress in deducing the consequences of the DIA in the quasilinear limit. The DIA is the natural first approximation in the renormalized perturbation theories of Kraichnan, Martin, Siggia and Rose (MSR)36 and others. The general conditions for the convergence of the MSR expansion are completely unknown; probably only convergence in the sense of an asymptotic expansion is expected at best. It appears to be a possible, but extremely tedious, task to examine the size o f the leading vertex corrections (in the sense of MSR) to the DIA. There is some Q that in quasilinear ordering these vertex corrections may be negligible. This hope is based on the following points: i) In the imposed field case the DIA theory becomes exactly quasilinear theory (QLT) in quasilinear ordering; the corrections in this case are o f order (T~/T<~,) << 1. ii) In the self-consistent field case the DIA differs from QLT in quasilinear ordering; the differences due to nonlinear polarization effects are the same order as the QLT terms. iii) If self-consistency also does not change the order of the vertex corrections, the corrections to the DIA will also be of order (T~/T<~,) as in i). 37 To make a point of contact between this formalism and the results of Adam, Lava1 and Pesme, we write an equation for the time evolution of ClbEl 2k>:

where (5.36)

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

This can be shown t o be an exact consequence o f eq. (5.26a)

’ 9 , i s t h e zero it,) = 0. I n t h e found t h a t Qk and 9 ,

where Qk + E(k,lk

+

provided

461

pk<
o f t h e d i e l e c t r i c f u n c t i o n i n t h e complex w plane:

l i m i t of q u a s i l i n e a r o r d e r i n g ( r h < < t c f > , kS>>l)

d i = w2+3k2Ve2, P = y i e L * . I n conventional q u a s i l i n e a r theory i t i s assumed t h a t Sk a r i s e s 2 2 from mode-mode c o u p l i n g e f f e c t s o f o r d e r <16Ekl > and i s thus n e g l i g i b l e . it i s

are given by t h e i r l i n e a r values:

9,

However, t o account f o r t h e r e s u l t s o f Adam, Laval and Pesme discussed i n Section 4 i t i s necessary t h a t t o l e a d i n g order

(5.37) where y y p i s t h e observed growth r a t e .

We see t h a t Sk i s r e l a t e d t o t h e r i g h t

hand side o f eq. (5.14). Comparison can be made w i t h t h e i t e r a t i v e c a l c u l a t i o n o f Laval and Pesme (see Section 5.3) by i t e r a t i n g Eq. (5.34) s t a r t i n g w i t h <6f16fl,> = [ r i g h t hand s i d e o f eq. (5.34)] and i n s e r t i n g t h i s formal expansion i n powers o f <6E6E> i n t o eqs. (5.31)

and (5.29).

The combination o f t h e l a s t

t w o equations then leads t o a formal expansion o f < 6 f ~ n c 6 f ~ : c > and a l s o o f

2 3 t h i s expansion agrees w i t h t h e r e s u l t s o f Laval and Pesme which was o u t l i n e d i n Section 5.3; because Sk i n powers o f .

To terms f o r m a l l y o f order <6E >

o f wave-particle resonances f o r yk<(k2DQL)1’3

t h e terms i n Sk o f t h i s formal 2

order appear t o be a c t u a l l y o f order <16Ekl >, a r e s u l t which i s c o n s i s t e n t w i t h eq. (5.37).

A complete s o l u t i o n o f t h e D I A equations f o r < 6 f ~ n c 6 f v l n C > k w

remains t o be c a r r i e d out.

The s e l f - c o n s i s t e n t ( n o n l i n e a r p o l a r i z a t i o n ) terms

2

o f the D I A do n o t appear t o change t h e order i n <16Ekl > o f q u a n t i t i e s such as Sk b u t they are n u m e r i c a l l y important and must be included, f o r example, t o pre-

serve energy conservation. Conc 1 us ion I n t h e f i r s t p a r t o f t h i s paper we have reviewed t h e r e s u l t s o f i n t r i n s i c s t o c h a s t i c i t y t h e o r i e s i n the case where t h e imposed e l e c t r i c f i e l d has random phase F o u r i e r components. Whenever t h e mode amplitudes a r e w e l l above t h e t h r e s h o l d f o r i n t r i n s i c s t o c h a s t i c i t y , an e x c e l l e n t agreement has been demons t r a t e d between t h e numerical experiments and t h e quasi1 i n e a r p r e d i c t i o n s f o r t h e d i f f u s i o n c o e f f i c i e n t and the Kolmogorov entropy. I n t h e second p a r t we have considered t h e Vlasov-Poisson turbulence i n which t h e e l e c t r i c f i e l d i s s e l f - c o n s i s t e n t l y r e l a t e d t o t h e p a r t i c l e s motion through Poisson’s equation. We have f i r s t observed a s t r i k i n g d i f f e r e n c e compared w i t h t h e imposed e l e c t r i c f i e l d case: t h e numerical s i m u l a t i o n s e x h i b i t a s i g n i f i cant discrepancy w i t h t h e q u a s i l i n e a r p r e d i c t i o n . The o r i g i n o f t h i s d i s c r e pancy has then been shown t o be a consequence o f a s t r o n g enhancement o f t h e wave-particle i n t e r a c t i o n . As a r e s u l t , the s t a t i s t i c a l p r o p e r t i e s o f t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d are c h a r a c t e r i z e d by a second c o r r e l a t i o n time Self-consistency (k2D)-1’3 much longer than the usual c o r r e l a t i o n time t leads thus t o a s o r t of s e l f - o r g a n i z a t i o n o f t h e turbulgnce. Dupree’s theory now appears t o be a p a r t i a l summation o f the i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n s o f t h e e l e c t r i c f i e l d which are neglected i n t h e simply renormalized t h e o r i e s . I t has, however;a remaining l a c k of consistency i n t r i n s i c t o a l l quasigaussian

.

0. PESME, D.F. DUBOlS

462

theories. We have then displayed the c h a r a c t e r i s t i c features o f D I A a p p l i e d t o Vlasov turbulence and we have shown how t h i s c l o s u r e takes i n t o account i n a s e l f - c o n s i s t e n t way the strong wave-particle i n t e r a c t i o n and t h e r e s u l t i n g nongaussian p r o p e r t i e s o f the e l e c t r i c f i e l d . ACKNOWLEDGEMENTS We are pleased t o acknowledge h e l p f u l comments from D r . J . C. Adam, D r . G. Laval, D r . H. Rose and O r . T. O'Neil. Conversations w i t h D r . F. Doveil, D r . 0. Escande and D r . A. Rechester are a l s o acknowledged. REFERENCES

52,

1.

Chirikov, B. V.,

2.

Escande, D. F. and Doveil, F. , Phys. L e t t . 83A, 307 (1981); Doveil, F. and Escande, 0. F., Phys. L e t t . 84A, 399 Escande, D. F. and Doveil, F., J. Stat. Phys. 6, 257 ( 1 9 8 1 r

3.

Greene, J. M. , J. Math. Phys. 2, 760 (1968). Lieberman, M. A. and Lichtenberg, A. J . , Phys. Rev. A 2, 1852 (1972). Rosenbluth, M. N., Phys. Rev. L e t t . 2, 408 (1972). Greene, J. M., J. Math. Phys. 20, 1183 (1979). Schmidt, G., Phys. Rev. A, 22, 2849 (1980). For references on a p p l i c a t i o n s see, e.g., Smith, G. R. , and Kaufman, A. N . , Phys. F l u i d s g ,2230 (1978).

4.

Rechester, A. B.,

5.

Rechester, A. B., Rosenbluth. M. N., and White, R. B., Phys. Rev. A Cohen, R. H., and Rowlands, G., Lawrence Livermore National Laboratory p r e p r i n t UCRL-85600, U n i v e r s i t y o f C a l i f o r n i a (1981). Karney, C. F. F . , Rechester, A. B., and White, R. B. , Princeton Plasma Physics Laboratory Report P.P.P.L. 1752 (1981). Cary, J. R. , Meiss, J. D., and Bhatta Charjee, A., I n s t i t u t e f o r Fusion Studies Report IFSR-6, A u s t i n

Phys. Rep.

263 (1979).

(lm);

and White, R. B., Phys. Rev. L e t t .

44,

1586 (1980).

g,

2664 (1981).

(1981). 6.

Rechester, A. B.

42, 1247 (1979).

,

Rosenbluth, M. N., and White, R. B., Phys. Rev. L e t t .

7.

Rechester, A. B., Rosenbluth, M. N., and White, R. B., i n : I n t r i n s i c S t o c h a s t i c i t y i n Plasmas, e d i t e d by G. Laval and D. G r e s i l l o n (Les Ed. de Physique, Orsay, 1979), p. 239.

8.

Molvig, K., Freidberg, J. P., Potok, R., Hirshman, S. P., Whitson, J. C., and Tajima, T., I n s t i t u t e f o r Fusion Studies Report, A u s t i n (1981).

9.

Flynn, R. W.,

10.

Doveil , F. , and G r e s i l l o n , D., i n Proceedings o f t h e I n t e r n a t i o n a l Congress on waves and i n s t a b i l i t i e s i n Plasma, Palaiseau, France 1977, p. 51.

Phys. F l u i d s

14,956

(1971).

11. Selected Papers on Noise and Stochastic Processes, e d i t e d by N. Wax (Dover, New York, 1954), p. 255.

12.

Gary, S. P., and Montgomery, D.,

13.

Graham, K. N., and Fejer, J. A.

,

Phys. F l u i d s

11,2733

(1968).

Phys. F l u i d s

2, 1054

(1976).

TURBULENCE AND SELF-CONSISTENTFIELDS IN PLASMAS

463

14. I n t h e complete p h y s i c a l problem o f a c o l l e c t i o n o f d i s c r e t e p a r t i c l e s f(xo,vo) (and t h e r e f o r e , f ( x , v , t ) ) i s a very s i n g u l a r o b j e c t . We w i l l consider here o n l y the Vlasov approximation which assumes t h a t f(xo, vo) i s a smooth f u n c t i o n o f xo and vo; t h i s i s known t o be v a l i d i f

3

nhD>>lwhere n i s the plasma d e n s i t y and AD t h e Debye length.

15. Sturrock, P. A.,

Phys. Rev.

141, 186

(1966).

16. Thomson, J. J., and Benford, G . , J. Math. Phys. 17. Dupree, T. H., Phys. F l u i d s

2 , 334

14,531 (1973).

(1972).

18. Pesme, D., and B r i s s e t , A., t o be published. 19. Adam, J-C., Laval, G., and Pesme, D., i n Proceedings o f t h e I n t e r n a t i o n a l Conference on Plasma Physics, Nagoya, Japan, 1980. 20. O ' N e i l , T. M., and Malmberg, J. H., Phys. F l u i d s

11, 1754 (1968).

21. Bezzerides, B., Gitomer, S. J., and Forslund, D. J., Phys. Rev. L e t t . 44, 651 (1980). 22. Drummond, W. E . , and Pines, D., Nucl. Fusion, Suppl. P a r t 3, 1049 (1962). Vedenov, A. A., Velikhov, E . P. and Sagdeev, R. Z . , Nucl. Fusion, Suppl. P a r t 2, 465 (1962).

z,

1816 (1964). Zaslavsky, 23. Aamodt, R. E . , and Drummond, W. E . , Phys. F l u i d s G. M., and C h i r i k o v , B. V., Usp. F i z . Nauk 2,195 (1972) [Sov. Phys. Usp. 2 , 549 (1972)l. 24. O ' N e i l , T. M., Phys. Rev. L e t t . 25. Adam, J. C . , Laval, G.,

33, 73

and Pesme, D.,

26. Laval, G., and Pesme, D., Phys. L e t t e r s

(1974).

43,

Phys. Rev. L e t t .

E, 266

1671 (1979).

(1980).

27. Rudakov, L. I., and Tsytovich, V. N., Plasma Phys. 13, 213 (1971). D., and Horton, W. J r . , Phys. F l u i d s 11, 2048 (1974r 28. Adam, J. C . , Laval, G., (1980).

Choi,

and Pesme, D . , Suppl. J. de Physique 4 l , C3-383,

29. DuBois, D. F., Phys. Rev. A

2,865

(1981).

30. Kadomtsev, B. B., Plasma Turbulence (Academic Press, New York, 1965). 31. Kraichnan, R. H., J. Math. Phys.

2, 124 (1961).

32. Orszag, S. A., and Kraichnan, R. H., Phys. F l u i d s 33. DuBois, D. F . , and Espedal, M., 34. Krommes, J. A.,

and Kleva, R. G.,

Plasma Phys.

20, 1209

Phys. F l u i d s

35. Krommes, J. A., and Similon, P., Phys. F l u i d s 36. M a r t i n , P. C.,

lo, 1720

Siggia, E. D., and Rose, H. A.,

(1967).

(1978).

2,2168 (1979).

23, 1553

(1980).

Phys. Rev. A

8,

423 (1973).

464

0. PESME. D.F. DUBOIS

appears to be only a restriction on the level 37. The condition of electric field fluctuations whereas the condition for smallness of the vertex corrections to the D I A must also involve some restriction on <6f2>. The fluctuation level <6f2> is asymptotically in time proportional to <6E2> and both are bounded. Convergence would seem possible for sufficiently small and smooth initial fluctuations < 6 f2>.

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (edc.) @ North-Holland Publishing Company, 1982

465

SEW-FOCUSMG TENDENCIES OF 'RIE NaJLMEAR SCHF~DINGERAND ZAKHAW~VEQOATIONS HARVEY A. RC6E AND D. F. WBOIS

IDS A l m s National Laboratory

P. 0. Box 1663 LOS AlamoS, NM

07545

The Zakharov equations and an associated nonlinear SchrUdinger (NLS) equation are models of long wavelength, interacting Langmuir waves. Although there seems to be a parameter regime i n which these models consist of weakly interacting waves, t h i s regime is pre-empted by self-focusing; which, i n the case of the NLS equation, can generate a singularity i n a f i n i t e time. 'Ihe implications for a s t a t i s t i c a l theory are discussed.

The problem of Langmuir turbulence i n plasnas is one of the conceptually s h p l e s t and most studied problems in nonlinear plasma physics.1,2,3 'Ihe physical situation is that of a c-letely ionized, urmagnetized plasm w i t h electron temperature mch higher than the ion temperature where the linear collective modes of the system are longitudinal Langmuir modes and ion-sound modes and transverse electromagnetic modes. A useful &el which we w i l l use here and which describes many of the interesting non-linear features of the system is the model of Zakharov4 which explicitly separates the f a s t and slow time scales of the system. Weak turbulence theory is built on the linear collective modes of the system. It has been studied since the early 1 9 6 0 ' ~ .It~ is well-known1r2 that t h i s theory predicts a transfer of Langmuir wave intensity to low wave nunbers by a cascade of three wave processes of the form 8 8' + s, i.e., the three wave process which converts a Langrmir wave ( 8 ) w i t h wave nunber into a Langmuir k' and vice wave ( 8 ' ) w i t h nunber k' and an ion-sound wave w i t h wave nunber versa. 'Ihe frequency matching condition for these processes is the well-knm one

*

-

H.A. ROSE, D.F. DUBOIS

(1.1)

112

Langmuir frequency and cs = (Tdmj is the ion-sound speed where we express temperature in energy u n i t s . If Langmuir energy is injected a t some f i n i t e k >> (1/3) ( m d m i ) 1/2km = k* then Langmuir i n t e n s i t y cascades t o lower k ' < k u n t i l i.) the intensity is low enough for some f i n i t e $in > k* so that the mode hin is stabilized by linear collisional damping or ii.) unstable modes for k < k* are excited for which further decay by t h e process (1.1) are not allowed. "his piling up of Langmuir i n t e n s i t y a t low k has been called the "Langmuir condensate." The question then arises as t o how t h e energy i n t h i s condensate is dissipated. A t t e m p t s have been made to invoke weak turbulence processes such as wave-wave scattering, i.e., Ll + L2 = a3 + i4. However, as we w i l l show here explicitly the weak turbulence scheme breaks down for low wavenumbers k < k*. is that coherent or incoherent Langmuir oscillations a t long wavelengths are unstable t o modulational instability i n which regions of excess Langmuir oscillations deplete the local electron and ion density by action of t h e ponderomtive force, and the density depletion through its effect on t h e Langmuir wave phase speed, a t t r a c t s the trajectories of nearby waves. *is instability involves the competition of linear dispersive and dissipative terms w i t h the nonlinear "self-focusing" effect of the ponderomotive force; t h i s is a case of effective "Reynold's number" of order unity. The linear stage of the modulational instability does not directly transfer energy t o the high k region where Landau damping can be an effective dissipation process. It

The conceptual

picture of the nonlinear evolution of the modulational instability derives from a combination of computer experiments and special analytic solutions.' In the case of one dimension, it a w a r s 5 t 6 that the modulational instability can develop into a set of spiky structures which may be loosely related t o the known soliton i n i t i a l value solutions of the undriven, When the system is undamped Zakharov or nonlinear SchrUdinger equations. driven, e.g., by a uniform pump, there is conflicting evidence5 that the& spiky I n two or three structures may, i n fact, undergo a singular collapse. dimensions it is'well-known from t h e pioneering work of Zakharov4 and m r e recent work by Goldman and Nicholson6 that a localized Langmuir wavepacket w i l l tend t o collapse i f the electrostatic energy i n the packet W s a t i s f i e s

SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EOUATION

W >

nTe

467

dk

f(-)2

kD

where Ak is the width of the spectrum in wave nunber and f is a constant of order unity. The ultimate evolution of the collapsing wave packet presumably involves strong wave particle interaction i n which the energy of the packet is given up t o very energetic electrons. In two or three dimensions it appears that direct collapse can be initiated without t h e intermediate stage of modulat ional instability

.

is conjectured that the nonlinear stage of the modulational instability results i n snall scale structures of the order of several Debye lengths which It

dissipate energy by wave particle interaction. Galeev, e t al.' argue that snall scale density fluctuations are created by sound radiation from supersonically collapsing wave packets; these snall scale density fluctuations provide an enhanced absorption (analogous to inverse Bremsstrahlung) of longer wavelength Langmuir waves which terminates the collapse. In our research we have attempted t o formulate a systematic s t a t i s t i c a l theory of Langmuir turbulence using the "strong" turbulence or renormalization methods of Kraichnansrg which were originally applied to incompressible fluid turbulence. The simplest renormalized theory, the so-called Direct Interaction Approximation (DIA) has been successful i n describing moderate Reynold's number fluid turbulence so it seemed reasonable t o apply it t o the Langmuir turbulence problem w i t h "Reynold's nunber" of order unity.

In applying these methods t o the Langrmir turbulence problem one is innnediately faced with problems not encountered i n the u s u a l applications t o isotropic, homogeneous Navier-Stokes turbulence. Among these problems is the existence of weakly damped, oscillatory modes corresponding i n the l i m i t of large k t o the unperturbed (Langmuir and ion sound) modes of the system and the existence of nonlinear normal modes due to self focusing, i.e., modulational instability which goes over t o transient self trapping i n the steady state. These problems corrplicate both the nunerical solution of the DIA equations and further analytic approximations such as the "pole approximation" discussed below.

H.A. ROSE, D.F. DUBOIS

468

2.

THE

ZAKH Aw ) v

EQUATIONS

L e t $ ( x , t ) be the (complex valued) low frequency envelope of potential. %en the Zakharov equations a r e 4 1 1 r 2

the e l e c t r o s t a t i c

where $*(x,t) is the c q l e x conjugate of $ ( x , t ) . are using a dimensionless set of u n i t s here’ i n which distances a r e measured is the electron Debye length, t h e is i n u n i t s of 32(mi/nQ 1/2X where X 3De De measured in units Of T(mi/%)upe-l where is the electron plasma frequency, We

WP,

e l e c t r i c f i e l d strengths a r e measured i n u n i t s of (me/mi) 1/2(16nno~J3) where no is the average ion or electron density and Te is t h e electron temperature and the density fluctuation n is measured i n u n i t s of %(4m$3mi). Phenommological Langmuir wave damping ve and ion wave damping u i ( i n the above units) have been added to the equations; we can be chosen to be nonlocal i n real space to mimic Landau damping i f desired. We w i l l not discuss the conditions for v a l i d i t y of these equations here since they have been given elsewhere.’ Cur goal is to study the s t a t i s t i c a l theory of these model nonlinear equations; our hope is t h a t further physical refinements can be added l a t e r in a straightforward way. They can be c a s t i n t o Hamiltonian form’ when the d i s s i p a t i v e coefficients, ve and wi, vanish by formally regarding $ and $* as independent f i e l d s arid by the hydrodynamic flux p o t e n t i a l u:

introducing

(2.3a)

(2.3b)

-an =

at

0%

(2.3~)

SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EQUATIONS

469

(2.3d)

Define the Hamiltonian H a s the volume integral of t h e density h.

+ '[2

h = V2$V2$*

(_OU)~

+

n2]

+ n_O$*_O$* .

Then equations ( 2 ) a r e equivalent, when ue = u i = 0, t o the canonical system.

-an = -

u -a =

6H/6u

at

at

6H/6n

In addition t o H , equations (2.5) conserve: the plasna nunber N,

N = I_O$*_O$*dz= IE*E*dz

(2.6)

where E is the corrplex e l e c t r i c f i e l d envelope,

P, =

J{+

i(q,,aE;Z/ax,,,

-
-n

and the angular momentun

E

= /{i

x

E* + 5 x p}dg

where p is the linear mmentun density.

- I$; the linear

au/ax,}dx

momentun

H.A. ROSE. D. F. DUBOlS

470

3.

THE "LINEXR SCHR~DINGER EQUATION

The NLS equation;

is derivable from the Hamiltonian density

I t has the p l a m n nunkr as an invariant, as well as momentum

I n special dimension, d , equal t o one, it has an infinite nlanber of invariants. There is a close relation between (3.1) and the adiabatic l i m i t of ( 2 . 1 ) , (2.2) :

sound waves enables the density fluctuations to adjust rapidly t o the instantaneous pondermtive force so that (2.2) can be replaced by

a t long wavelengths (>>1) the f a s t phase speed of the ion

When t h i s is substituted into (2.1) a NLS equation is obtained (3.1),

but

which

resembles

for which there is no self focusing theorem w i t h the generality of

(5.1).

4.

FAILURE OF A WEAKLY INTERACTING WAVE APPROXIMATION

If the electric field has a snall amplitude, can the treated as a correction to Ho,

IEI4 term in

(3.2)

be

SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EQUATION

471

Even i f t h i s is true i n i t i a l l y , the evolution of (3.1) can lead t o a spectral distribution [of the fourier transform of E , E ( k ) ] which violates l H I l << Ho. The general tendencies for spectral evolution can be inferred from t h e conservation laws for H and N. L e t u s define the spectral function

where do is the solid angle differential.

If [ H I ]

<<

HO, then

m -

H

Ho =

/0

k2C(k)dk,

and, i n general,

I f , a t t = 0 , C ( k ) and k2C(k) are peaked a t a wave nlnnber k0, the evolution of the NLS equation cannot lead t o a spectral function similarly peaked a t a k f ko because the area under t h e peak must be constant t o conserve N , while its k 2 moment must be constant t o conserve Ho. Nor can there be a symmetric broadening of the peak since t h i s would increase Ho. Adepletion of the peak w i t h a stronger transfer of C ( k ) towards k < ko is the only kind of consistent w i t h conservation of Ho and N.

spectral solution

I n the framework of weakly interacting waves, a kinetic equation for the evolution of C ( k , t ) , whose collision term only includes resonant wave interactions, is known as weak turbulence theory (WIT). The scaling solutions (C(k) l / k n ) for the NLS equation have been discussed by Z a k h a r o ~ ~who ,~~ showed that these solutions have a flux of ‘N towards the infrared, and a flux of

-

Ho towards the ultraviolet.

For t h e Zakharov equations, t h i s tendency has been seen i n nunerical simulations.6 These are two confirmations of the tendency deduced from general principles i n the preceeding paragraph. (A similar general tendency was pointed out by Kraichnan” for t h e two dimensional Navier Stokes equation.)

472

H.A. ROSE, D. F. DUBOIS

T h i s transfer of N t o ever

longer wavelengths, w i l l eventually lead t o a spectral distribution which is inconsistent w i t h the ansatz l H I l << Ho. *is follows from the gradients ( k 2 factor) i n Ho, and the absence of gradients i n HI. Cne specific mechanism which w i l l come into play is the modulational instability. 'Ihis instability is easily seen i n the special case of a spacially uniform solution t o t h e NLS equation (i.e. a spectrum peaked a t k = 0 , w i t h a peak width of zero). Aside from a time dependent phase factor, solutions is characterized by I E I 2 = W = N/unit length

t h i s class of

The linearization of the N U equation about t h i s solution yields an eigenvalue The eigenvalue has an problem, w i t h eigenfunctions, 6E, of t h e form 6E eih. imaginary part, y , for k2 < 2W (we specialize here to d = 1)

-

y =

.

k(2N-k2)1/2

If the spectrun has a wave nunber width A , then for A small enough so that there is a modulational instability,

Y

- A*A]

k [ (2W-k2) 'I2

where A is a constant of order unity.

5.

~ L L A P S I N GSOLUTIONS OF THE NLS EQUATION

Goldman and Nicholson6 have proven the following theorem for

solutions to the NLS equation, (3.1). If t h e solution is spacially localized, so that various surface integrals a t infinity can be ignored, then

= [2HD

-

P 2 / N 2 ] t 2 + Bt t

t (2-3

0

It'd"' 0

t = O

SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EQUATION

473

where d is the spacial dimension,

(5.2)

(5.3)

This implies that i f

H

<

P2/2N

(5.4)

then for d > 2

GIt =

tc = 0

(5.51

where tc is t h e collapse the, O
unless

the solution E ( x t ) develops a

-

If t h e f i r s t singularity occurs a t tc, then for t + tc, IEl is going to zero everywhere, except a t r=F(t), where IEl + because N is a constant. "his is aptly described as "collapse." Unless the i n i t i a l conditions preempted by a l e s s violent

have a special symmetry, we believe that tc is k i n d of singularity. Consider an i n i t i a l

condition, Eo, which is the superposition of two fields, El and El'. each of 0 0 the l a t t e r two are spherically symmetric, spacially localized and identical t o

is centered a t E = (-L, 0, O ) , and the other a t (L, 0 , 0 ) . As L + m , it is plausible that these two spheres evolve as i f its twin were absent, especially i f each sphere by i t s e l f s a t i s f i e s H < P2/2N. Therefore the time for t h e formation of a singularity should be t h e same as for a single sphere i n i t i a l condition. However, the i n i t i a l value of 6r2 is the only parameter i n (5.1) which approches infinity as L does (it goes l i k e L 2 ) . Therefore (5.5) implies each other

except

that

one

474

H.A. ROSE, 0.F. DUBOIS

The basic reason why tc is hardly ever attained is that t h e NLS equation is spacially local, while the collapse condition (5.4) is a global property. Since the follcwing transformation takes solutions of the NLS equation into another solution:

E(x,t)

+

S*E(Sx,S2t)

(5.6)

where S is a scale factor, there are only intrinsic measures of distance, one of which is determined by the field strength

and another is the gradient scale length

Perhaps the collapse condition (5.4) can be improved by replacing the global integrals of h ( x ) , p ( x ) , and I E I 2 ( x ) ( t o give H, p, and N) by local integrals of linear extent EE O K Ep. The condition obtained by using is consistent w i t h our intuition concerning t h e two, well separated, sphere i n i t i a l condition. course, the physics of plasnas would invalidate the N L S equation i f field strenqths and gradients become two large. However, it is conceivable that w i t h i n i t i a l plasna conditions which were characterized by very weak, long wavelength fields (field energy << thermal energy, gradient scale lengths >> Debye radius) the plasna could evolve t o a state which is strongly collapsed relative t o the i n i t i a l conditions, but still be i n a physical regime where the NLS equation is still valid.

Of

As collapse continues i n time t h e density w i l l not keep up w i t h the rapid variations of E, and the adiabatic relation (3.3) m u s t be replaced by the Zakharov equations. These too w i l l become invalid as collapse intensifies, and t o properly account for the interaction between fields and particles the Vlasov equation is needed.

SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EOUATION

475

i n the s t r i c t sense that (6rI2 + 0 , then the singularity would be a t a p i n t , i.e., a set w i t h dimension zero. Even though t h i s many not be realized, it still m u s t be true that whatever the nature of the singularity, infinite field strengths are present because otherwise the dispersive term i n (3.1) would dominate t h e nonlinear term, and the resulting linear SchrUdinger equation is w e l l behaved. Since N is conserved, t h e regions where IEl is becoming stronger and stronger must f i l l a snaller and snaller fraction of t h e spacial volume, and t h i s is what we mean by intermittency. If collapse did occur for solutions of the NLS equation, -

i n the n e x t section can accurately describe a strongly intermittent solution of the N L S equation. However, there are physically relevant parameters r e g h e s i n which collapse is cut off by dissipative effects so that the intermittency of the solutions to the highly idealized NLS equation is not actually realized. S t i l l , there is a tendency to self-focus, so that a theory that is more comprehensive than weak turbulence theory is called for. I t is unlikely that the s t a t i s t i c a l theories of the k i n d

6.

discussed

STATISTICAL THEORY OF THE ZAKHAKN AND NLS EQUATIorUS

Since the NLS equation is a long wavelength l i m i t of the Zakharov equations, it is possible t o develop a s t a t i s t i c a l theory of the former by taking the long wavelength limit of the l a t t e r . Conversely since, by solving (2.2) for n in terms of IVJ1I2, and substituting i n (2.1) , the Zakharov equation for J, h a s the form of a generalized NLS equation, a s t a t i s t i c a l theory of a NLS equation could be used for the Zakharov equations. has been found that t h e DIA12 as applied t o the Zakharov equations is essentially equivalent13 t o the Screened Potential Approximation for the NLS equation ( i - e . , the classical field theory analog of the quantun mechanical plasna approximation that is needed t o understand electrodynamic screening of charged particles). The propagation of infinitessimal disturbances i n t h e density field n, d,of (2.2) (disturbances of I E I 2 i n 3.1) is modified by polarization-like effects, which for the Zakharov equation DIA, amounts to replacing k 2 , t h e fourier transform of (-a2/ax2) i n the inverse of the sound wave propagator, It

H.A. ROSE, D.F. DUBOlS

476

by k2

+

k20(k,w)

where o(k,w) is t h e transform of o ( x , t ) ,

-

o ( x , t ) = i[R$J(xt)C*(xt) c.c.]

and where R$J is the mean propagator of infinitessimal disturbances i n

$J,

and

c(xt) = - y. = x, T - T * = t. Hombgeneous turbulence has been results presented for d = 1, w i t h similar results for d > 1.

with y

While

the

traditional

polarization of

assmed

charged particles

and

the

involves a

Debye rearrangement of charge density over an infinite distance (i.e., screening) when a local charge inhomogeneity l a s t s a long time, i n the Zakharov and NLS equations, there can be transient but long lived (or even unstable) trapping of Langmuir waves by density inhomogeneities i f the wave nunber bandwidth of t h e spectrum of fluctuations of E is mall compared t o (W)ll2. These are the self-focusing and modulational instability mechanisms mentioned earlier. For the special case of a narrow bandwidth spectrum of Langmuir wave fluctuations centered a t k = 0, we have compared the linear response functions a s predicted by the DIA with a mnte Carlo simulation of a few fourier modes extracted from t h e Zakharov equations and found qualitative agreement. l 2

As an aid i n understanding t h e structure of the DIA, and also for the purpose of finding a method of solution that is more efficient than nunerical integration of the DIA, we12 have developed a consolidated continued fraction technique that is applied to the space and time fourier transformed correlation and linear response functions C(kw) and R ( k w ) . L e t u s recall that the form of the DIA is

477

SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EQUATION

where G is a generalized propagator that includes both C and R; and quadratic functional. Equation (5.1) is replaced by

1

is a

Under some mild restrictions it is easy t o show that each iterate, Ga(kw) is a rational function of w and hence is characterized by a nmber, n ( t ) , of poles and residues. Since n ( a ) is a rapidly growing function of a , a nunerical investigation of the possible convergence of (5.2) must involve some kind of consolidation of the poles a t each iteration. For have examimed, t h i s technique (using rough, ad hoc appears to succeed i n the sense that convergence the fixed point solution t o (5.2) i n good agreement DIA, determined by nunerical integration.

the particular cases that we methods of consolidation) is frequently obtained, w i t h w i t h the solution t o the

I t is natural t o interpret the collection of poles of G(kw) for fixed k , a s nonlinear normal modes of the turbulent system because they depend on the strength and spectral distribution of fluctuations. I n addition t o modes t h a t correspond t o turbulently modified continuations of the linear normal modes (these are the only ones allowed by the various weak turbulence theories) there are new modes that correspond t o the dynamics of self-focusing and modulational instability.

Although t h i s technique is still nunerically ccinplex, it should serve as an intermediary i n the construction of a simpler phenanenological theory a t the level of the Eddy damped quasi normal approximations"+ that have proved t o be of great u t i l i t y i n the study of fully developed fluid turbulence.

REFERENCES 1.

w

useful review a r t i c l e s on the subject of Langmuir turbulence are: and D. ter Haar, "Langmuir Turbulence and Modulational Instability", Physics mports 43c (1978) 43 and L. I. Fudakov and V. N. Bytovich, "Strong Langmuir Turbulence", Physics &ports 40 (1978) 1. S. G. Thornhell

H.A. ROSE, D.F. DUBOIS

478

2.

B. B. Kadomstev, "Plasma V. N. Tsytovich,

Turbulence",

"Theory

Academic

Press,

of Turbulent Plasma",

New York,

1965.

Consultants Bureau, New

York I 1977. 3.

A. A. Vedenov and L. I. Rudakov, Sov. Phys. Doklady 9 (1965) 1073.

4.

V. E. Zakharov, Sov. Phys. JEW 35 (1972) 908.

5.

J. J. Thomson, F. J. Faehl,

6.

M. V. Goldman

and W. L. Kruer, Phys. Rev. Lett. 3 1 (1973) 918. G. J. MDrales, Y. C. Lee and R. B. white, Phys. Rev. Lett. 32 (1974) 457.

Astrophysical

J.

Phys. Rev. Lett.

D. W. Nicholson,

and

D. R. Nicholson,

M. V. Goldman,

223 (1978)

P. Hoyne,

and

605. D. R. Nicholson

41

(1979)

406.

J. C. Weatherall,

and

M. V. Goldman,

Phys. Fluids 2L (1978) 1766. 7.

A. A. Galeev,

R. 2. Sagdeev,

V. D. Shapiro,

and V. I. Schevchenko, JEW

Letters 24 (1976) 21. 8.

R. H. Kraichnan, J. Fluid Mech. 3 (1959) 32.

9.

R. H. Kraichnan, Phys. Fluids 7 (1964) 1163.

10.

V. E. Zakharov

and E. A. KuznetSov, JEW

11. R. H. Kraichnan, Phys. Fluids 12.

(1978) 458.

(1967) 1417.

D. F. Dueois and H. A. Mse, " S t a t i s i t i c a l Theories of Langmuir Turbulence:

The Direct I n t e r a c t i o n Approximation Responses, Los A l m s report LA-UR-80-3603 (accepted f o r publication i n Phys. Rev. A ) . I.

13.

D. F. Dubois, D. R. Nicholson, and

H. A. Mse,

"Statistical

Theories of

Langmuir Turbulence 11 ( i n preparation). 14.

For a review of these kinds of turbulence approximations see: H. A. Mse and P. L. S u l m , Journal d e Physique 2 (1978) 441.

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (4s.) @ North-Holland Publishing Company, 1982

479

CHAOTIC OSCILLATIONS I N A SIMPLIFIED MODEL FOR LANGMUIR WAVES P. K. C . Wang

School o f Engineering and Applied Science University o f California Los Angeles, C a l i f o r n i a U.S.A.

The r e s u l t s o f a f u r t h e r study of the chaotic s o l u t i o n s o f t h e single-mode equations corresponding t o Zakharov's model f o r Langmuir waves i n a plasma are described b r i e f l y .

Recently, i t was found [ l l t h a t the following single-mode equations corresponding t o Zakharov's model f o r Langmuir waves i n a plasma have nonperiodic chaotic solutions

where

Here, a - b denotes he usual scalar product o f two vectors and b i n the r e a l Eucl idFan space IRt; $, i s the orthonormal ized eigenfunction o f the Laplacian operator corresponding t o the eigenvalue < 0; Ym and m r are t h e phenomenol o g i c a l damping c o e f f i c i e n t s ; E and nm are r e s p e c t i v e l y t h e c o e f f i c i e n t s o f density n i n terms o f $,, defined on expansions f o r the e l e c t r i c f i d d E and a bounded s p a t i a l domain Q C R N . E EIR represents the normalized amplitude o f an external s p a t i a l l y homogeneous3lectric f i e l d (pump) o s c i l l a t i n g a t the electron plasma frequency. I n 111, chaotic s o l u t i o n s were discovered through numerical experimentation and b i f u r c a t i o n analysis w i t h Il&lp as a v a r i a b l e parameter. The s t r u c t u r e and the onset o f these s o l u t i o n s were n o t explored i n detail. Here, we summarize the r e s u l t s o f a f u r t h e r study o f t h e chaotic solutions o f ( 1 ) - ( 3 ) .

icp

1.

B i f u r c a t i o n o f E q u i l i b r i u m States

For fixed 6,,, > 0 and v a r i a b l e parameter

Il&11',

the t r i v i a l s o l u t i o n 2 = (nmm.~m,E+,)

(O,O,Q+ iQ) i s the o n l y e q u i l i b r i u m s t a t e when 0

ri

11&,11'

< E&,

where

480

P. K.C. WANG

A t 11&112 = E&, a n o n t r i v i a l e q u i l i b r i u m s t a t e z emerges and i t b i f u r c a t e s i n t o two d i s t i n z t e q u i l i b r i u m states and 3 when *I&,IIZ > E&. For t h e case where 1 1 h 1 1 i s f i x e d and ym i s a v a r i a b l e parameter, t h e t r i v i a l s o l u t i o n i s the For 0 < y, < y,$, and y, # where o n l y e q u i l i b r i u m s t a t e when ym = 0.

%,

we have, i n a d d i t i o n t o the t r i v i a l e q u i l i b r i u m state, two d i s t i n c t n o n t r i v i a l e q u i l i b r i u m s t a t e s and they merge i n t o one when y = yc. When ym increases beyond ,y; the n o n t r i v i a l e q u i l i b r i u m s t a t e s abruptyy digappear. A t y, = o n l y a t r i v i a l and a n o n t r i v i a l e q u i l i b r i u m s t a t e s e x i s t .

%,

2. Chaotic O s c i l l a t i o n s

I n the case where ym and rmcorrespond t o Landau damping, Y v a r i e s r a p i d l y f o r small p m X ~> 0, where AD i s the Debye length. It i s interest t o determine the threshold value o f Ym f o r the quenching o r onset o f c h a o t i c o s c i l l a t i o n s . A d e t a i l e d numerical study was made f o r the one dimensiona w i t h fi = ]O,L[, , !-11= n/L, L = n/m, r l = 2, a1 = ( 2 /L)/(3n) and 91(x) = JB = -8nm!?L!0r the t y p i c a l case where :E = 2.669, we have =,.5.3358. I i can be v e r i f i e d t h a t Hopf b i f u r c a t i o n ( w i t h respect t o the parameter Y = l / v l ) takes place about the e q u i l i b r i u m s t a t e z+ when Y YH = 4.35. As Y i s decreased from YH, i t was found t h a t the soeutions a b i u t zg change from b e r i o d i c t o almost p e r i o d i c and become c h a o t i c when Y Y =1.55. The power spectra o f the e l e c t r i c f i e l d evolve from e s s e n t i a l l y d l s z r e i e spectra t o broadband spectra as Y crosses YT. The phenomena appear t o be more complex than simple p e r i o d We a l s o obtained numerically the p o i n t s doubling as observed i n many systems. generated by Poincar6 maps corresponding t o various hyperplanes i n z-space which Typical r e s u l t s are sFown i n are transverse t o the c h a o t i c t r a j e c t o r i e s . Figures 1 and 2. Evidently, t h e p o i n t s appear t o l i e i n the neighborhood o f some curve i n R 3 , implying t h a t the Poincar6 map i s e s s e n t i a l l y one-dimensional However, t h e graphs o f t h e Poincar6 maps i n terms o f some curve i n nature. A more d e t a i l e d d e s c r i p t i o n o f t h e aboveparameter are n o t r e a d i l y obtainable. mentioned r e s u l t s i s given i n reference [Z].

o'P

$-F"

YT

REFERENCES

[l]Wan ,P.K.C. ,"Nonlinear O s c i l l a t i o n s o f Langmuir Waves", J. Math. Physics, 21 81980) 398-407. [2] Wang,P.K.C. ,"Further Study o f Chaotic O s c i l l a t i o n s o f S i m p l i f i e d Zakharov's Model f o r Langmuir Turbulence", Univ. o f C a l i f . Engr. Rprt. No. UCLA-ENG8013 (March,1980).

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+

0

0

0

0

N 0' 0 0

N

0 0

W

N 0

0

N 0

W 0

0

0

0

0' 0

0

m

rn'

0

0

0

m0

m0

3 '"'

-c

0

0

0'

c

0

0

-

y. 0

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0

-=ln -I

+ 0 0

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ION DENSITY

0- 00.00

P. K. C. WANG

I00

40

-20

n -80 ..

.:.

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.

."

-1 40

-200

-100

-80

-60

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-40

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Fig. 2: Isometric representation of points generated by Poincar; map (for the solution shown in Fig.1) onto the hyperplane: Im E 0 in z-space.

-

483

INDEX OF CONTRIBUTORS

Andrei, N., 169 A r i s , R., 247 Bardos, C., 289 Bishop, A.R., 195 Brauner, C.M. , 335 Campbell , D. K. , 195 Chang, S.-J., 395 Crandall, M.G., 117 Dubois,

D.F., 425, 465

Emery, V.J., 159 Etemad, S . , 209 Feigenbaum, M.J. , 379 F i f e , P.C., 267 Flaschka, H., 65 Foias, C., 317 Fremond, M., 109 FriaZ, A., 109 Gollub, J.P., Greene, J.M.,

403 423

Heeger, A.J., Holland, R.R. Howes, F.A. , Hyman, J.M.,

209 229 329 91

Jeandel, D.,

335

Kumar, P.,

229

Leach, Lewis, Lions, Lions,

P.G.L., 133 H.R., 133 J.-L., 3 P.L., 17

Mathieu, J., 335 McCormick, W.D., 409 Newell, A.C., Newman, W . I . ,

65 147

Orszag, S.A.,

367

Palmore, J.I., 127 Patera, A.T., 367 Pesme, D. , 435 Raupp, M. , 109 Rice, M.J., 189 Rose, H.A., 465 ROUX, J.C. , 409 Ruck, H.M., 237 S a t t i n g e r , D.H., 51 S c h r i e f f e r , J.R. , 179 Singer, I.M., 35 Swinney, H.L., 409 Temam, R., 317 Turner, J . S . , 409 Wang, P.K.C., 479 Wortis, M. , 395 Wright, J.A., 395

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