Algebraic aspects of nonlinear differential equations

ALGEBRAIC ASPECTS OF NONLINEAR DIFFERENTIAL EQUATIONS

Yu. I. Manin

UDC [512.88+517.946]:530.145.6

INTRODUCTION 0.1.

This article is an attempt to systematize and display the basic structures in a

vast, beautiful, and important mathematical literature which has accumulated during the last decade as a result of the boom surrounding the Korteweg-de Vries equation ut =6uux—tixx*. This equation was proposed and investigated in 1895 for describing long surface waves in a channel with rectangular walls. Korteweg and de Vries also obtained its first solutions: "cnoidal wave" — an infinite periodic wave train moving with constant speed without changing form — and its limiting case — the solitary wave or soliton (the terminology of Kruskal and Zabusky). As early as August 1834 Scott Russell observed a soliton on water.

He subsequently

described his observations in the "Report on Waves" (see the reference in [47]) which is these days widely cited everywhere from the SLAM Review to the Young Technician.

The lively

image of him galloping along the banks of a channel in pursuit of a solitary wave evokes in the modern reader a mild and pleasant nostalgia harmonious with the general style of a bygone era. It is not often that a single problem brings to life an entire theory. we are presenting its inception.

It seems that

The number of publications on this topic is now counted in

the hundreds; what is more important, in the theory of nonlinear differential equations a clear shift of interests and a reevaluation of priorities are taking place. For the fundamental equations of classical fields, special hopes rest on the existence of soliton solutions and their peculiar nonlinear superpositions.

The elementary particles

are possibly related to such solutions. There are still no general notions of what soliton and multisoliton solutions really are, which classes of equations have them, and how they are related to the presence of an infinite sequence of conversion laws and the so-called Backlund transformations.

There is

only a rich experimental material which remains to be sorted out. The word "experimental" is here meant in the broad sense to include direct observations, calculations on the computer, and investigation of particular interesting classes- of equations.

It is clear that the

deciphering of the structure of these phenomena must play a fundamental role in understanding them.

A good and to considerable extent algebraic and geometric theory is needed, and this

remains to be created.

Translated from Itogi Nauki iTekhniki, Sovremennye Problemy Matematiki, Vol. 11, pp. 5-152, 1978.

0096-4104/79/1101-0001$07.50

©1979 Plenum Publishing Corporation

1

We shall briefly describe certain properties of the Korteweg-rie Vries equation which may serve to clarify the plan of the present article. 0.2.

Derivation of the Equation.

written in the form

H/(—-C M.V.V = 0.

form u = f(x+ct)+g(x—ct) stant speed right.

2

The usual linear one-dimensional wave equation is

Its general solution is a sum of two waves of arbitrary

one of which moves to the left and the other to the right with con-

c . We consider the equation

ut+cux = 0, which distinguishes waves moving to the

Among its solutions are the harmonic waves u = expi(t<>t—kx) where the frequency w and

the wave number k are related by u = cA, or for waves of both types by co2 = c2A2, where

c is

characteristic of the medium. If the wave equation remains linear but includes derivatives of higher order, then the relation between the frequency and the wave number of a harmonic wave may have the more general form u£ = f(kr),

where / is not necessarily a linear function.

In the approximation

of long waves, i.e., small k, we may restrict attention to the first two terms of the Taylor series for / and write u ^ c ^ - f ^ or n s r f - L - i L .

Waves with this dispersion relation

(dependence of the frequency on the wave number) are described by the equation tit-\-cux — e „ ~2c uxxx — " •

On the other hand, the simplest nonlinearity enters if it is assumed that the speed depends on the amplitude u . For waves of small amplitude it may be assumed that the dependence is linear, and the equation can be written in the form

ut-\-(c-\-au)ux=Q.

The dependence

of the speed on the amplitude for suitable sign of a may cause the crest of the wave to move faster than the trough, i.e., curling of the front occurs with subsequent formation of breakers and decay of the wave. Simultaneous consideration of dispersion and nonlinearity, leads to the equation ut-\-cux — iuux — 4- ur(X = 0 . If we go over to a system of coordinates moving to the right with speed c, the term eardrops out, and we arrive at the Korteweg—de Vries equation up to a normalization constant which can be changed by scaling u, x, i

(Making use of this fact, in

the main text we shall often write this equation with different coefficients.) This derivation is good in that it nowhere appeals to hydrodynamics and indicates the universal applicability of the Korteweg-nde Vries equation to one-dimensional media where the essential features are only weak dispersion and weak nonlinearity. 0.3.

Cnoidal Waves and the Soliton. We shall seek a solution of the Korteweg-nde Vries

equation «( = 6«M'—u'" in the form of a traveling wave u(x,t)=U(x—vt),

where U is the wave

form and v is a constant speed (we recall v is really the speed by which the traveling wave exceeds the wave speed in the simplest approximation ut-\-cux = 0). For U we obtain the equation U"-\-a 3

—vU'=WU'—U'".

Integrating we obtain —vU = 3LT2 —

, where a is a constant. Multiplying by U' and integrating again, we find — o^~ =

U — ^-U'' -\-aU-\-b

, where b is a new constant, or U'*=2U3+vU2+aU-{-b.

Up to a normaliza-

tion constant, the general solution of this equation is the Weierstrass

function U(x — vt) =

cl$>(x — vt)-{-C2, the periods of which are the periods of the elliptic curve

T: Y2=2X3 + vX2 +

2

aX-\-b

; here ct, and c2 are suitable constants. This is a cnoidal wave train if the dis-

criminant of the curve is different from zero; the period of the wave train is the real period

r

, i.e.,

, where u> = dX(2X3-\-vX--\-aX-\-b)

w

2

,and I is a real cycle on the

Riemann surface of r \ The soliton is obtained for the curve with a double point at the origin: Y2 = 2X3-\-vX'2, a = b=0

. The explicit formula for it has the form

U(x — vt)= — -|-ch~2M-^-(JC — vt)\.

This

is the solitary wave (in the present normalization it is rather a "solitary well") with trough at the point x=vt arbitrary.

. The depth of the well is proportional to its speed which may be

The soliton is the limit of the cnoidal wave train when its period tends to

infinity. 0.4.

Superposition of Solitons and Quasiperiodic Solutions.

Since solitons decrease

at infinity and large solitons move faster than small ones, we may attempt to consider the solution of the Cauchy problem for which

u(x, 0) is the sum of two widely separated solitons

of which the left is larger than the right and therefore begins to move almost independently of the right soliton and strives to overtake it. After this occurs a period of essentially nonlinear interaction ensues, and it is of interest to consider what form the solution may have at a later time.

Contrary to usual expectations, numerical experiment showed that

after a rather long time the solution is nearly the sum of the same two solitons of which the greater has already overtaken the smaller, and the result of the collision is found only in a shift of phase but does not affect their form or speed. This provoked attempts to analytically prove the existence of the superposition of solitons.

Lax in [42], which had a great effect on the subsequent development of the theory,

established, in particular, the existence of the two-soliton solution, and almost simultaneously explicit formulas were found for the superposition of any number of N solitons. These formulas have the form u(x, t)— — 2-r-;logdet(E-\-A), matrix and the element i = l, .. ,.,N,

Zj=£i.j

where E

is the N\N

identity

_1

.A of the matrix ij is C;Cy (a^ + ay-) exp [(a'-fa^) t— (ct;-{-ay).x:], aj>0,

for i=f=j . Asymptotically for |f|-»-oo this solution decays into a sum of

N solitons arranged in order of decreasing (as t-+ — oo ) or increasing (as t->eo)

amplitudes

and speeds. After some time solutions were discovered which are related to the Af-soliton solutions in the same way as cnoidal waves are related to the single soliton solution (for the history of this discovery see the survey of Dubrovin, Matveev, and Novikov [8]). They were found to be related to the Riemann theta function for hyperelliptic curves of genus N 2

Y = F (X),

where F is a polynomial of degree 2N-\-1 ) . The explicit formulas have the

form u(x, t) = — 2-pjl.og6(-ica-M,3-{-Tr)-f-(:onst vectors. work.)

(with equation

, where a, j3, i are certain N-dimensional complex

(For the details see the review of Matveev [44] and Sec. 7, Chap. 4 of the present

The A^-soliton solutions are obtained in the limit as the hyperelliptic curve de-

generates to a rational curve with N double points. Partial degeneration (with reduction of the genus) leads to a "multisoliton solution on the background of a quasiperiodic solution."

3

0.5.

The Conservation Laws. {

K(u, a', .. .,u

U)

) lti

A conservation law for the evolution equation

ut=

= —-) i s a r e l a t i o n of the form Tf-\-Xx = 0 , where T and X are func-

tions of «, y > 0 , which follows formally from the equation. equation which i s rapidly decreasing at i n f i n i t y , then

If

u i s a solution of the

-^ ^ Tdx = — ^ Xxdx = 0 , so that T — 00

—00

is the density of a quantity conserved in time. The first three conservation laws for the Korteweg-de Vries equation are obtained without difficulty: they have the form

ut +

(-3u2-{-uxx)x=0;

(u?)t+( - 4a3 -f 2uux -

(u3 + ±uZ)t + [-^u*+3u2axx-6uul

ux)x=0;

+

uxuxx-±uQx=0.

They can be interpreted as the laws of conservation of mass, momentum, and energy.

However,

the Korteweg—de Vries equation has an infinite sequence of conservation laws which are polynomials in the u1^ . They were first written out by the method of undetermined coefficients; this work was practicable up through the ninth law.

According to Miura [47], "in the summer

of 1966 the rumor circulated that only nine conservation laws exist." Miura killed this rumor by spending a week of summer vacation computing the tenth law; after this a machine program was written which computed the eleventh law consisting of 45 terms.

(With the pre-

vious program all storage capacity was used already at the sixth conservation law.) Very transparent proofs of the theorem on the existence of an infinite sequence of laws and results on their structure were then obtained in connection with important theoretical progress: the discovery of the Lax representation and the applicability of the technique of the inverse scattering problem. 0.6. The Lax Representation and the Inverse Problem. de Vries equation can be written in the form 3

d

i

d

d

Lt = [P, L], where

L= —-^--\-u

{x, t) and P =

*

— A-^r-\-Z\u(x,t)-^-4—^-u(x, ferential operators; Lt

Lax [42] observed that the Korteweg—

t)\.

Here [P, L\ is the commutator in the ring of linear dif-

is the coefficientwise derivative of L

with respect to the "param-

eter" t . Equations of this form for flows in Lie algebras have been known for a long time; the best known of these is the Schrodinger equation in the Heisenberg representation where P

is the energy operator of the system and L is any observable.

Classical Hamiltonian

equations in the Lie algebra of functions on phase space with the Poisson bracket and also the equations for the rotation of a'solid body can be written similarly. There is a simple formalism of conservation laws connected with such equations: if there is a linear representation

This development led to the formulation of the extremely important

method of inverse scattering theory.

4

In general outline it reduces to tracing the evolution

in t of the space of solutions of the linear problem Lty = l\£ ( A a constant) in terms of the "scattering data." The scattering data consist of the discrete part of the spectrum of L , the normalization constant of the eigenfunctions, and also the scattering matrix for the (rapidly decreasing) potential u from the continuous part of the spectrum.

For further

details see the survey of Faddeev [23], the extensive literature, and also Sec. 4 of Chap. 4. 0.7. The Variational Formalism and the Hamiltonian Property. x

equation can also be written in the form „ _J?
The Korteweg-rle Vrles

where T T = 7 (~^)'^!T17T

is the

Euler—Lagrange operator or the variational derivative. This is a somewhat unusual Hamiltonian form: the standard form of the equation of a Hamiltonian evolution for a vector-valued function a of dimension In is /i, = i Mi

aw

.

.

&H

Pn !——

•'fiW

. where E is the (ftX«) identity matrix, // = fiff

''

n (u)") is the Hamiltonian, and — — = ,-?—,-•.,;—.. Various aspects of the Hamiltonian property were investigated in the important work of Gardner [36], Lax [43], and Zakharov and Faddeev [12]; it was shown, in particular, that the conservation laws commute in the sense of the Hamiltonian formalism. M, = - J ^ - —

Equations of the form

, where T,-{-Xx — 0 is some conservation law for ut = 6uux — uxxx,

have received the

name of higher Korteweg—de Vries equations. The solutions described in part 0.4 were characterized invariantly as flows induced by the Korteweg-de Vries equation on the stationary manifolds of conserved quantities defined by the ordinary differential equation -4—=0. 6u There are other equations. Practically all the effects described above for the Korteweg— de Vries equation were subsequently found for a large number of physically interesting equations including the sine-Gordon equation, the nonlinear Schrbdinger equation, the equations for self-induced transparency, etc. For other "suspect" equations part of the properties have been verified sometimes by numerical experiment. 0.8.

The Plan of the Article.

In the vast majority of studies pertaining to the

Korteweg—de Vries equation and its analogues, a substantial role is played by a system of purely algebraic structures connected with these equations which do not depend on assumptions of analytic character, the choice of function spaces, existence and uniqueness theorems, etc. The principal aim of this article, as previously mentioned, consists in displaying and systematically expounding the origins of the theory of these structures. This objective has determined the choice of the material as well as the order in which it is introduced. The first chapter is devoted to the foundations of the variational calculus with higher order derivatives which is necessary for the natural introduction of the conservation laws and the Hamiltonian structure. Here an attempt is made to follow the invariant interpretation of the variational calculus in terms of differential forms and vector fields on spaces of jets withoutwhich the formulas, which become more complex with increasing order of the derivatives, are hard to interpret and work with in a practical manner. Special attention is focussed on the basic facts of the Hamiltonian and Lagrangian formalisms. 5

In the second chapter a detailed study i s made of the structure of general Lax equations as well as of an enigmatic system of wave equations of Benney which displays many of the features of the systems described above but so far does not f i t into the general theory.

We

consider i t an i n t e r e s t i n g object for future i n v e s t i g a t i o n s . The t h i r d chapter i s devoted to the Lax equations of multisoliton and quasiperiodic type.

Here we have also strived to display in the clearest possible way the mechanism of the

appearance of the algebrogeometric structures in the theory of the equations without writing out e x p l i c i t formulas for t h e i r solutions which i s done in a number of other surveys and i s b r i e f l y considered in Sec. 6 of Chap. 4.

Exceptions are the "solitons of higher rank" which

are here obtained by algebrogeometric methods for the f i r s t time.

One common feature of a l l

problems solved should be emphasized: the introduction of an auxiliary fiber bundle over an algebraic manifold and the i n t e r p r e t a t i o n of the equation as the problem of finding a connection in t h i s bundle with certain additional p r o p e r t i e s .

Recently the problem of "instanton"

solutions of the Yang-Mills equations (more p r e c i s e l y , the duality equations) in Euclidean field theory with the group SU(2) has been reformulated and advanced in t h i s manner (Atiyah following preliminary work of Penrose, t ' H o o f t , A. S. Shvarts, Polyakov, and o t h e r s ) .

It

3

reduces to the c l a s s i f i c a t i o n of two-dimensional complex vector bundles over ^ (C) , which are t r i v i a l on a certain class of lines in P3 . F i n a l l y , facts regarding p a r t i c u l a r i n t e r e s t i n g constructions and methods which have not yet been s u f f i c i e n t l y thought out or subjected to systematization are collected in the fourth chapter.

The exposition here follows the sources cited in the corresponding s e c t i o n s ;

proofs for the most part are omitted.

The only exception which deserves mention i s thebeginning

of Sec. 3 of Chap. 4 where an attempt i s made (not e n t i r e l y successful) to invariantly define a very i n t e r e s t i n g Lie algebra introduced by Estabrook and Wahlquist in connection with t h e i r theory of "prolongation s t r u c t u r e s " and generalized conservation laws. The reader should not take the scattered references and credit of authorship for various r e s u l t s too s e r i o u s l y . Many similar works were done almost simultaneously and almost i n dependently; many approaches revealed a parallelism unknown to t h e i r authors; many ideas hung in the a i r and continued to hang in the a i r some time after formal f i r s t publications.

The

history of our question, i f i t deserves such, remains to be w r i t t e n . In s p i t e of the length of the a r t i c l e , many i n t e r e s t i n g facts have remained beyond i t s scope. F i r s t of a l l , the analytic theory of the method of inverse s c a t t e r i n g has been omitted completely in s p i t e of i t s importance and the fact that i t motivated the inception of many of the purely algebraic constructs described here. Regarding t h i s question, the reader may find abundant information in the l i t e r a t u r e c i t e d . Secondly, very l i t t l e attention i s devoted to specific solutions of p a r t i c u l a r equations or to t h e i r physical i n t e r p r e t a t i o n . Third, we have l e f t untouched the i n t e r e s t i n g p a r a l l e l theory of discrete systems such as the "Toda l a t t i c e " and such of i t s principal applications as the explanation of the Fermi—Pasta-Ulam paradox. Fourth, i n t e r e s t i n g investigations of flows of Lax type in finite-dimensional Lie algebras and the many-particle problems related to such algebras have been omitted. The 6

informed r e a d e r w i l l p r o b a b l y d i s c o v e r s t i l l more omissions v o l u n t a r y or i n v o l u n t a r y . Among t h e r e s u l t s n o t c o n t a i n e d i n t h i s work b u t n a t u r a l l y r e l a t e d t o i t mention should be made of the i n v e s t i g a t i o n of B o g o y a v l e n s k i i and Novikov [1] and of G e l ' f a n d and D i k i i [3] on r e s t r i c t i n g Hamiltonian flows t o s t a t i o n a r y manifolds of c o n s e r v a t i o n l a w s .

They a r e of b a s i c im-

p o r t a n c e for u n d e r s t a n d i n g t h e r e l a t i o n between s o l i t o n s and c o n s e r v a t i o n laws and m e r i t generalization to the multidimensional case. We have n o t endeavored t o compile a complete b i b l i o g r a p h y .

In p l a c e of t h i s the b i b l i o g -

raphy i n c l u d e s surveys w i t h l a r g e b i b l i o g r a p h i e s and c o l l e c t i o n s devoted to s p e c i f i c of the t h e o r y [ 2 , 8 , 2 3 , 2 6 , 3 3 , 4 4 , 4 7 , 5 0 , 5 3 ] .

aspects

Beyond t h i s , p a p e r s having a d i r e c t

rela-

t i o n to the q u e s t i o n touched on h e r e have been s e l e c t e d as have s e v e r a l works to which we do n o t r e f e r b u t which, i n our o p i n i o n , deserve s p e c i a l 0.9.

attention.

I t i s i m p o s s i b l e t o o v e r e s t i m a t e t h e r o l e t h a t t h e a u t h o r ' s many c o n v e r s a t i o n s

w i t h I . M. G e l ' f a n d and a l s o the work of G e l ' f a n d and D i k i i [2-5] p l a y e d i n the design and plan of t h i s p a p e r . The s e l e c t i o n of m a t e r i a l f o r the p a p e r was made d u r i n g a s p e c i a l course which t h e a u t h o r gave i n the mechanics and mathematics department of Moscow S t a t e U n i v e r s i t y i n 1975/ 1976 and from t h e i n t r o d u c t i o n t o a seminar i n 1976/1977.

The p a r t i c i p a n t s i n t h e c o u r s e and

seminar p r o v i d e d t h e a u t h o r w i t h a g r e a t d e a l of m a t e r i a l which d i r e c t l y o r i n d i r e c t l y

af-

f e c t e d the c o n t e n t of t h e p a p e r . In p a r t i c u l a r , a l a r g e p a r t of t h e new r e s u l t s of Chap. 1 belong t o B. A. Kupershmidt; t h e i r p r e s e n t a t i o n i s based on h i s p u b l i s h e d p a p e r s and n o t e s which were k i n d l y given t o the author before t h e i r p u b l i c a t i o n . c a r r i e d out j o i n t l y by B. A.

The i n v e s t i g a t i o n of Benney's e q u a t i o n s i n Chap. 2 was

Kupershmidt and t h e a u t h o r .

The a l g e b r a i c reworking of the

Gel'fand—Dikii t h e o r y i n Chap. 2 d e r i v e s from a r e p o r t of M. S. Shubin i n t h e seminar i n which t h e s i m p l i c i t y of t h e formalism of p s e u d o d i f f e r e n t i a l o p e r a t o r s over a one-dimensional b a s e was r e v e a l e d .

The r o l e of bimodules and c o n n e c t i o n s to which Chap. 3 i s devoted was

c l a r i f i e d by V. G. D r i n f e l ' d .

From t h e r e p o r t of S. I . G e l ' f a n d t h e a u t h o r f i r s t

understood

t h e t e c h n i q u e of Estabrook and W a h l q u i s t , w h i l e t h e e x p o s i t i o n of the r e s u l t s of Lax in Sec. 5 , Chap. 4 i s based on t h e n o t e s of I . Ya. Dorfman.

F i n a l l y , c o n v e r s a t i o n s w i t h B. A. Kuper-

s h m i d t , M. A. Shubin, V. G. D r i n f e l ' d , V. E. Zakharov, I . Ya. Dorfman, and D. R. Lebedev were v e r y u s e f u l t o the a u t h o r .

I am happy t o e x p r e s s my d e e p e s t g r a t i t u d e t o them a l l . CHAPTER I THE VARIATIONAL FORMALISM

1.

Differential Equations: Three Languages 1.1.

The Classical Language.

In this language we first of all choose a notation for

the independent variables, say, xlt...,xm

, and for the unknown functions, say itt, ...,#„. m

Let £ = (&i,..., £m) where kt>0 KJ*> the derivative

are integers, and let \k\=£kl.

We denote by the symbol

^ — r — . A system of differential equations relative to {«J is a 7

collection of relations of the form Fj(*i

«<*>,..., a<s»)=0,

*«: tiu...,un;

(1)

where the Fj are some functions. It is sometimes convenient to distinguish one of the variables, say, t (the "time" as opposed to the spatial coordinates xx

xm), and to consider a system of evolution equa-

tions of the form

-gL^tij.t^Fjixu

...,xm,t;

a,

an; «<*>

«(*>).

(2)

We point out that Fj does not depend on the derivatives of ut with respect to t\ j = l, ...,

n.

1.2. The Language of Differential Algebra. Let ^ be a ring, and let M be a left A module. We recall that a differentiation of A into M is any additive mapping d:A^>-M with the property d(ab)=adb + bda for all a, b£A .

The algebraic analogue of the system of equation (1) is a structure consisting of a ring A , some Lie algebra D with a differentiation into itself, and an ideal Id A , for which Dial

. More precisely, we suppose that Ft

in (1) is infinitely dlfferentiable in all its

00

arguments and set A = U C~(jd, ... , xm\ «<*> 11 < / < « , \k\ are formal indepenm

dent v a r i a b l e s .

Let f u r t h e r

D=

'^Adj,

where

3; t a k e s

X; i n t o

8,7,

and

u<*> i n t o

ullk+ei) ,

l-i

s; = (0.. .1.. .0) n

d*>...da Fj

(1 sits at the j-th place), and let / be the ideal in A , generated by all

. The structure (A, D, I) corresponds to (1). Any smooth solution of (1) corre-

sponds to a homomorphism A ^ C 0 0 ^ , ,.. , xm) = K, which is the identity on K, contains / in its kernel, and takes d,

into T— .

'

axj

I f t h e r i g h t s i d e s of (2) do n o t depend on t

e x p l i c i t l y , then the a l g e b r a i c analogue

of (2) i n t h i s c o n t e x t can be c o n s t r u c t e d by n o t i n t r o d u c i n g (A, D)

and t h e a d d i t i o n a l d i f f e r e n t i a t i o n

[X,
for a l l

t e x p l i c i t l y : i t c o n s i s t s of

of " e v o l u t i o n " X : A-*-A , d e f i n e d by t h e c o n d i t i o n s

/, and XUJ = FJ ( t h e r i g h t s i d e s of (2) for

/= 1

n).

of (2) corresponds t o a /(-homomorphism A-+C™(Xi,..., xm, t), which t a k e s X into

dj i n t o

d/dx,,

and

d/dt.

V a r i a t i o n s a r e p o s s i b l e i n the d e f i n i t i o n of t h e r i n g A . (1) do n o t depend e x p l i c i t l y on /4 = R[«,-""] is

Any smooth s o l u t i o n

x,

For example, i f the Fj

in


and a r e polynomials i n u, , i t i s p o s s i b l e t o s e t

and c o n s i d e r t h e a l g e b r a i c o b j e c t modeling (1) t o be a minimal i d e a l i n A, which

£>-closed and c o n t a i n s

Fj .

I t i s a l s o p o s s i b l e t o t a k e A t o c o n s i s t of a n a l y t i c

func-

t i o n s , meromorphic f u n c t i o n s , germs of f u n c t i o n s , e t c . , depending on t h e p r o p e r t i e s of Fj and the s o l u t i o n s of i n t e r e s t t o u s . 1.3. class

The Geometric Language.

C°° ) f i b r a t i o n

played by a p o i n t on

n : N-+M ,

Here we s t a r t w i t h some l o c a l l y t r i v i a l , smooth ( i . e . ,

The r o l e of t h e independent v a r i a b l e s

M , and t h e r o l e of t h e unknown f u n c t i o n s

smooth s e c t i o n s : M-+-/V of t h e f i b r a t i o n 8

it .

uu...,un

xx

xm

i n (1) i s

i s assumed by a

I n o r d e r t o d e f i n e a geometric o b j e c t

correspond-

ing to the system (1) i t i s necessary to introduce the tower of j e t spaces of the fibration n .

We r e c a l l the corresponding d e f i n i t i o n s . Let s,

at a point

and

s2 be local sections of the fibration

n. .

They are tangent to one another

|eiV to order fe^sO, i f they pass through t h i s point and t h e i r Taylor s e r i e s in

any local coordinate system for n(£) coincide through order k .

A k-jet

at the point | i s

the equivalence class of local sections which are tangent to one another at the point \ h

order k .

Let J n be the set of a l l

k -jets.

to

I t i s equipped with a n a t u r a l smooth manifold

s t r u c t u r e and there i s a tower of smooth fibrations

• •. /^JI-*/ 6 - 1 !!-*-.. .-*-/°jt = JV--»-M .

analogue of system (1) i s then a closed subset in a s u i t a b l e story of the tower

An

/*n , p r e -

scribed by the vanishing of the r i g h t sides of (1). More p r e c i s e l y , the connection of the c l a s s i c a l notation (1) with geometry i s realized by means o-f a choice of consistent local coordinate systems on a l l the spaces J"~ . with a choice of a pair of neighborhoods *&VcN tion

and ir(;)6(/ = 7:(V)cM, such that the r e s t r i c -

~:V-+U i s diffeomorphically equivalent to the projection

diffeomorphism i s a local chart for

We begin

R mn ->R m .

The choice of t h i s

r.\ prescribing i t i s equivalent to prescribing a local

coordinate system in N of the form («.,...,«,,; x,, . . . , xm) such that the r e s t r i c t i o n s of

(«/)

to each fiber of ~ give a local chart in t h i s f i b e r , and the (•*/) are l i f t s from the base. With respect to t h i s local chart a local chart i s canonically constructed in each J"x with a coordinate system which i s denoted by (u(/\ manner.

Let s be some local section of s

by the functions coordinate

!

u], ... , u neC' °(xu • • • , •*«)•

.x,| |/| < fc) and i s uniquely defined in the following it|y , which i s represented in a local chart of it I t s fe-j.et i s a point of

«(," at t h i s point i s by definition

'—r .

of the section s at a l l i t s points i s a smooth section of to

J"- , and we denote i t by the same l e t t e r 5 .

J"r..

The value of the

Obviously the collection of

fe-jets

Jkr. ; i t i s called the l i f t of s

The l i f t s of sections are compatible with

the projections of the j e t spaces. I t i s now clear that the system (1) defines some subset <1» in

Jkr. , and i t s solutions

are those sections of IT, whose l i f t s l i e in <1>. The connection of the geometric language with the language of d i f f e r e n t i a l algebra i s as 00

follows: A = U

C°°(y*it) (imbeddings with respect to the natural projections), D = DC

is the

fc-0

Lie algebra of "horizontal" differentiations of A (for the precise definition see the following section),

I cA

is the ideal of functions which vanish on the "lifts $ " (the geometric

definition of a lift is most simply introduced by the condition of differential closedness of the corresponding ideal). 1.4.

A Comparison of the Languages. The three languages briefly described are not

equivalent either mathematically or esthetically; each has its advantages and shortcomings.

9

The classical language makes the fewest explicit assumptions on the form of the functions Fj,

the sense of the derivatives «/*', the domains of existence of solutions, etc. It

affords the freedom of interpreting, e.g., the uj

as various types of generalized solutions

(and therefore, possibly, not elements of any natural ring).

It also permits when necessary

accentuating other algebraic structures which are essential for investigating the system and its solutions, for example, linear topological structures. However, this language is not suitable for investigating global properties of solutions related to the topology of the manifold of independent variables and fibrations of the unknown functions.

Further, this

language may poorly express the invariant properties of Eqs. (1) and (2) and their consequences . The language of differential algebra is better suited for expressing such properties and puts at the disposal of the investigator the extensive apparatus of commutative algebra, differential algebra, and algebraic geometry; this is especially true if the Fj in (1) and (2) are polynomials, and we are interested in special classes of solutions. The numerous "explicit formulas" for the solutions of the classical and newest differential equations have good interpretations in this language; the same may be said for conservation laws. However, the language of differential algebra which has been traditional since the work of Ritt does not contain the means for describing changes of the functions «j and the variables xt

and for clarifying properties which are invariant under such changes. This is one of the

main reasons for the embryonic state of the theory of so-called "Backlund transformations" in which there has been a recent surge of interest. The geometric language is especially well suited for formulating and clarifying global and invariant properties of general systems of equations and for applying to them the theory of differential-geometric constructs and ideas.

Its main drawback is its generality which,

on the one hand, requires a rather lengthy development of foundations without concrete applications and, on the other hand, creates the risk of overlooking interesting properties of special classes of equations connected with fortuitous additional structures. For these reasons the present paper is written in a broken and somewhat eclectic jargon in which modes of expression from all three languages are mixed in those proportions which to the author seemed most suitable for the object of study.

Equivalent or comparable formula-

tions of the same facts and constructions in different languages are often presented. 1.5.

The Lagrange and Hamiltonian Equations.

The classes of equations which will be

of main interest to us in this work are engendered either by a Lagrangian or Hamiltonian formalism.

Their choice and investigation is strongly motivated by the finite-dimensional

case which corresponds to the projection n: N-*- (point).

In this case the analogue of our

"Lagrange" problem consists in choosing a smooth function L : N-*-R and finding its stationary points grad

L = 0 , i.e., from a general point of view it is not intrinsically a problem in

differential equations.

The "Hamiltonian" problem is obtained if a Hamiltonian structure is

given on N which makes it possible for each Hamiltonian H : N-t-R to construct the appropriate vector field XH

10

on N', it is required to investigate the properties of its trajectories.

If the base M i s not a point then the role of the finite-dimensional configuration (or phase) i s taken over by the infinite-dimensional space of sections s:M-+N

of the fibration

ic , and the f i r s t problem i s to choose a s u i t a b l e class of functionals on the sections from which the Lagrangian and Hamiltonian can be selected.

Here we adopt the point of view of

the c l a s s i c a l v a r i a t i o n a l calculus according to which the i n i t i a l functionals have the form u)(s)=Jco*

, where » in a local chart has the form L(xt; uf^dxiA-

• • A'dxm- and ms i s the

M

r e s t r i c t i o n of to to the corresponding l i f t all

s .

(If M i s not compact


<«(s) are defined on

to — on s u f f i c i e n t l y rapidly decreasing

sections.) In the Lagrange problem to i s called a Lagrangian density (or simply the Lagrangian) ,
8\to* = 0

in c l a s s i c a l notation.

M

In the Hamiltonian problem i t i s , in addition, necessary to give a Hamiltonian s t r u c t u r e : o>*+X~ , where X~ i s the d i f f e r e n t i a t i o n of evolution corresponding to the Hamiltonian Not every such (linear) mapping defines a

Hamiltonian s t r u c t u r e .

to .

In analogy with the

finite-dimensional case, we introduce on {«>} the Poisson bracket by the formula {to,, u>2}=X-2» and we require t h a t the mapping ^Xs

be a Lie-algebra morphism: -^{o,, 5,} = 1-^,1 -X~l (the

commutator of the evolution f i e l d s ) . I t i s well known that by means of integration by p a r t s the condition

8\
to a system of d i f f e r e n t i a l equations (the vanishing of the v a r i a t i o n a l derivatives or the Euler—Lagrange operators of the form to). bracket

X^2

.

This same mechanism works in studying the Poisson

The formalism of the v a r i a t i o n a l calculus i s based on the fact that e s s e n t i a l -

ly a l l computations are carried out with the integrands, i . e . , in the algebra of certain operators on forms on the space of j e t s . derivatives in xt

Since for us Lagrangians and Hamiltonians with

of a r b i t r a r i l y high orders are e s s e n t i a l , i t i s important to clarify the

invariant meaning of the c l a s s i c a l constructions and formulas;, dealing with these objects becomes complicated as the order of the derivatives increase, while invariance under change of coordinates i s almost not amenable to v e r i f i c a t i o n . f i r s t chapter.

This i s the basic objective of the

In Sec. 8 we r e s t r i c t our consideration to the case of a one-dimensional base

and introduce the class of Hamiltonian structures which are important in the sequel. In conclusion we remark that the choice of the class of functionals i s not the only possible one nor even the most important.

{to} on the sections

In field theory, for example, the

class of functionals of basic importance are those generated by i n t e g r a l transformations of polynomials of the form P{us(xM)

«*(*<">)) , where (*<",..., *<">) €MX • •. X M=-MN , and us

are the coordinates of the section.

They include, say, Fourier transformations of the co-

o r d i n a t e s , and, i f generalized kernels are admitted, also a l l functionals of the form


Moreover, they form a r i n g , while functionals of the form «i constitute only a l i n e a r space. A systematic development of the v a r i a t i o n a l formalism in t h i s class has apparently not been carried out in s p i t e of the importance of t h i s problem. 11

§2.

Fields and Forms on the Space of Jets The Basic Model. We consider a fibration N-*M, locally diffeomorphic to the

2.1.

projection Un+m-+Rm , and we describe a number of concepts and constructions related to differential forms and vector fields on various stories of the tower of jets.

The defini-

tions and formulations of results are given in invariant terms which automatically ensures their consistency in local models (an exception is the Legendre operator for which see Sec. 4).

For this reason computations may be carried out in local coordinates if desired which

clearly indicates the connection with the classical formalism.

We thus introduce a number

of bundles and homomorphisms between them, but we carry out almost all computations in a and a diffeomorphism with projection R n+m ->R" 1 ), and we assume

single local chart ( KV'-V-+U

as in Sec. 1 that A'=C0O(i/), At = C°°{J'r.v), i>0, and A_l=^K(zA(j'^Al(z s

we identify with the induced homomorphism A-+-K': P>-+P . C

*«; up\\k\
°°<*>

dxk°

2.2.

Vector Fields on Jets.

In a local chart we have AL =

«; for any s.

dxkn

1

s:U-+V

A section

m

If I is a smooth manifold, B = C~(L) , we denote by D(B)

the Lie B-algebra of vector fields on L, considered as differentiations of B into itself. If C-*-B

is a homomorphism induced by a smooth mapping we denote by D(B/C) the subalgebra

of fields of D(B), which are trivial on the image of C. k>l

For

a

Xk\A,=X'[ D(Ak)

a field Xk£D(Ak)

. Since J*KV = J'*vXR i

is called an extension or lift of a field X,eD(At), , any field Xt extends to D(Ak) .

denote by D(A) the set of all differentiations X:A-*A a k>i,

for which X\AfiD

tion

_d_

|/|
, such that for each i there exists

^-module freely generated by the partial derivatives

Any element of D(A) can we w r i t t e n u n i q u e l y a s an i n f i n i t e l i n e a r combina-

X= ZJP, J— +2jQk,i—TTT!

T

'' tt

sively restricting

Pn Qi.&A.

The c o e f f i c i e n t s

P-Qk.i

a r e determined by s u c c e s -

du

k

X t o A, w i t h

i-+oo.

Obviously,

Pj = XXj, Qk,i=Xuk'K

v e r i f i e d t h a t D(A) forms a Lie A-algebra. 2 . 3 . LEMMA. L e t PQA b e such t h a t P'=0 f o r a l l s e c t i o n s Proof.

I f P(Xj, uk»)^0

for functions

Finally, we

(A(, Ak).

In a local chart D(At) is a free I d_

D(AltAk)(z

We denote by

the submodule generated over Ak by all elements of D(At) in D(Ak) .

if

, then t h e r e e x i s t

s.

lp v«'>€R , such t h a t 1

uycK , such t h a t a t t h e p o i n t (£,) we have {d^ldx ^..

I t i s easily

Then P = 0 . P(Zj, v<'>)=£0.

l

s

.dx jn)u t(Zj)=vk'K

We look They d e f i n e

s

a s e c t i o n s, f o r which P ^f=0. 2.4. s:U-+V

LEMMA.

There e x i s t s a unique mapping D(K)->D(A):

and f o r a l l

Proof.

XeD(K),

PeA

we have (XPy = XP> .

I t i s an imbedding of Lie

D(K) i s f r e e l y g e n e r a t e d o v e r Af by the f i e l d s

2 «<*+e')dlduf*£D(A), where ei=(0... 1.. .0) (one sits at the i,*

12

X<-*X such t h a t f o r a l l

dldxt

.

sections /f-algebras.

We s e t dldxl = di =' ^-+dxt

i-th place) and we extend this

definition by /C-linearity to a l l of D{K) .

Obviously, dt\K = dldxlt (<J,aJ*>)5 = ( a ^ ' ) J , = ^ («}*>*) •

From the fact that P^*PS i s the i d e n t i t y on K and from formal properties of derivatives i t follows that (XPy = XP* for a l l The uniqueness of

X

The fact that X^-X

XqD(K)

and

P£A.

follows from Lemma 2 . 3 . i s a Lie-algebra homomorphism follows from the computation [X, Y\ P° = XYPS - YXPS = X {YPy - Y iXPf = =(XYpy - (Yxpy=([X,

and the uniqueness of Obviously,

Y] py

[X, Y] .

dj i s the " t o t a l derivative" with respect to Xj in the c l a s s i c a l terminology.

We denote by BcczD(A)

the image of D{K) under the imbedding

forms a Lie subalgebra in D(A).

X+-+X .

I t i s clear that ADC

In a local chart we s h a l l write P<" = (?{<.. .6'mP.

This i s

s

consistent with the notation tttp and the homorphisms P*+P . 2.5. [F, X]=0

LEMMA.. There e x i s t s a unique imbedding of AD(AQlK) into D(A):Y*+Y such that for a l l XeD,. and YP =YP for a l l P ^ V

fields which commute with Proof.

Dc and are t r i v i a l on K .

The action of

Y on xt

and

uk coincides with the action of

for | £ | > 1 i t i s defined by the con dition s i b l e formula for

The required properties of

l)

{,)

[K, £>c]=0: Yai = (Yuk) . i)

7 has the form

hence valid everywhere.

I t s image i s the Lie subalgebra of a l l

7P = ^l{Yuk) ' ^T

Y, and on aj,°

Therefore, the only pos-

.

Y axe. obvious on the generators of Dc and A0 and are

Any field which commutes with De and i s t r i v i a l on K, can be r e -

presented in the form 2 Q*° d (/> » and therefore has the form of

Y .

Finally, such fields

obviously generate a Lie algebra. We denote by

DevdD(A)

the image of AD(A0jK) in D(A).

the system of evolution equations t i v e " by v i r t u e of t h i s system.

Assigning to each Y&Dev

Uj,t — YUj , we see that Y:A-+A i s the " t o t a l time derivaThe condition [Dc, P] = 0 means that the t o t a l time and space

derivatives commute. 2.6.

LEMMA,

algebra in D{A). with

a) [Dev, ADc\dADc

. b) Dev + ADc = Dey®ADc

(sum of spaces) i s a Lie sub-

The r e s t r i c t i o n of Dev + ADK to A0 defines an isomorphism of- t h i s space

AD(A0). Proof, b)

a) For 7&D„, X&)e,

The r e s t r i c t i o n of

P$A we have {?, PX\ = (YP) J?GADC, since

J Q*1"^" + 2

#/dfcD.y + ADe to >t0 coincides with 2 Q * 5 5 T

S S i ^ + S " ! ' ' ^ ) - ^ uniquely determined for any f i e l d of AD(A0),

[7,JC]=0.

Therefore,

Q.and

+

/?, are

whence b ) .

13

For any element Dtv-\-ADc . .

X&AD(A0) we denote finally by X&D(A) the corresponding element of

As i s evident from the proof of Lemma 2 . 6 , on D ev and Dc t h i s canonical l i f t

operator coincides with those defined in Lemmas 2.4 and 2.5 respectively. Although Dev-\-ADc far from exhausts D(A),

these d i f f e r e n t i a t i o n s completely suffice in

order that i n t e r i o r products with them should uniquely determine the d i f f e r e n t i a l forms on the space of j e t s (see below). 2.7.

Indeed, we have the following r e s u l t .

LEMMA. Fields of the form X + Y , where X&D(A0lK),

YeD(K\ form a basis of the

k

tangent space at any point of J %v • Fields X\Ak have the form ^ ( Q i ) u •- ^

Proof.

used in Lemma 2.3 shows that for any point prescribed values

;>

Qi

at t h i s p o i n t .

, and an argument analogous to that

ZGJknv i t i s possible to find

Q&A with any

Thus, X generates the spaces of v e r t i c a l tangent

v e c t o r s , while the spaces generated by Y for Y^DIK) project onto the e n t i r e tangent space O at it(c) and therefore give the lacking horizontal complement. ( I t should be mentioned that Y takes Ak onto Ak+l, so t h a t , s t r i c t l y speaking, Y does not define an ordinary tangent However, we s h a l l use t h i s lemma to verify that each d i f f e r e n t i a l form on Jkr. i s

vector.

determined by i t s values on f i e l d s of the form X-\-Y, and for t h i s purpose our argument suffices.) 2.8. of

The de Rham Complex.

For i>— 1 we denote by QAt= ©2M, the e x t e r i o r algebra

C°° d i f f e r e n t i a l forms on J'IC with d i f f e r e n t i a l rf:2M,-> 2*+M,-.

At-*-Ak (k>i)

The canonical imbedding

define a system of imbeddings QAi-^QAk consistent with d .

We set

QA =

lim2^ = u QAh 2 M = Z&A • —•

i—i

i—t

In a local chart Q»A i s freely generated over A by elements of the form atttl Adu^Adx^A w&JAt

••• Adxib,

i, < . . . < * „ t a + 1 < . . .
For any vector field XeD(At)

A---

and forms

u

the standard compositions ixuigS^Mj ( i n t e r i o r product) and Lx > = (ixd+dix)u>dQJAi

(Lie derivative) are defined.

If k>i,

then tjfio = fx« and L^—Lx^.

Therefore,

possess the usual p r o p e r t i e s .

In p a r t i c u l a r , the Lx (respectively,

X, and are d i f f e r e n t i a t i o n s

X&D(Ak), i s an extension of XQD(At), and w&i(A,)czQ(Ak), ix and Lx are defined for X(*D(A),
(respectively, a n t i d i f f e r e n t i a t i o n s ) of the algebra QA .

over, \Lx,d\ = 0, \LX,LY\=L[X,Y\;

Lpx--=PLx+dPAix\

[Lx, iy] = iix,ri', ipx = Pix-

More-

We s h a l l sometimes

write Xm in place of Lx® . We now concern ourselves with the r e s t r i c t i o n of forms of

QA to sections.

Any section

s defines an algebra homomorphism QA-+QK- «>*-*•«>*. 2.9. Proposition, a) There e x i s t s a unique operator -z-.QA^-QA with TQA
i s the ideal in 2^4, generated by KertnS'-A.

d) t/.jf=*TijfT for a l l

14

XeDev + ADc; tZ.]f=Z.jFT for a l l

XeDeV.

that

e)

xrfx = xd

Proof,

and

xdxd=0 .

a), b).

We d e f i n e an .A-homomorphism of a l g e b r a s

QA->AQK on t h e g e n e r a t o r s

m

^{du^)=^u^+ti)dxi.

by the formulas *(dXj)=dxt,

It is clear from the definitions that

xQAczAQK, (T(I>)*=U)*, and x2 = x. Uniqueness follows from the fact that AQK module with generatorsrfjCj.A-• • /\dxik mined by its

is a free

A-

, and therefore any of its elements is uniquely deter-

s-images according to Lemma 2.3. m

We remark t h a t

tdP^^djPdx,

for a l l

x2 = x , we have

c) Since

and i n d u c t i o n on

t show t h a t

follows from t h e f a c t t h a t

PeA

Ker x = Im(l —x).

.

The i d e n t i t y

Im(l — t)n2 i -AcQ i _ 1 A(Im(l — x)n2M) .

Kerx i s an i d e a l c o n t a i n i n g

d) We s h a l l show f i r s t

o>Av — T(«)A V ) =

that

t i o n and Ker x i s an i d e a l , i t s u f f i c e s t o v e r i f y t h i s i n c l u s i o n i . e . , on

«<*' — 2 tlf^^dxj

The r e v e r s e i n c l u s i o n

Im(l — r)f\Q1A

L-^KeiiCzKeT x for X&Dey -\-ADc .

.

Since

Lj

is a

differentia-

on t h e g e n e r a t o r s of

. Because of t h e a d d i t i v i t y of L^, w i t h r e s p e c t to

t o c o n s i d e r ^f s e p a r a t e l y for

(U) — •M>)AV + ' W ) A C ' — X *)

Ker t,

X , i t is possible

and A'g.AD,.; we have

XGAD(AQ!K)

= <%«{*> - ^ (f.jU^dXj

+

U^dLjXj).

i )

the

For A'=2.Q{* -T»)

r i g h t

s i d e

i s

equal

t0

0—T)<*Q{*)eKerT.

For X=Pdt

the right side

i s e q u a l to P(\ — x) rfw^+VgKer x. Now xZ.^x — xLjf = x£^(x — 1) and -X'g.AD (,40/A') , then Ljf(xrfa<*))=xrfQ(*> .

i.e.,

rfKerxcKerx

If,

moreover,

^ = 2Q{*'—J$\

Ij f x=xZ.^x=xIj f .

xtf —tdx = x<* (1—x) . and hence

If

u>* = 0

xtf(l—x)=0.

for a l l

s , then (rf
Multiplying the equation

s,

xd = xdt on t h e r i g h t by

2

d , we f i n d t h a t 3.

Xjflmxclmx , s i n c e by t h e p r e c e d i n g c a l c u l a t i o n ( f o r Therefore

e) Obviously,

ImZ.y(x — l)clm(x — l ) = K e r x , whence T-Lj = tL-gt.

(xrf) =0 .

Integration by Parts 3.1.

Probably a l l t h e i n v a r i a n t i n f o r m a t i o n c o n t a i n e d i n the c l a s s i c a l p r o c e d u r e of

i n t e g r a t i n g the v a r i e d Lagrangian d e n s i t y by p a r t s i s c o n t a i n e d i n the n e x t p r o p o s i t i o n . Here we do n o t even assume t h a t the "form i n t h e v a r i a t i o n s " u>(*QlAQmK (du\k) i s t h e " v a r i a t i o n " of

«<*>) i s o b t a i n e d from a L a g r a n g i a n ; t h i s c a s e w i l l be d e a l t w i t h i n the n e x t s e c t i o n . 3.2.

to3^Q1AQm~1K a)

u) =

Proposition.

Let

<»&ilAQmKc:QA.

There e x i s t forms

a>l£AQlA(fimK, ^1&QxAQmK,

w i t h the f o l l o w i n g p r o p e r t i e s : o>j - | -
b) For a l l

X£Dev

we have

ix
The forms ml, w2 a r e determined u n i q u e l y .

The 15

form to3 can be n o r m a l i z e d by the following a d d i t i o n a l r e q u i r e m e n t : we choose

vQA£m/(

arbitrar-

i l y and impose on w3 t h e c o n d i t i o n : c)

tio 3 = v.

Then u)3 e x i s t s and i s u n i q u e l y determined i n t h e case m = l and a l s o i n t h e case a>($ZlAiQmK, i f we r e q u i r e , i n a d d i t i o n , t h e i n c l u s i o n Proof.

Existence.

I t suffices

IO&A&AQQ^K-

t o v e r i f y t h e e x i s t e n c e of

<»i, u>2 w i t h p r o p e r t i e s a ) , b)

on t h e a d d i t i v e g e n e r a t o r s of t h e group Q1AQnK which we t a k e t o be t h e forms dX\/\...f\dxM t h e forms

in a local chart.

We s e t

k(.i) = \k\ and c a r r y out i n d u c t i o n on £(<»). For A(u>)=0

«>,=u), u> 2 =u) 3 =0 s a t i s f y a) and b ) .

Suppose t h a t £ ( « ) > l a n d l e t £ / > l . -d,Prf« J ( *- , <» ArfJC,A • •.

o ) = p a « ( * ) A r f x , A . . . ,\dxm= + d, (Pduf-'S

)/\dxxA

- . . Arf.*m=u>' + v.

and t h e s e l i e i n t h e d e s c r i b e d subgroups of ix*-=h(dipdu?~'l)

Ad.*:A• • • /\dxm) e

= df ( X ^ f - ' ' dXl/\...

Adxm+P

(Xud^dx,

= dt(P(XUif- ' e )

=(-l)'- xrf/ 7 (Pda ( !*- /

2J4 .

3

Further,

+i^{Pdu\*>Adx1A.../\dxm)

e}

1

Then

Adxm+

By t h e i n d u c t i o n h y p o t h e s i s i t i s p o s s i b l e t o choose w[, u^ and u>3 w i t h for a l l XQD^,

Pdu^A

A ...

=

Adxm=

)dXlA-..Adxn=

Adx'tA...

Adx,_lAdxl+lA---Adxn

=trfty.

I t remains t o s e t o>1=2 = a>j-j-v, a>3 = 3-f v\ Without d e s t r o y i n g p r o p e r t i e s a ) and b) we may add t o o>3 any element the p r o p e r t y

*jf v "=0

p l a c e of (i>3 f o r any

for a l l

X^Dev.

v^AQmfC, s i n c e


In p a r t i c u l a r , i t i s p o s s i b l e t o t a k e a>3 —f(o3-|-v i n i-^(AQmK)=[0}.

Then

TO>3

i s r e p l a c e d by TV-=V, which gives

condition c ) . Uniqueness.

Let io = |-}-^

and i-^ = zdi-^a>'3 f o r a l l X , i . e . ,

of forms which s a t i s f i e s a ) , b ) , c ) .

Then f o r a l l

X^Dev

^J, «>2, <«3 i s a n o t h e r

triplet

we h a v e , on s e t t i n g o)j=(oi—OJ,',

i-£»\ — — i jf<»2 = — ~di~x<j>3. From t h e proof of Lemma 2 . 7 i t follows t h a t any form v a l u e s of

t^

f o r a l l X(*D{A0lK),

i t suffices

of

x :

QlAQmK i s u n i q u e l y determined by t h e

s i n c e such ^ g e n e r a t e a b a s i s f o r t h e v e r t i c a l

v e c t o r s a t any p o i n t of t h e s p a c e of j e t s . 2

(3 )

t o v e r i f y t h a t i^u>j = 0.

tangent

T h e r e f o r e , t o e s t a b l i s h t h e uniqueness of <»! and

For a l l s e c t i o n s

s we have by (3) and t h e d e f i n i t i o n

(«V»i)s=— dVx^zYWe choose any point xf*M. , a small open neighborhood U of it with boundary dll, and we consider the fields X for AT with support strictly inside rc_1(£/). By Stokes' formula J M m rf t

( jf">3)*=

I (fjea>3),=0> dCJ

16

f o r

a11 such

~X-

I f

%

2iPtctiliAdxlA~-Adxm

^>i = (=1

and Xu, = Qi, then

(2PiQi)i|c/=0

for a l l

Q.GAi > whence Psi\u=0

s and

and t h e r e f o r e

t h i s c o n c l u s i o n remains i n f o r c e i f i t i s assumed t h a t 0^=0)2=0 , and a c c o r d i n g t o (3) xcf/yio3=0 for a l l Even t h e n o r m a l i z a t i o n of n e s s of

u>3 .


u>3 depends on

X.

We remark t h a t Finally,

X(*Dev.

u>3 by c o n d i t i o n c ) , however, does n o t g u a r a n t e e t h e u n i q u e -

We cannot d e s c r i b e t h e complete k e r n e l _fl

Kerxrfj^, b u t i t c o n t a i n s ,

e.g.,

*6D, ev

m 2

dxdQ ~ K . I n d e e d , •zd [ijd\d)=i

(dij) dzd =

= t (Ljf — i-^d) did =

-X-^l-cd=iL-gidid=0,

a c c o r d i n g t o P r o p o s i t i o n 2.9 d) and e ) . I t remains t o c o n s i d e r c a s e s i n which we a r e a b l e t o g u a r a n t e e t h e uniqueness of The c a s e

m= \ .

Up t o a term i n AQmK, which i s u n i q u e l y n o r m a l i z e d by c o n d i t i o n c)

and l i e s i n t h e k e r n e l of a l l t h e «i3=2/>A*dtt/*)"PA*^-

w3.

If

fdi^,

t h e element

then

T<%C73=0,

a>3 can be r e p r e s e n t e d i n t h e form

xd^Pj&fj^dAP/kQj^dx^O

for a l l

Q} =

XufiA,

k

i . e . , V P^QJ** = const for any The c a s e

w^A^K,

Qj. .

m£AQlA0Qm-lK.

i s obvious from t h e c o n s t r u c t i o n of t h e element u>3(
rfa

a)

From t h i s i t

e a s i l y follows t h a t

The e x i s t e n c e of

u>3 i n t h i s subgroup f o r

o>3 a t t h e b e g i n n i n g of t h e proof.

t h e d i f f e r e n c e of two c h o i c e s of

m l

&AQ - K.

Pjk=0.

")3

(a^A^K

Up to a term of

AQmK

can be r e p r e s e n t e d in t h e

Further,

i j ^ 3 = 2 A a i < » / = 2 XUi ( ^ P^^XiA<

• • A<£xt A • • • A ^ ) ,

i

x d i ^ 3 = 2 i-lY-% (%*iPt,i) dxi A • • • Ad*mi.t

If the last expression is identically zero, then, since Xut =QiQA may be chosen arbitrarily and independently, we find that for each i

the sum V ( — l)'-1Xdj(QPi,;)does not depend on Q .

It is easy to see that this is possible only for Pf ,=0,

which completes the proof.

We denote the forms o^ and a>3, constructed in Proposition 3.2 on the basis of the form respectively.

We shall indicate their explicit forms in co-

ordinates. We begin with the following classical lemma. Let B be a (not necessarily commutative) ring, and let
For any sequence of numbers tl§ i2, i3,..., l
...dt :

d(0) = id . 3.3. LEMMA.

For any x, y£B

we have

17

d{h,.. .,ik)xy-(-\)kxd{ik,..

.,h)y=

k

= 2 < - xrldt

(dv**' ••••'*)*w»*' • • • • 'i)y)fl

0-1

The proof is obvious.

3.4.

COROLLARY. We assume t h a t m

C (/;*, / ) > 0 , l < / < m ; k,l$N ,

[d„dj] = Q for a l l 1 < i , y
Then there exist numbers

such t h a t for a l l seN™,
w

d**«/-(-l) jed^=

2

(- 1 )" I + 1 C(/;*, l)dj(d*xdly).

Proof. We write out the formula of Lemma 3.3 for all sequences {h, • •., ijS\) with

w 2 e ' = s = s » take their arithmetic mean, and collect like terms.

We remark that for m = l we have C{J, k,l) — l.

3.5.

We can now write down formulas for 8a> and §vw.

Let u>= ^ ^ j sdtl\s)dx\A-..

Adxn .

We set */»* =rfjc,A •.. Adxm, dfx=afJC, A • • • Arf-«/_i Arf-*7+i A • • • Adxm. Then by 3.4 we find that l»=^i(-\fd*Pljlu1Ad'"x, s

• _ fc. -

2

•( -

1

) w + l c
@'Pi,Mk))^dmx-

For any ~X&Dtv we have i^ (d^d'P, /f^*'Adm.*)) = *<% X(—1) /-1 (&Pl%sdu^)Adfx.

Replacing here du\k) by (rf — •td)aj*) » w e do

not

change the r i g h t s i d e .

i t i s also possible to take du\k) — u\kJre^dXj in the term zero.

After t h i s change the form under the sign idij

Therefore

In place of (d —•zd)u\k'>

d"*x ', the remainder terms give

i s in the kernel of

x.

Finally,

adding v to i t in order to s a t i s f y condition c) of Proposition 3.2, we obtain f i n a l l y

$v<°=* +

2

M) , " + 'C-(/.-Mtf , / > , 1 ,W

-u^)dxj)AdTx.

It+l+tj-S

4. The Euler-Lagrange Operators and the Legendre Transformation 4.1. THEOREM. Let
form which belongs locally to AQ Kl

m

m l

TZ:N^>-M

, i.e., a

Then there exist forms 8m and Su> , locally belonging

to AQ AQQ K and Q^AQ ~ K , respectively, such that for all Xf*DeY + ADc we have (locally)

18

xl^u) = x (ijd + di Y) o) = x (i jf 8iu 4- dijS*) and, moreover,

(4)

xSu> = u>.

The form 8«> i s always u n i q u e l y d e f i n e d by t h e s e c o n d i t i o n s , w h i l e the form Sco i s u n i q u e o r i n t h e case ^€AlQmK i.f i t i s a d d i t i o n a l l y r e q u i r e d t h a t

l y defined i n t h e case m=\ SueA&A&^K Here

(locally'). 8 i s c a l l e d t h e Euler—Lagrange o p e r a t o r , and S i s t h e Legendre t r a n s f o r m .

Proof,

a) The l o c a l c a s e .

We s e t 5u>=8*flfu>, Sio = Sadw .where 8 and ^ a r e defined as i n

P r o p o s i t i o n 3 . 2 , where v i n c o n d i t i o n c) i s

w.

We then o b t a i n a c c o r d i n g t o 3 . 2 :

rfu)==8a)-)-u)2; i^tOo = xrf/^S'j) for X^Dev\

From t h i s we deduce t h e i d e n t i t y (4) for X^Dev If

X(*DeV,

then

and

XQADC

xSu> = co.

(5)

individually.

whence a c c o r d i n g t o (5)

£^J>=0,

xL^io = x/-^cfu) = •zi-^ (Sin -\- co2) = xt^8a) -f- idijSin. The uniqueness properties of l

m

is noted that do, b^AQ AlQ K, i^du>, ij^Sa) for

8J> and St> follow from those proved in Proposition 3.2 if it so that these forms are uniquely determined by all values of

X(*Dev , and these values lie in AQmK ; therefore, x is the identity on them,

and hence the decompositionrfio= Sw-f-u>2 with the properties postulated in Proposition 3.2 follows from (4) for X&Dev . It thus remain to verify (4) for X(
We shall first establish that x(iy2m+1i4) = 0,

i.e., (i-^Qm+1A)s — 0 for all sections s . Since (QmnA)s={0}, s

(ijfv) = tA-v*

for Xi*D(K)-

Since

ij

it suffices to verify that

is a differentiation and restriction to s is a homomor-

phism of the algebraof forms, it suffices to verify it on the generators. coincide.

On A both sides

Further, (ijdXjY = {XxjY = (**,)* = ix (dXjY,

(M«J«)-(^J*') , -(S»J* 4 * , ) ^«)'-2^ + * , , , ^I.

Using this, we see that it is necessary to verify the formula idij^a = -zdi-^Sio. We have xd ipSui = -zL-^S'-a—i(ijdS)=xZ.j-.Su>. From P r o p o s i t i o n 2 . 9 we have 19

But TSU> = CO, which gives the required result. b) The Global Case. Since izing it are consistent

8co is locally uniquely defined and the properties character-

with the operation of restricting to a subdomain, all local con-

structions automatically fit together. where uniqueness is guaranteed.

For So an analogous argument goes through in cases

Aside from these cases, as has been shown, it does not hold;

e.g., it is possible to add locally to Sco any element of dxd(AQ™-2K) without changing the values of idixSa Sa

for all XeDev+ADc

. Therefore, to prove the existence of a global form

a special treatment is necessary in order to establish the possibility of consistent

local constructions. This has been done by B. A. Kupershmidt, but we shall not reproduce the proof, since it will not be needed below.

It is essentially a question of the vanishing of

a certain cohomological obstacle, and the problems which arise here merit special attention. The question as to what addition conditions (functorial property, for example) it is necessary to impose on So> in order that S© may be uniquely chosen has not been solved. 4.2. Classical Formulas. We shall apply the formulas of 3.5 to the case a>=PdmX, dm = 2^) r f »j"Atf m JC

. We obtain "«£.A. *./«. 6P

V/

iv>,W dP \(s)

Further, S«_.+ V

^(rf«|«-U^Adx^dfX,

where

_« i7r _2(-.)««c ( y, 4 .o( si ^ w )«» (it can be shown that C(J;fc,l)

depends only on A + e y ,/).

5. The Variational Complex 5.1.

Definition of the Complex. The beginning of the complex is locally as follows: A^AQtK*...

'*AOi*Kt*AQlAfir"K.

The fact that (trf)2==0, is proved in Proposition 2.9 d).

The equality *wf=0 is established

as follows. The operator *Lxd on AQ^K can be transformed in two ways. xljfrf=T (ijd -f- di y) d=id

20

(6)

(%rf).

First of all,

Secondly, by formula (4) and P r o p o s i t i o n 2.9 d) for XeDew + ADc xLJd = xL^xd = xi^xd + xd{i-RSxd). From t h e u n i q u e n e s s of t h e E u l e r - L a g r a n g e o p e r a t o r we t h e r e f o r e f i n d t h a t We s h a l l now i n d i c a t e how t o extend complex (6) to t h e r i g h t .

8tfif=(X

The e x t e n s i o n o p e r a t o r

w i l l a l s o be e s s e n t i a l l y an Euler—Lagrange o p e r a t o r but i n a l a r g e r bundle « X l d

:NXR-*-MXR-

L o c a l l y we d e n o t e by xm+l

Km

C°°(Xi, ...,xm+l),

t h e c o o r d i n a t e on t h e a d d i t i o n a l f a c t o r

AI^UC"^,

n;k£Nmn,

a<*>|t = l

|ft|
R , and we s e t

=

The c a n o n i c a l inbedding

AaAW,

c o r r e s p o n d i n g t o t h e p r o j e c t i o n of « X i d onto *, i n l o c a l c o o r d i n a t e s can be i d e n t i f i e d w i t h t h e imbedding a<*)*-»fcttj*,0\ which t h u s h a s an i n v a r i a n t meaning: t h e c o o r d i n a t e s "along t h e fiber"

ut

i n t h e e x t e n s i o n t o NxR

The imbedding lift

from it t o

a r e assumed t o be independent of t h e new v a r i a b l e .

A-+AM d e t e r m i n e s t h e imbedding QA-+QAW which a l s o c o r r e s p o n d s t o t h e

*Xid .

QlAQmK w i t h i t s image i n 21.A(1>QmA'(I,, we may c o n s i d e r

Identifying

t h e composition of t h e o p e r a t o r s 1

QlAQmK^QiAWQmK^-^AO)Q^K^)-1lQiA^Qm+1Kil),which

1

d e n o t e simply by BCt* ' (B* ' i s t h e E u l e r - L a g r a n g e o p e r a t o r , and fibration

i:Xid ) .

t

(1)

we

is the operator

x

for

the

This c o n s t r u c t i o n can o b v i o u s l y be i t e r a t e d which l e a d s t o t h e sequence (7) L^(2)01 J 4(2)gm+2/^-(2)_ > _

5.2.

THEOREM.

#

_

Sequence (7) is a complex which we shall call the (local) variational

complex of the fibration T..

The global complex is a bundle complex the local sections of

which have the form (7). Proof. We first of all introduce a new operator x+:Qm*tA-+ Q^A^K for all Xu . ..,XkeD(AlK)

by the condition:

m+

and t»eQ *A i*,. • •f'xft-+U) = t (**,.. -fXftW).

The e x i s t e n c e and u n i q u e n e s s of t h e o p e r a t o r

t + a r e o b v i o u s , s i n c e a form i n QkAQmK can be

c o n s i d e r e d a skew-symmetric, m u l t i l i n e a r f u n c t i o n of k m

Q K

.

I t is clear that

5.3. Proof.

LEMMA.

x* i s

(8)

f i e l d s i n D(AlK)

with values in

-A-linear. o>£AQmK we have 8 .

For any Lagrangian

I t s u f f i c e s to v e r i f y t h a t £^8u> = xi-^dS® for a l l X^DeV,

because of Lemma 2 . 7 .

According to formula (4) and P r o p o s i t i o n 2.9 d ) , we have / , - « ) = i L - a=xdi-^Sm + i % 8u>=x (Lj — i^d) Sio + i jj-B J> = =xL-g xSu> — xi-^dSo + ix&M=iLj a — xijdS®+i-j^-oTherefore, 5.4. Proof.

xi^dSia = i^o, LEMMA.

as required.

t0> =*<>>•:+ on

I n view of t h e

QtAQ^K-

A - l i n e a r i t y of both o p e r a t o r s , i t s u f f i c e s to v e r i f y k)

m

e q u a l i t y on g e n e r a t o r s of t h e form du\ f\d x kind t h e o p e r a t o r

x

+

is the identity.

(

and /\da f>/\d™x.

this

On forms of t h e

first

F u r t h e r , a s i s n o t hard to check, 21

z+ (rfu<*> Aduf

Adfx)

whence i t f o l l o w s t h a t t h e v a l u e of

= ( - l ) ' " 1 (u^du**

af+t')duip)f\d«x,

-

x(')t + on t h i s form i s e q u a l to (even i n

(-ly-^u^u^^-a^u^^dx^Adx.A... On t h e o t h e r hand, t h e v a l u e of

QAm)

Adxm.

t(') on our form i s equal to

The l a s t two e x p r e s s i o n s o b v i o u s l y c o i n c i d e . 5.5.

Completion of t h e Proof of Theorem 5 . 2 .

According t o Lemmas 5 . 3 and 5.4

8 < I V 1 > 8 = 8 ( 1 W r f S = 8<1>,(,)dS. But

8 (1) -: (I, rf=0,

A(l).

by t h e argument a t t h e b e g i n n i n g of t h i s s e c t i o n a p p l i e d to t h e r i n g

Thus, (7) i s a complex a t t h e term QlAiimK and hence a t a l l remaining new t e r m s : 3(a+I>,:j<«»)__ 0

for a l l

verified

The f a c t t h a t i t i s a complex i n t h e terms t o t h e l e f t of QlAQmK was

a>0 .

earlier.

The e x a c t n e s s of complex (7) i n t h e terms to t h e l e f t of

AQmK

Rham complex t o t h e j e t s ) was r e c e n t l y proved by A. M. Vinogradov.

( t h e " l i f t " of t h e de The e x a c t n e s s of t h e

g l o b a l complex (7) i n t h e remaining terms was e s t a b l i s h e d by B. V. Kupershmidt.

We s h a l l

r e s t r i c t o u r s e l v e s to t h e proof of t h e c l a s s i c a l p a r t ( " i f t h e v a r i a t i o n a l d e r i v a t i v e of t h e Lagrangian i s equal to z e r o , then i t i s a d i v e r g e n c e " ) . 5.6.

THEOREM.

Proof.

Ker6=ImTd. .

We c o n s i d e r a c o n t r a c t i v e homotopy V, t&[0,1] of t h e open s e t

which t h e complex (6) i s

0 < * < 1 , tpo X, that for

= 1

d and

-t)~2\).


(point).

be i t s c a n o n i c a l l i f t

t o D(A) m

tZ.jjf<»=I]f(fl f o r a l l * < 1 , and hence

<»eA2 K.

o*Lj—-j
t o formula ( 4 ) , we find

, over

defined: *,:(.*„ Uj)~(tXi, uj exp[l - ( 1

I t i s obvious t h a t

Vc.N

Let XfiD(AQ)

where

with

be t h e c o r r e s p o n d i n g v e c t o r f i e l d , and l e t

a c c o r d i n g t o Lemma 2 . 6 . Since Xt

onto i t s e l f

Since Xt(*D(A0) , i t i s obvious

i s t h e d e r i v a t i v e of




Applying

9*

that

^zdilljSv+rfljt*, since, as i s e a s i l y seen, the field

Xt

i s not defined, i t s l i m i t

t e g r a t i n g (9) on t. from 1

9*T = T9* ( t h e mapping

1

0 to

?}£x<» e x i s t s a s

<->l

for a l l

and

a\k)) .

m.

Therefore,

Although in-

1, we f i n d 1 _

_ a , = \ A-y*dt = trfC 9 J i 7 S a d t + x\
22


1 _

' _

def

= xrft\ y*ijSmdt + %jj w*i^Udt = — «%(w) — ij>2(8
since ~d = tdx.

Thus, 1

o

and we find that if

8u = 0,

'

then 1 (a>) which proves the theorem.

We remark t h a t if the form
«<*> ( r e s p e c t i v e l y , in «<*>, JC ; ), then

so are also Sw, hm polynomials by v i r t u e of the formulas of 4.2 and with these also


k

t|i2(8u))

, since
The Formula n

Fur ther

*<»**,=2 u^dx,

8
a>=Pdmx .

We s e t

Then 8a>= 2 ,_,

^ldul/\dnx.

+ «{em+«' dxm+u whence

i-i

n 6P ( W dx mviAdxiA T(»> 8
.• • A^»-

i-i

Therefore, 8(0,(08(0 = 2 gF/(S "i*" 41 ')

4*iAdxm+lAdax=

rt

'

/An \^rn+v

,|t W <S

= 2 ( 2 m ( - W (^r>5T, i

] ' JujAdx^Adnx ~ 2 IBSJ)

dalAdxnAAdxlA ... A ^ J .

Further,

Therefore, after division by dxMlrXf\dmx

the formula

8(1)T<1)8O)==0

assumes finally the following

form:

We recall now that over A the variables dkufm+l)

are free.

Therefore, for all i, j

we ob-

tain a family of equalities among differential operators which is equivalent to the formula 8(OtO)8=0 . 5.8. LEMMA. For any P&A and l
We shall need this lemma and its corollary in Sec. 7 to investigate the Hamiltonian structure. 5.9. COROLLARY. For any

XeDer

23

Proof. 6.

We apply the i d e n t i t y of Lemma 5.8 to

Xuj and sum on /'.

Nother's Theorem and Lagrangian Conservation Laws Let (a=PdmxeAQmK be some Lagrangian.

6.1.

t h a t the equation

s

(6co) =0

I t i s evident from the formulas of 4.2

r e l a t i v e to unknown s e r i e s

system of Euler-Lagrange equations for the functions

s s

coincides with the c l a s s i c a l un$ with Lagrangian

ux

P .

Its

solutions are called extremals of the Lagrangian w. 6.2.

I n t e g r a l s , Flows, and Symmetries; Analytic Version. m 1

integral for the form u> i s a form M^AQ ~ K such t h a t
tt(t')=0

A conservation law or an

for any extremal s of the Lagrangian

t h i s means t h a t vgi4 i s a functional of the space of j e t s which i s constant along

any extremal.

In the general case for

I 2(—1)'-1
v = £ Ptdfx

the r e l a t i o n

rf(vJ)=0

has the form

along any extremal. m

A flow for

a) i s a f i e l d

Y(
If <s>=Pdmx, F = 2 Q A

iym i s an i n t e g r a l .

2—1

m

then ty(i)=^| (— \)l~lPQldfx

. It is evident from this that if P does not vanish on its ex-

i-i

tremals, then for any conservation law v there is a unique current / such that v = i-pio. The classical Nother theorem affords the construction of conservation laws on the basis of a Lie group O, which acts on K:N-*-M If X

action s^co*.

and preserves the Lagrangian

io or even just the

is an element of the Lie algebra of the group G, considered as a field

fur

on N , then the G-invariance of o> implies that 1 ^ = 0 . More generally, let X(*AD(A0)

be

s

a field on the jets such that (L^
Integrals, Flows, and Symmetries: Algebraic Version.

The definitions of the

preceding section appeal to the set of all extremals of the Lagrangian u>, which may be empty or very complicated.

In practice integrals, currents, and symmetries always satisfy the

following algebraic version of the definition whose application does not require vanishing Dc -closed ideal in ^4, generated by expressions

on the extremals. Let /(8a) be the minimal of the form ixh,.. minimal

•*>„>>, where X£D(AlK),

£>e-closed ideal in the ring

K„ .. .,YmeD(K).

QA,

Let J(h^aQ'AQ^K

generated by /(&«>) .

We call v an algebraic conservation law iffi?vg./(8u>). We call if iyia

is an algebraic conservation law.

be the

Finally, X&AD(A0)

Y

an algebraic current

is an algebraic symmetry if

L-jfwQj (8
We point out that the use of this definition makes it possible to obtain additional information. dv

For example, if v is a conservation law, then the explicit representation of

as an element of /(8a>) determines important invariants; the theory of characteristics of

Gel'fand and Dikii [2, 3] is based on this. 6.4.

THEOREM,

NSther theorem).

24

a) If A" is a symmetry, then ti^Sa

is a conservation law (the formal

,

Y

b) For any flow Proof.

the field Y is a formal symmetry.

We write the proof in the analytic version; the algebraic version may be con-

sidered analogously. a) Since X is a symmetry, for any extremal s we have (Lj
s

from (4) that (i^w) -\-d(i^Sj>) =0

It therefore follows

on the extremals, whence d{ijSo>Y<=d(iixSa>Y= _(/x8
The last inequality follows from the fact that locally 8o> = 2 So" ^^mjc

»

so

that

'x*""

is a

linear combination of the g^-r which are zero on the extremals. b) Since Y

is a flow, we havetf(t7co)I=Oonthe extremals, so that (L7u>Y=(diy» +

iyd*»y=(i7du>Y.

But t h e l a s t e x p r e s s i o n i s zero a s v e r i f i e d i n t h e proof of Theorem 4 . 1 a ) . 7.

The H a m i l t o n i a n S t r u c t u r e 7.1.

We s h a l l g i v e a l o c a l d e f i n i t i o n of t h e Hamiltonian s t r u c t u r e on

TZ;N-+M\

global-

i z a t i o n goes through a u t o m a t i c a l l y . 7.2.

Definition.

Let

Y

b e an R - l i n e a r o p e r a t o r

T:AQmK-+bev

.

We d e f i n e a

m

b i l i n e a r c o m p o s i t i o n law on AQ K, by s e t t i n g {
(10)

The operator T gives a Hamiltonian structure if it takes { . }r into the commutator in the Lie algebra DeY ,

r{u)„ 2)], and i t s k e r n e l c o n t a i n s 7.3.

Comments.

(11)

Imxrf=Ker8.

The m o t i v a t i o n for t h i s d e f i n i t i o n was given i n Sec. 1; i t i s p a r a l l e l -

t o one of t h e c h a r a c t e r i z a t i o n s of a f i n i t e - d i m e n s i o n a l Hamiltonian s t r u c t u r e . here

m

^AQ K

i s c o n s i d e r e d a s a r e p r e s e n t a t i v e of t h e f u n c t i o n a l

u)(s) = W

In p a r t i c u l a r ,

on s e c t i o n s .

M

Since

tdv = 0 ( f o r a form

we r e q u i r e t h a t 7.4.

vg.i4Qm-1/f w i t h compact s u p p o r t o r on r a p i d l y d e c r e a s i n g s e c t i o n s ) ,

KerTiDlm'tt/ .

The f o l l o w i n g r e s u l t i s obvious from t h e d e f i n i t i o n s .

(We sometimes omit V i n

t h e n o t a t i o n {«>i, «>2}r) • 7.5.

THEOREM.

If

r:AQ m K-*• D eT d e f i n e s a Hamiltonian s t r u c t u r e , then for any
we have {•"I.^ + H .

•"ljGKerr,

U)

{">!. {<%, ^ i ) + K . {<°i. 2>} + K . K - <°i}}GKer r .

25

In p a r t i c u l a r , the operation { . }r induces on AQmK/KerY

the s t r u c t u r e of a Lie algebra,

m

and T induces an imbedding of the Lie algebra AQ K/Ker T->Dev . A

n

7.6. Since KerficKerT , we can always represent T as a composition AQmK->Im8-*• D,ev Indeed, lm^CzAQ1A(^mK , and our operator B in a l l concrete cases w i l l be a d i f f e r e n t i a l operator defined on the whole module AQ1A(j2mK. We c a l l such an operator a Hamiltonian if r = 5 o 8 defines a Hamiltonian s t r u c t u r e . 7.7.

LEMMA. For any a d d i t i v e operator 5:Im8-»-Z)ev we have £8 {«!, m2}B6 = B8 (/Btol8u>2).

Proof-.

For any X^Dtv,

(12)

by formula (4) and item 5.1 we have Ijf<»2 == ijf8
Setting here JT^B^ 7.8.

and applying (10) with r = 5 8 , we obtain (12).

We rewrite the Hamiltonian condition (11) of the operator T = 5 8 in terms of

structures related to the choice of a local coordinate system (Xj, at). Let us f i r s t agree on the following notation. the module of column vectors of height

Let 5s be an s

n with elements in 3 - .

A-module. 8

If

3*" denotes

P6S " , then

P'

denotes

the transposed row vector ( t w i l l generally denote the transpose of a matrix of any s i z e ) . If QeAn, P£$>n, then Q'P^

•7=:A-*-An takes

The operator /6P

^=|

W 5u /.

we wrxte

denotes the scalar product

OH

P&A into the vector

2^p<•?= = (£)

8tf

\otti;

(cf. 4 . 2 ) .

In place of

-=-.

6a'

Further, l e t A[DC\ be the ring of d i f f e r e n t i a l operators over A, generated by t o t a l d i f f e r e n t i a t i o n s along the base.

For any P&A we denote by D 0/ (P)(
Frechet derivative

^ < p > = 2 £ p 7 ^ *=aj....a!-,
as the

(nXtl)

matrix (not the determinant) with the operator Du.(Pi) at the site (//) . Its invariant interpretation will become clear in the next section. In order to write the Hamiltonian condition (11) in this notation, we identify the m

m

modules AQ K, A2M 0 2 A' and £>ev, respectively with A, A", A" m

mappings: PdxlA—Adxm^-P,^Pidutdx1A-Ad x^(Pt),X<^(Xai) l

n

amples lies in M„(A[DC\) 7.9. we have

26

Proposition.

by means of the following

for XeAD(AolK). n

operator B:AQ A0Q K-*-Dev

A-

will be represented by an operator B:A -*-A",

Then the

which in all ex-

. The operator B:An-i-An

is Hamiltonian if and only if for any P, Q(*A

Proof.

We s h a l l verify that the l e f t and r i g h t sides of (13) are i d e n t i c a l with the oh = Qdmx, <»2 = Pdmx

l e f t and r i g h t sides of (11), r e s p e c t i v e l y , for T=Bl,

According to (12), the l e f t side of (11) i s £3(iB^fi^) 8u>, i s represented by the vector

-=; ; the field

.

• According to 4 . 2 , the form

i s therefore represented by the vector

B&IDJ

n

B^S: . 6u

by

M8«>2)=2 AXuidmx

From the formula

^B~

.

i t follows t h a t the form

fZTi °"<

F i n a l l y , the operator fio = r

J'B&D.H

i s represented

i s represented by the operator B-= .

This

reduces the l e f t side of (11) to the l e f t side of (13). In order to compute the r i g h t side of (11), we note f i r s t of a l l that the field Bow2 8P

i s represented by the vector

—

—

Br= .

Further, for any R&A and X^D&V

we have XR —

n

2 — % {Xut)W = ^ Du(R)XUil i.fc

du

<-l

i

this explains the meaning of the Frechet derivatives. Thus,

'

_ _ for any vector R^A" we have XR = D(R)Xu

.

Applying t h i s to the case X=Bhml

we write BIUUBOM^U in the form £ ) [ £ — ) B - S .

Similarly, the second term of the commutator

gives the second term of the r i g h t side of (13). 7.10.

— gp and R = B^ ,

This completes the proof.

We s h a l l now concern ourselves with transformation of the c r i t e r i o n (13). We

r e c a l l f i r s t the formalism of the adjoints of d i f f e r e n t i a l o p e r a t o r s . have A[Dc] = A[du...,dn],

[dhdj]=0.

Let L£Mn {A [Dc]) 5

which act in the natural way on S ", where 5

s

(the ring of

i s any

In a local chart we

(nX n) matrix operators

A [D c ]-module).

We define an additive

+

mapping Z.—Z.+, by s e t t i n g for any matrix aeM„(j4):(ad*) = ( — l)l*id*°a', where

a' i s the transpose

of the matrix a . The following r e s u l t i s c l a s s i c a l (and can be deduced without d i f f i c u l t y from 3.3 and 3.4). LEMMA, a) For any L, M we have (Z.+)+ = L, (LM)+=M*L+. b) For any PeA",

7.11. where

5

S

Q&\

i s an A [Dc\ -module we have P(LQ-

{L+pyQedlg;>+... +dng>.

The operator L i s called formally symmetric (respectively, skew-symmetric) if (respectively,

IS=L

L*=—L ) .

The next r e s u l t i s obtained immediately from Lemma 5.8 and the d e f i n i t i o n of the Frechet derivative. 7.12.

LEMMA. For any element P£A the operator D (^=\ i s symmetric.

I t i s moreover clear from the considerations of 5.7 that the part of the v a r i a t i o n a l complex AQmK^-AQ1A0QnK-*AWQlAWQm+1Kil) with the i d e n t i f i c a t i o n s of 7.8 may be replaced by

27

6/4a D(-)-D+()

A+A* + M„(A[Dt]), and the theorem of B. A. Kupershmidt on the exactness of t h i s complex means that D+(Q) = D(Q), if and only if there e x i s t s a PQA, such that Q=?=T=- There i s thus an effective c r i t e r i o n _ 6u for determining if a vector QcA" i s a vector of p a r t i a l derivatives of some element of A. Suppose now that BQM„(A[DC]) i s some d i f f e r e n t i a l operator. 7.13.

THEOREM,

a) If B i s skew-symmetric and BeM„(K[Dc\) , then B i s Hamiltonian.

b) If B i s skew-symmetric and Bf<Mn(A0[Dc]) , then i n order that B be Hamiltonian i t i s necessary and s u f f i c i e n t that for a l l P, Q(*A the following i d e n t i t y hold:

[D,B}^B^-[D,B]^B6Z=B[^-d-i^\

(14)

We begin with t h e following lemma. 7.14.

LEMMA. Suppose that B&Mn(A>[Dc]) i s skew-symmetric.

±mB^)=D(^)Bb4-OmB^

Then

+8Zd-*%

(15)

where the last term is a column vector with coordinate - = 3 — = at the i-th row, and 3— is flu d"/6«

<*"<

the r e s u l t of applying ^— to the coefficients of the operator B. Proof.

By t h e d e f i n i t i o n s

Further,

'©*3*-)-['w*s+^«'§+ff'ttg-* m

2 ^ iditias u

C7ZJ

an

d 2i4 i s considered as an A-bimodule with m u l t i p l i c a t i o n

u>P=Pa> for

any PGA, a> Therefore, for any X(
Using the formula XR=D(R)Xu

, we rewrite the right side of (17) in the form

[(*SiO (I) *">S>(1)«+ (SIf)'*}

<«>

We now note that for any R&A we have djRdmx = ( — l)'-ITflf (/?#"*). Therefore, transposing the operators i n (18) while replacing them by t h e i r a d j o i n t s , as i n Lemma 7.11 a ) , we do not 28

change the value of the r i g h t side modulo Imzd. In the f i r s t term of (18) we transposed the operator operator 5£>(^£) .

D ^ = l a n d in the second term the

Recalling t h a t B i s skew-symmetric while D ^ = ) and £> ^ £ ) are symmetric,

we obtain f i n a l l y

From the uniqueness of the Euler-Lagrange operator i t follows that the form following on the r i g h t side of (19) i s id (*£t B^Ld^x} 7.15.

Proof of Theorem 7.13.

If BeMn(K[Dc]) , then to the vector Xu£A\ sions coincide, since

^f=0XeDeV,

.

%

This and (16) give the assertion of the lemma.

By formula (15) the l e f t side of (13) i s equal to

Moreover, in t h i s case they give BX^

BD

[%)=D(B^)*

and XB ^ p ,

K[DC] commutes with Z>ev .

s i n c e when

a

PPlied

respectively, and these expres-

Thus, the c r i t e r i o n (13) i s s a t i s f i e d in

t h i s case. Using the information obtained, for B^AQ\DC\ formula (14) obviously coincides with c r i t e r i o n (13). In Sec. 8 we s h a l l apply the c r i t e r i o n (14) to prove that two special operators B over a one-dimensional base are Hamiltonian: the Gel'fand-ilikii operator and the Benney operator. 7.16. r.:N->M .

Hamiltonian Conservation Laws.

Suppose V defines a Hamiltonian structure on

For any form m^QrfjdA . . . /\dx„£AQmK

system of evolution equations u,*=T(
io or Q

i t s Hamiltonian and B

The form ux = Pdxl/\

to the field

I »

there corresponds the

in the notation of the preceding s e c t i o n s .

the corresponding Hamiltonian operator.

. . . /\dxm , or the coefficient P, i s called an integral or a conserva-

tion law for t h i s system if

{
t h i s means that for an evolution

In the corresponding analytic formulation

s, because of our system,

dt

M

M

M

i.e., \<»l is a quantity conserved in time.

In the notation of 7.8-7.9 P is an integral

M

for the Hamiltonian Q, if

8u \ 6a'

Su J

29

The integrals Pi and P 2 are said to commute if {<«i,<»2}r€Ker8, i.e., ou V, ou1

ou I

We note in conclusion that if {<«>, ..., {u>,«>,}..;}gKer 8 (k>2

factors
sections s, which are uniquely included in the flow with Hamiltonian co, it is possible to \

find k— 1 quantities conserved in time as follows. We have \-§r) '

{(o.u),}.. .}=0 . Therefore W

M

M

is a polynomial in t of degree k — l in general.

M

Normalizing

M

the initial time on a given trajectory of the flow so that the (k — 2)-th coefficient of the polynomial is zero, we find that the remaining k — 1 coefficients are invariantly defined, conserved quantities. 7.17.

Stationary Manifolds of Integrals. Let wt be a conservation law for a Hamilton-

ian system of evolution corresponding to a field

r(i») . By definition Zr^.i^Glm"^ or

ir(»)8«)iGlnitrf . The latter condition means that if s is an extremal of the Lagrangian

ID, ,

then within a short time in the linear approximation the evolution of s, under the field r(u>), will also be an extremal or that the "field extremals u>1 ."

T(m) is tangent to the manifolds of

It is possible to give a precise meaning to the last assertion and to prove

it at least for /re=l and polynomial integrals «>i, when the manifold of extremals W can be identified with a finite algebraic manifold by assigning to each extremal its value at x — 0. Then T (co) defines on it a flow which is also Hamiltonian with Hamiltonian computed on the basis of «) and m1. W

This assertion is nontrival because the natural class of functions on

is not obtained by restricting functionals W to

functions on

W in the natural structure of W.

T I it consists, for example, of smooth For a precise formulation and proof see

the papers of Bogoyavlenskii and Novikov [1] and also Gel'fand and Dikii [3], We shall make use of this remark in Chap. 3 where in place of the equations we will solve jointly the system ut=T(
and 8'J>1=0 , where wt

is an integral.

ut=T(m)u It is

just this procedure which distinguishes in an invariant way the class of multisolution and finite-zone solutions of the Korteweg—de Vries equation as already mentioned in the introduction. 7.18. Examples, a) Let

n = 2r , and B = \ E ~ Q \ * where E is the identity matrix of —•

go

order r . The system of evolution at = B—=- is traditionally called a Hamiltonian system 6u with Hamiltonian P in "canonical coordinates". It is also Hamiltonian in our sense according to Theorem 7.13 a ) , since B is a differential operator of order zero and B*=—B . Below we shall show how it is possible to define a "cotangent fibration" n:Ar->Af to any fibration ir:Af->./W, a canonical Hamiltonian structure on it, and to each local coordinate system (xu uj) on N a system of coordinates (xit Uj, Vj) on N, such that in these coordinates the operator B corresponding to this structure has the form ( E J.

30

b) Let

m=l

and B — d = di .

That equations of the form ut = d-s—, which include the

usual and higher Korteweg—de Vries equations, are Hamiltonian was established by Lax in the functional version and in terms of Fourier coefficients by Gardner.

Our proof of Theorem

7.13 i s a generalization of the proof of Lax as reworked by M. A. Shubin.

A formalized

version of Gardner's arguments w i l l be given at the end of t h i s section. 7.19. The Cotangent F i b r a t i o n . The cotangent fib r a t ion to TZ:N^-M we c a l l the fib r a t i o n lz:N-+M, where N = T*{N!M)®Amit*T* (M). Here T(N/M) i s the tangent bundle to N along the fiber

"*, T (M)

i s the tangent bundle to M , and the a s t e r i s k denotes dualization; _

_ a

it

the tensor product i s taken over vV ; v i s the composition N-+N-+M, where a:N-+N

i s the

natural projection. Sections of a a r e n a t u r a l l y identified (locally) with forms in Q1A0QmKLet [Xi, Uj) be coordinates on N. On the basis of these i t is natural to define coordinates (xt, Uj, vj) on tf (/ = 1> • • •.») as follows. If x£N l i e s on a section of o, c o r r e al sponding to the form ^lPjduJ/\d'nx , then vj{x)=P1{a(x)). i 1 _ " Let A) be the ring of smooth functions on ic-1((/), where U i s a neighborhood with coordinates

(x).

to the projections We consider section

Then A0 = C°° (xt; u}, vj),

and the natural imbeddings KdA0c:A0

correspond

i: and o . P = 2 VjdU]Admx(*AQQ1(A0)QmK. I t possesses the following property: if the

s of the f i b r a t i o n

a corresponds to the form

to see that t h i s property determines

p uniquely.

uigQ1 (Aa)QnK » then p*=
Thus, f

I t i s easy

i s defined canonically and

globally. Let

Dev be the evolution d i f f e r e n t i a t i o n s for

it" .

XQDey , then i-^d^lAaQmK

If

and

the formula i jd p = 2 (Xvjduj—Xttjdvj) d'x /-i

shows that t h i s mapping Dev-+QlAQQ"K i s an isomorphism of l

m

inverse operator, B:Q A0Q K-+Dev

A0-modules.

We denote by B the

, and we set r=-58:I0Qm/r-vDeV.

7.20.

THEOREM.

In the coordinates (xh a,, vj) on N with the conventions of 7.8 (as

z

applied to w ) the operator —B

i s represented by the matrix

/0 - £ \ I» n ) »

the

°Perator

^ is

Hamiltonian, and the system of evolution corresponding to the Hamiltonian Pdmx(
6P

«

*P

31

All t h i s follows readily from the definitions and Theorem 7.13 a ) . We note further that the c l a s s i c a l expression !-2fL.-Q--- PJL\dmx i s equal to iBta,B^m>2 where v\=Pdmx, Qdnx . 7.21.

The Hamiltonian Property in Formal Fourier Coefficients.


In t h i s and the follow-

ing sections we b r i e f l y describe the formalism of v a r i a t i o n a l calculus in terms of Fourier coefficients in the simplest case: M i s the unit c i r c l e , Af = vWXR .

We r e s t r i c t ourselves

to Hamiltonians which a r e polynomials in u and do not depend e x p l i c i t l y on s t a r t from the algebra A = C[u, u',...].

x , i . e . , we

We s h a l l f i r s t introduce formal Fourier coefficients oo

(•n„|ftgZ)

"Vne2Kinx. As the Fourier

, assuming that the Fourier s e r i e s of u i s F(u) = 2 n^—oo

s e r i e s for uu) i t i s then natural to take F(uU))=^(2un.yvne2lti',x, n U)

The Fourier coefficients of the polynomials in u i n f i n i t e l y many v a r i a b l e s B = C[v„] .

vn .

w i l l , however, be special s e r i e s of

In order to introduce the corresponding r i n g , we f i r s t s e t

In t h i s ring we introduce the order function

OTd('^cl.v±\=mla^otdv±\c±^'0) then obviously

.

B = © BN

i . e . , F(a (y) ) = (2ic/By©n .

If

BN

ord l©^'.. .ti°*) = |«i|+• • •+|«*| and

i s the space ot polynomials purely of order

( d i r e c t sum). We s e t £? = I I BN •

N,

The elements of the space

OO

B

are i n f i n i t e s e r i e s

2 fft> /N^BN •

Since

ord(/£) = o r d / + o r d g

for f£BM, g§BN, B

N=0

has a natural ring structure: (2/v)(2 ^ ) = 2 A * '

K^/K-MSM

'

We call the ring

5((^iJr)) = ( 2 / ^„2xinx

fn£B, zc>0,

n0>0,v\n\>nQ,

ord/„>c|«|} the ring of formal Fourier s e r i e s ; i t has the natural m u l t i p l i c a t i o n \^ifme2,tlm \^hpe2"'px

, where hp = y

fmgn

.

The convergence of the s e r i e s for hp in B

x

) (2w2"iw)= i s ensured

m+n=p

by the fact that OTd(fmgn)>c(\m\ 2 ix

B((e " ))

+ \n\)-+co together with

, because ord/t„> min c(\m\ +

\m\, \n\ .

The product l i e s in

\n\)>c\p\.

m+n=p

We extend to B((e2mx))

the differentiation -^-

by continuity: we extend from B to

B and then set -^-(e2ni"x) = 0 . This is clearly possible, since OTd[~-)>otd/~n.. We similarly define a differentiation d, which on B has the form dvn = 2itivn. We introduce a ring homomorphism fM-*^/ 1 8 ')), by defining F(a) = 2 " / ' " . It is not hard to see that F

is an imbedding which commutes with d. Let

for any P g A 7.22. LEMMA. For any P&4

32

we have

F

n[-^-)=-^-F0(P).

F(ua))=dJF(u)

F(P)=^Fm(P)e'2Mmx

Proof. We consider the two mappings of A-+B({en

They are both differentiations of (;)

Further, on the generators «

A

into the

)):

A -module B((e2Mx))

they coincide, taking

(relative to F ) .

a<'> into (2i:in.y'e2ainx'. They therefore

coincide everywhere, whence

We note now that F0°d = 0,

so that we may "integrate by parts" under the sign of

F0 , and

the last expression is equal to

'•(§'--<-•>'*«'(&))-M& which proves the lemma. 7.23.

Corollary.

6P The equation «/ = 1.

Thus, s e t t i n g pn=v_n,

4n = %^* H(p, q) = iF0(P), we find that on the "hyperplanes" z>0=const

our equation formally has the Hamiltonian form dp!L=_dH_ dt dqn'

dt

dqndH dPn

Further, the Poisson bracket may be w r i t t e n

^ S

= (2al^o(Q)e^

whence 00

FoiQ, -P} = 2(-2w«) a -^-F 0 (Q) 5 | x F 0 (P) = n—oo

in agreement with the c l a s s i c a l formalism. 8.

Special Hamiltonian Operators Over a One-Dimensional Base

8 . 1 . In t h i s section we s e t tn — l, Xi—x, d=dl=dldx, K=Caa{x), and we work with the sequence of rings -An = U C°°(JC; a<*> | 0 < / < r t , 0 < & < i ) , and also with t h e i r union A=\jA. . We point out t h a t the enumeration of the variables Uj begins with zero in contrast to the preceding conventions. In correspondence with t h i s the enumeration of the elements in the rows and columns and also the enumeration of rows and columns in different matrices also begins with zero. 33

We define the operator

Bn^Mn^ (An [
B « = 2 [Bn.id'-(-iyd'oB'Hil],

(Bn,i)ab

0 for a+b + i + l>n, (' + ") for a+b + i +

BnjeM^M,

a+b + i + l=n+2.

These operators were introduced by Gel'fand and Dikii [ 4 ] .

l^n+2,

Following their, work, we s h a l l

show in the next chapter t h a t the Lax equations Lt = [P,L\ , where I = d n + 2 + 2 utdl are _ «-o Hamiltonian in the s t r u c t u r e defined by the operator Bn, with a s u i t a b l e Hamiltonian depending on P. We further introduce the operator B^Maa(A[d]) by the formulas B=Bd + doB', BeM„(A), fl

(fiu={;0"a+6-i for Here Ma,{A)

(21)

for

a+b>l, a+b=0.

denotes the group of matrices with an infinite number of rows and columns down-

ward and to the right beginning with the zero (index) columns and rows. MociAld])

The elements of

are considered as left operators on the A -module Af of finitely supported

infinite columns with elements in A (finitely supported means that only a finite number of the coordinates are nonzero).

The range of such an operator lies in the A -module A°° of

all infinite columns. The scalar product PlQ is defined if at least one of the two vectors 6P

— —

P, Q

is finitely supported.

All vectors T = for P(*A are finitely supported, and there-

fore the B-= are defined but are not necessarily finitely supported. Frechet Jacobian D(P) ly large j

Each row of the

is finitely supported for any P£A°° , since DUi(Pi) = 0 for sufficient-

(depending on i ; see the definition in 7.8). Therefore, D(P) 00

to any vector of A

.

may be applied

The left and right sides of equalities (13) and (14) of Sec. 7 are

thus defined, and we take them as the definition that B

be Hamiltonian in the ring A

of

functions on the space of jets of the corresponding "infinite-dimensional" fibration (the projective limit of the finite-dimensional fibrations).

It is not hard to verify that the

remaining constructs of Sec. 7 with appropriate modifications carry over to A. The operator (21) was introduced in the work of Kupershmidt and the author [19] to study the system of equations for long waves with a free surface suggested by Benney. sequence of unknown functions

The infinite

u„, n > 0 , is, in fact, the sequence of moments of the horizon-

tal component of the velocity, and the system of Benney's evolution equations for it is found to be Hamiltonian with operator B

and a corresponding Hamiltonian.

Details may be found in

the next chapter. The following theorem is the main result of this section. 8.2.

THEOREM.

The operators

Bn and B are Hamiltonian.

The proof follows by means

of lengthy computations. Among the reasons for this is probably the fact that we do not know 34

an invariant (coordinate-free) characterization of these operators. We note that according to formula (20) there is also a formal "limit" Bx operators B n , which is a matrix operator of infinite order. Boo , by setting « n+2 =l, Uj=0 for j=n+l,.j>n+2.

from

The operator

of the

B„ is obtained

Although we are not in a position to

assign Boo a substantial meaning, in the computations to verify the Hamiltonian property it is possible to work directly with

Boo, since the identities of interest to us will

obviously be preserved under the substitution

«j = 0 for j>n, j^n+2,

These identities are obtained in the following manner. fiO and -a we consider the columns AT=(X0, Xu ...)'

u n+2 =l.

In place of the columns

and Y = (Y0, Y\, • • •)' with formal variables

as coordinates; the action of d* on them is interpreted as the conversion of Xt, Yk i

i

(

the formal variables d Xi = X i^, d'Yk=Y l>),

— 6u

into

independently of one another and of a<" . Each

element of- the left and right sides of (14) will then be a formal infinite sum of monomials in a, X, Y and their derivatives which is trilinear in u, X, Y ; we simply verify that the coefficients of each such monomial at corresponding places on the left and right coincide. (Here X and Y are not to be confused with the previous notation for fields on jets!) 8.3.

For the reformulation of the Hamiltonian criterion (14) which we shall use below,

we introduce some further notation. Let P,QdA DUj(PQ)=PDUi(Q)

+ QDu.(P)

. Since DUj is a differentiation, we have

, or in vector form D-t (PQ)=PD-t

(Q)+QD-t

(P), where D-,=(DB., D a „ ...).

Now let P, Q be columns. From the last equality it then follows immediately that D-,(P'Q) = P~'D(Q)+~QtD(P) .

Finally, let C

be a matrix and Z

a column.

Then CZ is a column, and

D(CZ)=CD(Z)+Z'D(C),

(22)

where the expression ZlD(
which contains at the

infinite column containing at the finite matrix with

t - t h row 2

th element the operator

d.

Gi={i-th place C)', D(C) i s an

i - t h place the matrix **

D{Ct), and Z'D(C)

i s an in-

( k-tb. row of -D(C,))=2 Z„D-t(Clk) and hence with

ij

d , the identity

d(Pdx) =^iDU[(P)dui/\dx

implies that DUl and

From (22) we therefore find that D(CdlZ) = CD(d'Z) + (dlZyD(C)=:Cdl°D(Z)+Zw'D(C). D (
where

ZW = d'Z and the circle following

example, the notation

-

^ ZkDUj(Cik) . *>o

Since d commutes with D commute with

z

f-th place

C is a transfinite object —

dl on the right is inserted in order that, for

dl°CD(Z) not be interpreted as (d'C)D(Z).

In this notation the criterion (14) after obvious calculations may be rewritten in the following manner.

35

8.4. (—lydloBj)

The Hamiltonian property of the skew-symmetric operator -^{Bfl1 —

Proposition.

is equivalent to the following identities between the infinite columns:

B (y,X'd-i/YU)-X'U)dlLy)

= (y XU)td-p--(-\)ldioXt%\BY

-(M same with X~Y).

(23)

Here the r i g h t and l e f t sides of (23) correspond to the l e f t and r i g h t sides of ( 1 4 ) , dBjldue i s the column of i n -

respectively, and the following additional notation i s used: f i n i t e matrices containing a t the i - t h place the matrix _ , _ the matrix with (t*)-element 2^ Xt ^ (Bj,u); B'j={B))~. 8.5.

4= ( i - t h row of B)) ; XU)t-=L da du'

is

We begin with the verification that the operator (21) is Hamiltonian. The

identity (23) to be verified assumes the form {X'—dX, etc.) (Bd+d°B')(X'¥Y'-X''dJ?v\=(xt'

§t+doX'

<24)

~)(Bd+d°B<)Y-(iiie*mevirhX~Y).

We expand here all the brackets. On both sides there are sums of monomials each of which contains A""' and YW (possibly transposed).

The ordered pair

ij we call the type of the

corresponding monomial. By collecting all terms of the same type, we bring (24) to the form BrX<M-Y'+(B+B')X'J^Y"-Bt'Xt'^Y+(B+B^Xi'-^Y,-{B+B<)Xi'i¥!rY'du

du

du

du

-(B+B')Xr^-Y=X'^rBrY+X'^r(B'+2Bl')Y' + Xr

+ X'^-(B+B')Y" du'

du

da'

+BI)

B''Y+Xr

dlB

du'

du'

d B

( ±B') (B + BWdu'

du

(25) + (the same with X~Y).

We shall prove that this is an identity by showing that the terms of each type on the left and right cancel. All calculations making use of the concrete form of the matrix B we reduce to the following lemma. 8.6. LEMMA. For any i>0, and any X and Y the following assertions hold: a) The expression X'-^-Bi(i)Y

is symmetric in X and Y .

du*

b)

(B+B')WX'-^Y=X'^r(B+B'Yl,Y.

c)

W

da

da' Ht)

B X'-¥LY=Y'£rB X. du

da'

d) The expression X'~(B+BlYl)

Y is symmetric in X and Y .

du*

Proof, a) and d ) . •S-(Btab)'=^iXl>bh,a+b-\

The matrix X'-^z-

du'

has at the ad -th place the element

, where 8 is the Kronecker symbol. Further, (B'^)dc = cu^lc_1 ADt

fore, the a-th element of the column

X'-^=r-Y

is equal to

ou*

2

b,c>0 d=a+b-l

36

bC *»*"»iVl K «=2 ^U+c-2X>Yc b,c>0

2 Xb

£5,

. There-

It is obvious that this expression is symmetric in X , Xl-^=-

Similarly, the matrix

Y.

has at the ad -th place 2 Xb-g— (Bfl(,)=2 Xba&d,a-y>-i ,

°~a

d

b>0

0 a

while the matrix (B+B')' t the dc-th. s i t e has the element element of the column X'-^-iB+B'Y^Y is du' »,cX>

6X>

(c + d) a ' ^ - i »

so

that

the

fl

~th

»,c>0

and this expression is symmetric in X, Y. b) The element at the d -th place of the column X'—=-Y Further, (B+B')$=(a+ d )"£V-r is equal to 2

b

is X'-*—Y

= ^, Xbbhd,b+c-iYc

.

Therefore, the a-th element of the column (B+£<) ( l ) X'-^LY

(«+ * + c ~ ! ) " i ' V * - ^ ' '

»,c>0

In analogy with the computation at the start of the proof, the a-th element of the column X'^-(B + B')WY i s *£-K da' 2

Xbb(c+d)tt^d_xYe=

»,cX> d—a+b-l

2b(a+b

+ c-l)u«J>b+c_2XbYc.

»,cX>

c) As above, the a - t h element of the column BWX'^M-Y

is

S abUaU+c-2XbYe . t>,c>0

On the other hand, the matrix y^ald,a+c-iYc 2

, and (B*)lJl = bui%d-i •

Kjafrtta+j+e-z^Tj .

Y'-^=j has a t the Therefore, the

ad - t h place the element a-th element of Y*~BmX

^.Ycr~{Bacy= i s equal to

The proof of the lemma i s complete.

b,c>0

8.7. Verification of the Identity (25). Type 00. On the left there are no such terms, while on the right there is X'-^-B'Y du' Type 01. We must verify that

— (the same with X**Y

B'X'6AY-^X'-^riB' du du1

+ 2BnY'-Y1'

d

) , i.e., zero by Lemma 8.6 a ) .

< g + B '> out

B'X.

On the r i g h t there i s the monomial X'—zr-B'Y' and a similar term with opposite sign and du' X ++K\ which cancel by Lemma 8.6a). The remaining terms we transform in accordance with Lemma 8.6 b) and c ) : X*2j!r(B,+Bt')Y'-(B,+Bl')X'-£-r\ du' du , i -Y' -^-B X=-B'X'M-Y'. du' du Their sum is obviously equal to the monomial on the left side. Type 02. We have the equality at once and it follows from Lemma 8.6 b) .

37

Type 10. We must verify that d

(B±~B">

-B'X'^Y^X'' du

By Lemma 8.6 a) the term X

BtY-Y'^-(B'+2Bt')X'.

du'

du'

y^-5 Y on the right cancels with the term symmetric to it. We

transform the remaining terms according to Lemma 8.6 c) and b ) :

X' '2i-BtY=B'Y' du'

-?2-X', du

-Y'-^rdu' (B'+B')=-(B'+B')Yt^-X'. du The sum of the right sides is equal to — B'Y'-Jl-X'. that Y'-j=-X'=X

The proof is completed by the remark

-jF^Yi which is immediately evident from the definition of these expressions.

Type 11. We must verify that

(B+B^X'-^-Y'-iB+B^X'-^Y'^X'^^-iB+B^Y'-Y'-

*&+»"> (B+B')X>.

By transforming the first term on the left according to Lemma 8.6 b) and cancelling with the corresponding term on the right, we obtain the equivalent identity:

-(B+B')X''d^LY'=Xt' du

*£-(B+B<)Y' -Yr du*

On the left we interchange X and Y, replacing B'

by

*@±]*l(BJr.B<)X\ du' * B,

again apply Lemma 8.6 b ) , and

cancel by Lemma 8.6 d ) : l 0=X''-¥L(B+B<)Y'-Y du'

'du' J!L(B+B<)X:

Type 20. We must verify that

-(B+Bf)X''^Y^ du On the left we make the change X"**Y

-Y' du' — iB+B^X".

and B'^-B ; applying Lemma 8.6 b ) , we obtain the re-

quired result. This completes the proof that the operator (21) is Hamiltonian. We now proceed to verify that the operator (20) is Hamiltonian. put B=Bca = llmBn

(formal limit), B,=llmB„,j

It

;X

In criterion (23) we

and Y are infinite columns.

n

The next lemma is verified by straightforward computations using (20), and we limit ourselves to formulating it.

38

8.8.

LEMMA,

a) I n t h e

a - t h element of t h e column ^.BXt^L-YW )

of a monomial of t h e form

only t h e

coefficient

8)

X

to + p + 6-/)! S r(a L aipi(6—/)!

(« + /)! (a + p + 6 - / ) ! (? + /)'1 at/! ~~ a! pi (6—/)! y!/I J*

Here and below a l l i n d i c e s a r e i n t e g e r s c o n s i d e r e d e q u a l to zero for y > 8 b) I n t h e

> 0 ; terms w i t h

(respectively,

-

(respectively,

-^=-K only t h e c o e f f i c i e n t of a monomial

)

of t h e form A tf>K
1VH.A+<+.yrr

K

iw

^LV

'

c) In t h e

(a + P + V + 6+Ti + l)!

a - t h element of t h e column

XWF<, 8) aW ry+ p +v+fl+n+2

(g + /)|

M « + T + /+1)1 PI («-/)«»ll

can be n o n z e r o .

For

, , l

«!/!

\^XW TJ = 0

(f5 —/)l ) a r e

7>P ).

^^BX(^J)

O-th element of t h e column

(8—J)[

(26)

(cc + p + v + 6 + ti + l)!

(? + /)! 1

, U / ;

M a + T + /+l)!8!(p-/)!tl! " W T J -

-rJj-BY

only t h e c o e f f i c i e n t of

this coefficient

(q+p)!r(a + ct + p + 6+l)! , a\f>\ L(B + a + P+l)!6! " ^

i s equal to

i s e q u a l to

n f t (T

+ 6)'l V! 6! J'

'

,00. ^Z8'

For r(^=0 this coefficient is equal to /

n„4*(T

v

'

+ 6 + r))!(q + P)! V!8!T)!

a!p! '

(29)

d) In the a-th element of the column ZJ(~iy+ldJ°X<-?J-BY X4e)^(6)Ba+a+B+V+6+T.+2' c a n V,

^6V

1Vu.fi+/+„+i

'

De

only the coefficient of ou'

•*••

nonzero, and this coefficient is equal to

(g + P + a-Z + iDICa + a + P + e + tn-l)! (a + a + p + 6 - / + r i + l ) ! a ! p ! ( 6 - / ) ! t ) ! / !

y J ^ ^

1tf+

'

^ ( « + P +fe+ Ql(V + 6 + Tl-fe-0l a! pi Y! ( 6 - 0 ! ( l - * ) l *l "

*

,^n.

K

* '

We remark t h a t p a r t s a) and b) of t h i s lemma r e f e r to t h e e n t i r e l e f t s i d e of ( 2 3 ) , w h i l e p a r t s c) and d) r e f e r only to h a l f of t h e r i g h t s i d e , t h e remaining terms b e i n g cons i d e r e d by permuting

X++Y , i . e . , the p a i r of i n d i c e s (a, B) **~(y, 6 ) .

We now proceed to d e s c r i b e t h e c a n c e l l a t i o n s of s i m i l a r t e r m s . 8.9.

LEMMA. + P + 6-/)! 2. (qa!p!(6—/)! 6

Proof.

A f t e r d i v i s i o n by

(a

g^

)!

(« + /)! _(a+P)! (a + a + P + 6 + 1)! etl/1 — a!?! (« + a + p + l)!6! *

t h e sum on t h e l e f t i s t h e c o e f f i c i e n t of xa^YaTb

in

t h e polynomial 2 {X + 7 T + P + 1 W {Y + T)a+t

=(X + T)a+*(K+Tf

(X + T)^-(Y

+ Tf+^

j<6

39

Separating from the two first factors the coefficient of XaJ^-rYa-s, s

l

and (Y + T?+l

the coefficient of X ' Y T i (after the binary expansion of (X + Tf+ by

X — Y) , w e obtain

{l+T^{\+TY+*(l

8.10.

2

and division

( t )l ,)'?)» which is the coefficient of T6 in the polynomial

+ Tr , i.e., just COROLLARY.

and from the third

{

i ; « ^ +l .

All terms of the form (26) cancel with the first term of (28) and its

transform under the substitution (a, P)*-*-(T. 8) . 8.11.

LEMMA.

2/

iw

(« + T + 6 + l)I

(a + /)l

.

1v>(Y +

8)!_

Proof. W e make use of the identity (q + y + 6+1)! (6-/)!(a + T + /+I)!

(tt + y + 6)! . (g+ Y+ 6)! (6-/-l)!(a + v + /+I)! "•"(6-/)! (a + y + /)! '

8.12.

COROLLARY.

The expression (27) is equal to

8.13.

COROLLARY.

The first sum in (30) is equal to fv _ 1V)+n+i (« + P + H)» (g + 6)l ' aipit)! «16t *

8.14.

COROLLARY.

The second sum in (30) is equal to , _ itf+j-H,(« + P + y + 6 + T| + l)l(« + p)l ' aipiaiT)l(« + p + 7+l)!

1

Proof. We first sum o n / for fixed k with the help of Lemma 8.9 (with different values of the parameters) and then sum on k using the same lemma. 8.15.

According to 8.12 and 8.14, the terms of (27) cancel in the second sum of (30)

and its transform under the substitution (a, B)-"-^, 6). The remaining term of (28) and the coefficient of (29) cancel with the transform of the first sum in (30) under the substitution (a, &)*-(y,

6) according to 8.13.

Finally, the last remaining terms cancel by symmetry also

by 8.13. CHAPTER II THE STRUCTURE OF THE BASIC EQUATIONS 1.

Introduction 1.1.

40

The principal purpose of this chapter is to describe the construction, algebraic

s t r u c t u r e , and conservation laws of some classes of nonlinear p a r t i a l d i f f e r e n t i a l equations and also ordinary d i f f e r e n t i a l equations r e l a t e d to them.

The introduction i s devoted to a

brief description of these c l a s s e s . 1.2. istic

Lax Equations.

Let

33 be some a s s o c i a t i v e algebra over a f i e l d of character-

k, which l i e s in the center of 53; l e t dr53->53 be a d i f f e r e n t i a t i o n of $ which i s

t r i v i a l on k .

The algebra of d i f f e r e n t i a l operators 53 [d] consists of a l l expressions of

the form 2 btdl, b0H , with the commutation r u l e d°b =db-irbd

.

If L=2Jbldl,

(—0

number N

bN=j=0 , the

1-0

i s called the order of the operator L and i s denoted by

known that the l e f t

S3-module of operators of order
ordi .

I t i s well

i s freely generated by l,d,...,d

•

Basic examples: a) variable

53 i s the ring (of germs) of smooth, analytic or meromorphic functions of the x, of the v a r i a b l e s

x, t , or of the variables

b) 53 = /fe[«<»|i=0,...,/t; j>0\;

d:u\'^a\'+^,

x, y, t; A=R or C; d = d/dx.

* = Q , R or C.

c) S = AJZ(5B0) i s the ring of (ZXO matrices over a ring 530 of the type described in a) or b ) ; d i s the unique extension of d from 530 to iW, (33Q) . We suppose further t h a t in $ there are defined two additional d i f f e r e n t i a t i o n s d2

such t h a t

d, dt, d2 are pairwise commutative.

b)

du d2 a r e some d i f f e r e n t i a t i o n s commuting with

t i o n f i e l d s of Chap. I .

dt and

In example a) di = d/dt, d2 = d/dy ; in example d , i . e . , formal analogues of the evolu-

We s h a l l usually write dt, dy in place of

For any L = ^btdl&^[d\ we s e t
d lt d2 in case b) as well. d2 .

The symbol

[P, L]

A pair of operators P, L&-3) [d] i s called a solution of the s t a t i o n a r y Lax equation (respectively, the Lax equation, the equation of Zakharov—Shabat) if [P,L]=0 diP=[P, L], diP+d2L=[P,

L]).

(respectively,

For b r e v i t y , we s h a l l often refer to a l l these equations as

Lax equations. We remark that in case b) (and similarly in the matrix case) the "unknowns" in the Lax equations, in addition to

P, L ± include the fields dlt d2, and solutions of these equations

in such rings correspond to what in algebraic geometry are called "generic points" of the algebraic manifolds represented by systems of algebraic equations.

From the more t r a d i t i o n a l

point of view, when working with these r i n g s , we shall investigate the s t r u c t u r e of the Lax equations themselves rather than t h e i r solutions in the sense of t r a d i t i o n a l a n a l y s i s .

The

l a t t e r case corresponds to rings of type a) and i s treated in Chap. I I I . The main r e s u l t s pertaining to Lax equations are proved in Sees. 2-5. These include the following: an e x p l i c i t description of operators P with the property ord[P, LJ^ord L—1 or ordi—2 ; establishing that the equations Lt=[P, L] are Hamiltonian over commutative rings of type b ) ; formalization of the method of Zakharov—Shabat. We make use of the

41

technique of fractional powers of Gel'fand and Dikii in a considerably simplified and formalized version. 1.3.

Another version based on the study of the resolvent may be found in [ 5 ] .

The Benney Equations.

This name we give to the following system of equations

for the two-dimensional motion of a nonviscous, incompressible fluid in a g r a v i t a t i o n a l field in the long-wave approximation: u(+uux—Uy^ux\

ht+\\udy\

d^ + hx=0,

=0.

Here — O O < J C < O O i s the horizontal coordinate;

the time;

u=u(x,

0<#

i s the v e r t i c a l coordinate;

t

is

//, t) is the horizontal component of the velocity a t the point (x, y)

at

time / ; h(x, t) i s the height of the free surface above the point notation

tit i s an abbreviation for

-gr u(x, y, t) , e t c .

(x, 0) a t time

The

The system of units is chosen so

that the g r a v i t a t i o n a l acceleration and the density are equal to one. \Ujcdfi a r i s e from the equation of continuity ux-\-vy=0

t.

Integrals of the type

where v i s the v e r t i c a l component

of the velocity and from the boundary condition v—0 a t y=0 ; these r e l a t i o n s make i t possible to eliminate v, by expressing i t in terms of «; •v(x,y,t) =—\uxdn o

•

For the r e -

maining details of the derivation see the work of Benney [27] who first discovered an unexpected property of the system: the existence of an infinite sequence of conservation laws for it. The Benney equations display a number of unusual properties. "We do now know of a Lax pair for them; the conservation laws in the interpretation of the present work (in contrast to the formal derivation of Benney) are obtained from a nonlinear integral equation with a parameter.

This equation enables us to obtain the conservation laws of Miura [46] for the

Benney system. We further establish the Hamiltonian character of the "reduced system" (with the additional condition

« y =0 ) and the commutativity of the reduced integrals. These

results are then generalized to the full system. Our exposition is based on the work of Kupershmidt and the author [19]. The main results pertaining to the Benney equations are formulated in more detail in Sec. 6; Sees. 7-13 are devoted to their proofs. 2.

The Commutator and Fractional Powers of Differential Operators *

2.1.

In t h i s section we begin the study of the Lax equations. If dtL=Lt = \P, L\ or

[P, £ ] = 0 , then in any case ord [P, L]
We therefore f i r s t investigate the conditions

under which the commutator of two d i f f e r e n t i a l operators has lower order. ff

We s e t

L=^undn, n=0

42

M

P= 2 m—0

v

mdm-

2.2.

LEMMA. Setting drtw = wia)

for any w(*S3, a > 0 , we have

i*4- 2

\[«)^)-(l)w%)}dm+a-a-

a,m,n>0L^

'

\ i

I

J

Proof. According to Leibnitz's rule d'(«zi)=2 (! a=0 *

\uwvu~a). '

2 . 3 . COROLLARY. The coefficient of dM+ff in [P, L] i s equal to [vM, uN] . ord[P, £ ] < M+N — 1, if «^6 i s contained in t h e center of 33 . 2.4.

[v, L] = [v, uN] d" +(Vu'N —NuNv)d"~l+

COROLLARY. For v£3d we have

order
In p a r t i c u l a r ,

(terms of

In p a r t i c u l a r , if the center of 33 i s infinite-dimensional over

k, then

the l i n e a r space of those P, for which ord [P, L\
COROLLARY.

on those coefficients

The coefficient of
[P, L] depends only

Vj of the operator P and t h e i r derivatives for which

j>k . The

terms depending on ?>*. Vk+i and t h e i r derivatives in t h i s coefficient have the form [**. «Ar]+ ['*+!• uN-i] + Mv^xUN-NaNv'k+v Below we shall always assume that duN—0).

(1)

ttN i s i n v e r t i b l e in 33 and i s a d-constant

(i.e.,

The operator L i s fixed, while P may vary.

2.6.

THEOREM.

If a v , a^_, l i e in the center of 33 and d i s the dimension over k of

the space of ^-constants i n 33 , then the dimension of the space of operator P&B\d\, for which

ordfP, L\
does not exceed

(M + \)d .

I t i s equal to {M + l)d, if d:33-+33 i s a

s u r j e c t i v e mapping ( i t i s possible to " i n t e g r a t e on x" in the ring 33 ) . Proof.

For k = M, M— 1, .. .,0, —1 we c a l l the "k-th equation" ( r e l a t i v e to N+k

condition t h a t the coefficient of d Af-th

in [P, L] vanish.

equation i s t r i v i a l l y s a t i s f i e d .

v;) the

I t i s evident from (1) that the

Further, for fixed vM,...,vk+2

the k-th equation,

according to ( 1 ) , has the form v'k+l = ii> , where w i s a polynomial in u^1, «*.a), vf), M>l>k

.

If i t i s solvable a t a l l , then any solution i s obtained from one solution by addition of any d -constant

in 33 .

I t i s c l e a r l y solvable if d:33-+33 i s s u r j e c t i v e .

This completes the

proof. 2.7.

COROLLARY.

If the d -constants in 33 coincide with

k ( e . g . , 33 i s the ring

;)

of germs of functions of x or 53 = fe[?ij ] ) , then the dimension of the space described in Theorem 2.6 does not exceed 7W+1 .

( I t w i l l be shown below that in the commutative case

i t i s equal to M-\-\, even if d:33-*33 i s not s u r j e c t i v e . ) For matrix rings 33 i t i s possible to obtain an analogous estimate under l e s s r e s t r i c t i v e assumptions which we now axiomatize. 2.8.

Let adu^:33-^33 be the operator

&&UN(p) = [u,N, b\ .

We s e t 33+=Ker adttAf, 33~—lm ad«j\r

and assume that the following conditions a r e s a t i s f i e d : 43

a) $ = $ + @ 3 r

(as a

b) The o p e r a t o r

^ + S _ c ^ " , S"S+CS".

k -space);

adUff-33~-*3B~ i s b i j e c t i v e .

c)
If for

WG33 we d e n o t e by

w + , w~ t h e components of •© i n

UN t h e c o n d i t i o n s a ) - c ) a r e s a t i s f i e d then we c a l l

commutative r i n g

SB every element i s s e m i s i m p l e :

+

$ = $ J , 98~={0} .

S + and 53", r e s p e c -

UN s e m i s i m p l e .

In a

i n t h e g e n e r a l c a s e our

terminology i s m o t i v a t e d by t h e f o l l o w i n g example. 2.9.

Example.

nonzero elements sites

Let

cfck

S3Q be commutative,

on t h e d i a g o n a l .

for which c^Cj,

and

$B=Ml(3B0),

Then

and

UN be a d i a g o n a l m a t r i x w i t h

SB* c o n s i s t s of m a t r i c e s w i t h zeros a t t h o s e

SBT of m a t r i c e s a t t h e s i t e s

ij,

for which

easy to v e r i f y t h a t a l l c o n d i t i o n s 2.8 a ) , b ) , and c) a r e s a t i s f i e d . semisimple m a t r i x of

Mt(k)

ct = Cj.

It

is

This i m p l i e s t h a t any

i s a l s o semisimple i n our s e n s e of t h e word.

As i s known, t h e

converse i s also t r u e . I n c o n d i t i o n s 2.8 we d e n o t e by d t h e dimension of

KetdnSB*

over k .

In Example 2.9

i t i s equal to t h e sum of t h e s q u a r e s of t h e m u l t i p l i c i t i e s of t h e e i g e n v a l u e s of 2.10.

THEOREM.

If

UN

i s i n v e r t i b l e , c o n s t a n t , and semisimple and

dimension of t h e space of o p e r a t o r s i s p r e c i s e l y equal to d(M+\) Proof.

P

with +

if

+

d:33 -+33

otd[P, L\<.N—2 is

d(MJr\)

.

It

The proof i s analogous t o t h e proof of Theorem 2 . 6 , b u t i t i s somewhat more

We do i n d u c t i o n k downwards assuming t h a t v~, w i t h i n d i c e s g r e a t e r t h a n o r equal to

&-fl.

vk+i

as w e l l a s

vk:

= w.

vk+2, .. .,VM a r e a l r e a d y defined from e q u a t i o n s Then t h e c o n d i t i o n for s o l v a b i l i t y of t h e

c o n s i s t s i n (by 2 . 8 b ) ) UN-I]+-(NUNV"II+1)+

[vM,

If i t i s s a t i s f i e d , Uff-iB^T

, then the

surjective.

\vk, uN\ + K + 1 , uN-i] - NuNv'k+l

vk

UN-X£3B~

does n o t exceed

complicated due to t h e f a c t t h a t t h e k - t h e q u a t i o n now c o n t a i n s

k for

uN .

then

. Therefore,

hypothesis.

Hence

v^

[vk+u v£+l

i t necessarily e x i s t s if

= W+.

i s uniquely determined.

UN-I]* = [V£+V

UN-I]+

and

v~+l

On t h e o t h e r hand,

is surjective.

Ujv-i]+ = 0, s i n c e

i s a l r e a d y d e f i n e d by t h e i n d u c t i o n

i s determined up to an element of d:ffi^ffl

[v£+v

K e r d f | 5 3 \ i f i t e x i s t s a t a l l , and

This completes t h e i n d u c t i o n s t e p and

the proof. 2.11.

Using t h e method of G e l ' f a n d and D i k i i , below we c o n s t r u c t e x p l i c i t l y for each

o r d e r an o p e r a t o r

P w i t h t h e c o n d i t i o n ord[P, L]
p r o p r i a t e f r a c t i o n a l power of We d e n o t e by with

3B) .

not of r i n g s .

44

L .

a s t h e d i f f e r e n t i a l p a r t of an a p -

To t h i s end we i n t r o d u c e t h e formal r i n g of symbols.

£ t h e f r e e v a r i a b l e and c o n s i d e r t h e polynomial r i n g

3B[Z] (;

The mapping L*+-L:£ bid'>-*£ bfi- induces an isomorphism of l e f t The t r a n s f e r a l of m u l t i p l i c a t i o n of o p e r a t o r s t o

commutes

3B -modules b u t

33[l] we s h a l l c a l l composi-

t i o n and denote i t by °: PL—P°L

.

In order to describe composition in

the inner s t r u c t u r e s , we introduce two d i f f e r e n t i a t i o n s in dl'^i toft t-*£ibfil~l 2.12.

.

53 [S] in terms of dbfi1

53[S]: d:2b^'-*^

and

The next lemma follows from the Leibniz formula.

LEMMA. PL = 2 •% d\PdaL. 0L>0

The idea i s now to use the well-known construct of extensions of commutative rings and extensions of d i f f e r e n t i a t i o n s to them in order to learn how to extend composition by the formula of Lemma 2.12 and thus to construct useful ring extensions for d i f f e r e n t i a l operators. ( * = ®((f1)) = J 2

For our purposes a single extension suffices:

*iE'|*ie«| (the ring of

U—oo

formal Laurent s e r i e s ) . formulas. . We s e t

Obviously d and

j

dj extends to 53((H"1)) by continuity with the same

o r d ( 2 ^ ' ) = max{/fe|&fc=£0} and further |j)) (we assume that

ordO= — oo) . The norm [| || i s non-Archimedean: ||a+6||<max(||a||, || 6||) . Moreover, ||afc||< IIaIIII* II. Ilflll = 1 for ae53\{0} and | | a | | = 0 if and only if a=0. The algebra ^((r 1 )) i s complete in t h i s norm. For any additive operator F:®-+@ we s e t ||Z71| = sup|J-Fa||/|i«II-J<1

.

Obviously, ||
33((E'1)) the composition

In analogy with Lemma 2.12, we introduce in

«•»•=!! i <£«*•*. aX>

o, by s e t t i n g (2)

2.13. LEMMA. Series (2) converges in norm for any a, 6e53((S-1)) and defines on 53((S-1)) the s t r u c t u r e of an a s s o c i a t e k-algebra. Proof. The convergence of (2) follows from N[j-d£a-0 as a-*oo . B i l i n e a r i t y in a and b i s obvious. The i d e n t i t y i s 1653 . Associativity i s verified as follows: {a°b)oc = \2±dZad°b}c= L<x>0

A>0

J

Lflx) "'

2 a.p,v<&

J

ZW-^tPM^bdtc, F

* "'

A,B,T
" '

I t i s not hard to see that the s u b s t i t u t i o n o = r, $ = A + B — T, f = ^ _ r defines a b i s e c t i o n of the terms of both s e r i e s and corresponding terms coincide. The proof of the lemma i s complete. We now suppose that M i s a l e f t 53-module to which the action d: d(bm) = db-m+ bdm has been extended for £653, m£M . We denote by M (<£-*)) the l e f t 53((S-1)) -module {^mfil\ m&M, 3i0 Vi > i0, nii = o} with the obvious action. We extend the action of d to M^V1)) coe f f i c i e n t - w i s e . I t i s not hard to see that formula (2) enables us to define on M((i~1)) a new s t r u c t u r e of a l e f t 53((?-1)) -module r e l a t i v e to composition if we take a653((t"1)), bqM((£-1)). 45

Right composition M((S-1))X33((;"'))-+M((<--% i s defined similarly if with action

d .

M is a right

33-module

-1

F i n a l l y , if

M i s a 33 -bimodule, then M((' )) with l e f t and r i g h t composi-

-1

tion becomes a 53((? ))-bimodule.

The only nonobvious a s s e r t i o n s here concern a s s o c i a t i v i t y ,

and i t s v e r i f i c a t i o n in the proof of Lemma 2.13 c a r r i e s over automatically in a l l the required cases.

We s h a l l need t h i s construction in Sec. 3.

An i n t e r p r e t a t i o n of the elements of the ring

33 ((S-1)) with negative powers of \

as

symbols of i n t e g r o d i f f e r e n t i a l operators i s given in Sec. 5. 2.14.

^ bfil = b i s i n v e r t i b l e in

LEMMA.. An element

m u l t i p l i c a t i o n and composition if and only if Proof.

b,03

33((I'1)) both with respect to

i s i n v e r t i b l e in 33, where ft=ord ( 2 btV).

I n v e r t i b i l i t y with respect to multiplication i s well known.

Further,

CO

(^b,ll)ob^lVn

= l + c, o r d c < —1, as i s evident from ( 2 ) .

^(—i)lc°l

Therefore,

e x i s t s in

oo

®((S-1))i

and i s inverse to \-\-c

c°' = Co...oc, i

times).

in the sense of composition:

ft„^',=(l+c)0(~1,

Then

inverse i s established s i m i l a r l y . term of X must be equal to

on

i s inverse to 2 * ' "

F i n a l l y , if

b„ $"*, so that

b°x=\

tne

right.

write

A left

or x°b = \, then by (2) the leading

bn i s i n v e r t i b l e .

We proceed to the extraction of r o o t s . xfi

(1 + c ) ° ' 2 ( ~ " " ) ' c ° ' ~ * ^ we

If the element c = ^ t t « S

N

and W = UN .

i>0

s e r t i o n , we introduce the following notation.

is

N-th

an

power of

In order to e s t a b l i s h the converse a s -

We c a l l an element

W03 Af-admissible (A r >0) ,

S3*+-33: X*+2J wlxwN~l~l i s a b i j e c t i o n . In a commutative ring 33 with unique <-o division by N precisely the i n v e r t i b l e elements are admissible. In a matrix ring the if the mapping

N-l

element

o> = diag(Ci, . . . , c,) i s admissible if and only if a l l elements

clrc^~l~l (r, s = l , . . . , /)

^ (-0

are different from zero. 2.15. LEMMA. Let

c = '^lu„ln, N>0.

Then for each

Af-admissible root of an

power w of UN there exists a unique element X£33((V~1)), for which

X

=d

N-th

and X = w\-\-

0(1) (O(S') denotes some series of order <£)• Proof. We apply the method of successive approximations. We set X_i = w\. N

N l

Xl.x=uN\ -\-0(\ ~ ) 1

Obviously,

. Suppose that for some r> — 1 we have already proven the existence of X?*=c+0(\N~r-2)

and its uniqueness up toO(r(,+1)) . We

X,e33((Z- )),

such that Xr = wl+0(\),

seek Xr+l

in the form X„x = Xr + xr+iZ~lr+1) . Using the distributivity of composition, we have Af-l

(X, + x r + 1 r ( r + I >)°" = X°N + 2 K'°x'r+t~(,+1)=Jc;<^-'-"

+ (remainder).

1=0 N.-l

We compute the terms on the r i g h t up to

N r 3)

0(\ ~ ~ ) .

The sum has the form 2

wx

' r+i'

w"-l-^N~r~'2 + 0(lN~r~3) . The remainder consists of a sum of products of j— 1. Hence, from the admissibility y
46

of w it follows that xr+l 0(lN~r~")

X°r+i=c-\-

exists and is uniquely determined by the requirement

. This completes the proof. w

2.16.

L='2iundn£33[d]

We now choose an o p e r a t o r

power w of

UN ( i f i t e x i s t s ) .

X*=w\ + 0(\).

We c o n s t r u c t a r o o t X of d e g r e e S

We d e n o t e by

and an AT-admissible r o o t of an JV-th

1°

for any S=PN~1GQN

= ZAM

N of L, as i n Lemma 2 . 1 5 ;

t h e element XP£33{(\~1))

.

Let

i
Assuming that w and L are fixed, we shall often write below Vj{s;w, L)=Vj(s).

For s>0

we

set

/-o 2.17. c e n t e r of 2.18. and i s an

THEOREM.

S3 , t h e n even COROLLARY.

S^QN , we have ord[ < Ls > , L]
s>0,

ord [ < I* > ,

k .

Suppose t h a t

Then for any

— 2, o r d P < A f

UN, »JV-I l i e i n t h e c e n t e r of 38, t h a t

M>0

w

is invertible, a l l

w , t h a t t h e s e t of

t h e space of o p e r a t o r s

P£33[d]


of o r d e r <'AJ

is

invertible

with the property 0<s<MN~l.

i s f r e e l y g e n e r a t e d by t h e o p e r a t o r s < L > ,

s

ttN

d - c o n s t a n t s i n 33

s

Indeed, s i n c e t h e o r d e r of t h e o p e r a t o r sN

If a l l i e s i n t h e

L]
N - t h power of an a d m i s s i b l e element

coincides with o r d [ P , L]
For any

( L s > i s e x a c t l y sN, and i t s l e a d i n g g e n e r a t e a space of dimension

coefficient

M-\-l .

It re-

mains to u s e C o r o l l a r y 2 . 7 . For m a t r i x r i n g s , however, t h e o p e r a t o r s

{ Ls ) , i n g e n e r a l , g e n e r a t e o n l y a p a r t of t h e

space of i n t e r e s t to u s . 2.19.

Proof of Theorem 2 . 1 7 .

l a r , commute w i t h

N

L= X .

s

Z] = [f, L° — < L") ~\ .

A l l powers X from 2.17 commute p a i r w i s e and, i n p a r t i c u -

Therefore, in the ring s

But ord(L° —

ord[ < Ls > ~, Z]<jV — 1 and i s even

s

(L )

33((l~1)) w i t h composition we have

~ ) < — 1 by d e f i n i t i o n of < Z> > •

< A 7 —2, i f

a^ l i e s i n t h e c e n t e r of

Therefore, 33 (use i s made of

-1

C o r o l l a r y 2 . 3 which remains v a l i d for composition i n S((S )). by formula 3.

(2)).

The H a m i l t o n i a n P r o p e r t y for t h e N o n s t a t i o n a r y Lax Equations and Their 3.1.

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

33 = k\uj%

where

[<£*)",

i = 0 , . . . , J V — 2 (N>2),

Integrals j>0;

d:a<»-*a<>+»,

AT-2

are algebraically independent variables.

u l

Let further L=tf*-\-2J

$-

We

choose

w=\

<-o and

for

any

SGQAT

we

set,

as

in

2.16,

Z ° * = 2 Vj(s)V,

Vj(s)e33.

MAT*

47

According t o C o r o l l a r y 2 . 1 8 , a l l s o l u t i o n s of t h e Lax e q u a t i o n s

have t h e f o l l o w i n g form

P = '^lcl (L ')

f e r e n t i a t i o n ( i . e . , i t commutes w i t h

, cfck, sfcQN, Si>0\

diL==[P, L] i n t h e r i n g

33

5 1 = 9 ] , p : ^ - > ^ i s an e v o l u t i o n d i f -

d and i s t r i v i a l on k ) u n i q u e l y d e f i n e d by the con-

ditions , i\, £ = 0

N—2.

We shall here prove the following basic theorem. 3.2.

THEOREM,

l^tCjiL'1)

a) The e v o l u t i o n d i f f e r e n t i a t i o n

> £J» i s Hamiltonian w i t h o p e r a t o r

£JV_2

du

d e f i n e d by t h e c o n d i t i o n

dxL =

( c f . Sec. 8 , Chap. I ) and Hamiltonian

Jgc) »-*<*+'> Sj+l b) A l l Hamiltonians

V-i(r), r( 0 , commute p a i r w i s e i n t h i s Hamiltonian s t r u c t u r e

and a r e t h e r e f o r e c o n s e r v a t i o n laws f o r any of t h e Lax e q u a t i o n s d e s c r i b e d . We b e g i n w i t h a number of a u x i l i a r y a s s e r t i o n s . 3.3.

LEMMA.

Proof. (2)

Let

a, b£33((£-1)) .

We s e t

res ( ^ btl') = b_i.

Then res(aob — boa)£d93 .

I t suffices to verify t h i s for

a=Vim,

b = u\n .

I t f o l l o w s e a s i l y from formula

that (1B+B+1)

if

m+«-f-l>0

cases.

and e i t h e r

Let us suppose t h a t

/ra>0,

or

m>0, /i<0;

[m + n + lj Therefore,

rt>0,

but

mn<0

, and res[a, &]=0

i n t h e remaining

t h e second a l t e r n a t i v e i s t r e a t e d a n a l o g o u s l y . Then

(m + n + 1)!

res[a, b] i s p r o p o r t i o n a l t o valm+n+1)—(

l

'

\m + n + \}'

— l)m*a*1tt'Oi'a+'t+1> and i s a t o t a l

derivative

by a lemma of Chap. I . 3.4.

I n o r d e r to v e r i f y t h a t t h e Lax e q u a t i o n s a r e H a m i l t o n i a n , we a d a p t t h e r e s u l t s

of Chap. I on t h e c h a r a c t e r i z a t i o n of v a r i a t i o n a l d e r i v a t i v e s To t h i s end we d e n o t e by • 2 ' ( S )

lLyUSla\h-

We

dldXi

extend

to t h e j e t s .

hard t o d e m o n s t r a t e

48

8:;S->2'(53) be t h e u n i v e r s a l

Q'(S) i s f r e e l y g e n e r a t e d over S& by t h e elements •$ t o a d i f f e r e n t i a t i o n

This i s a formal a n a l o g u e of t h e o p e r a t o r of

to our formal c a s e .

t h e u n i v e r s a l module of d i f f e r e n t i a l s of t h e r i n g

i n t h e a l g e b r a i c s e n s e of t h e word, and we l e t I t i s w e l l known t h a t

^-

Obviously,

L^^

diS1^)-*"8'^).

of Chap. I where d/dxt

ffilk

differentiation. 8tt<'>

and

8/> =

by s e t t i n g d(8«(»)=8«(M-i>. i s the canonical

8d =
u\^ .

lift

I t i s not

t h e v a l i d i t y of t h e f o l l o w i n g formal v e r s i o n of Theorem 4 . 1 of Chap. I .

3.5. LEMMA.

P&3B there exists a unique representation of SP in the form

For any

N-i

2

a ^ S 1 ^ ) ; for t h i s r e p r e s e n t a t i o n

Q M + ^>

Qlr°=6A

. l

i-0

( A l l formulas of S e c . 4 , Chap. I p e r t a i n i n g t o t h e computation of v a r i a t i o n a l t i v e s a r e to be " d i v i d e d by 3.6.

w 1

2u £ '

0)

We d e f i n e t h e

r e p l a c e d by

S3((£"'))-module

deriva-

Sw^' ) . Q1 (33)((;-1)),

con-

1

j62 (^) w i t h t h e u s u a l r u l e of m u l t i p l i c a t i o n on t h e l e f t and

The d i f f e r e n t i a t i o n s

d and

t h e second by t h e u s u a l formula. on

and a r e to have dti\

53((c_1)).

We now pass to t h e r i n g

s i s t i n g of s e r i e s right.

dxi», (tn=l),

n

d|

extend to

2 1 (53) ((s -1 )): t h e f i r s t c o e f f i c i e n t - w i s e and

This e n a b l e s us t o i n t r o d u c e a n o t h e r a c t i o n of

53((£-1))

21~(53) ((;"')) — t h e composition 1 •*•*. a! a»0

I

According to t h e remark f o l l o w i n g Lemma 2 . 1 3 , 2 1 (53)((S-1))

i s converted i n t o a l e f t

($((S -1 )). °) -

The composition mob d e f i n e d by an analogous formula c o n v e r t s Q1 (53)((£""')) i n t o a r i g h t

module.

($J((; -1 )). °) -module.

I t i n f a c t becomes a bimodule, s i n c e t h e formula of

associativity

i s v e r i f i e d i n t h e same way a s f o r t h e a s s o c i a t i v i t y of m u l t i p l i c a t i o n i n

6O(COCC) = (6O(»)DC

53 ((;-)) . We extend t h e d i f f e r e n t i a l 1

8 : $-> 2 1 (33) to

8:53 ((?->)) ->• 0 1 (33) ((S-1)) c o e f f i c i e n t - w i s e .

2 1

mapping res: Q (®)(('"'))-* . (33) p i c k s o u t t h e c o e f f i c i e n t of

The

l

\~ .

The e a s i l y v e r i f i e d c o m p a t i b i l i t y p r o p e r t i e s of t h e s t r u c t u r e s i n t r o d u c e d a r e c o l l e c t e d i n t h e f o l l o w i n g lemma. 3.7.

LEMMA,

a) The o p e r a t o r s

res, d, 8 a r e p a i r w i s e commutative ( i n t h e sense of the

commutativity of t h e c o r r e s p o n d i n g d i a g r a m s ) . b)

o(aob) = Uob + aoU

for any

a, *633((?-1)) .

The second assertion follows from the fact that 8 also commutes with d% . We now apply this formalism to the computation of —^-^ . 3.8. Proof.

LEMMA. 8©,,(s) = sres 8ZoZ° ( * _1) modImd Let

A, = I c( * f ~ ,) 1

We wish to e s t a b l i s h

s^pN-1

.

Then

for

s$QN,

s>0

.

and v_x(s) = resX° p

Z=X°"

.

that 8 res A'°" =~N ' res 8 (X°N)°X°(P

m

mod Im d

and NS res AT 0 "=p8 (A'°")oA'°(''-Ar, mod Im d. According to Lemma 3 . 7 , p-i

JV8 res X°" = iV res 8 (AT0")=JV res 2

X°l o8A^oA"'(p~'-1,

i-0

Further, X°l o8AVXo(p-/_1> = 8A'oA'°(p-1,-[8A'oA'°(p-''_1), AT0'].

It is shown by the same argument as

in Lemma 3.3 that the residue of the commutator on the right lies in

Imd.

Therefore,

49

JVSres XePmNpteshXoX'(p-l)mod

Im
On t h e o t h e r hand, pl(X°N)oXp-N=p

J

Jc'oS^oX 0 ( p - i - 1 )

and s i n c e X'°hX°XP-i-1

= lX°Xr-i - [ZXoX'C-1-",

X°l],

t h e same argument shows t h a t p res 8 (X°N)oX°(p-w=Np

res hXoX" u-» mod
This completes t h e proof of t h e lemma. 3.9.

COROLLARY.

For

s£Q„, s>0 »

and 0 < i < A T —2

we have

6w_, (s)

Proof. According to the preceding lemma,

8xu(s)=sresn2 8^')°( 2

M*" 1 ) «'))•*

^sres(2(«)^< ) (*-0^ +/ - a Ws2( l - + f /+ i)^ +y+1) (*- 1 )8" i niodIin(?, whence t h e r e q u i r e d r e s u l t follows i n view of t h e c h a r a c t e r i z a t i o n of v a r i a t i o n a l

derivatives

i n Lemma 3 . 5 . 3.10.

Proposition.

For a l l

s>0,

S6Q„ and 0 < i < A r — 2

we have

<4>

^-M(5-I)=I2(;J(-^-5^^^ Proof.

R e l a t i o n s (3) can be c o n s i d e r e d a system of e q u a t i o n s f o r

which has t r i a n g u l a r form and can t h e r e f o r e be s o l v e d by i n d u c t i o n .

y_i(s—\),...,v_N+l(s—\),

The c l o s e d formulas

(4)

a r e most e a s i l y o b t a i n e d by w r i t i n g (3) i n o p e r a t o r form

tofe&^sQ

+ doiyv^is-l),

where T is the operator for increasing the index by one: T (v^)

(5) =v(*l+1 .

Equation (4) then

becomes

*>_,._, (5-i)=4(i- and the fact that (6) is the inversion of (5) is obtained by induction on /.

50

(6)

3.11.

Proof of Theorem 3.2 a ) . It obviously suffices to consider the equation

,/.] . We rewrite it in the algebra by (L*)~—L°s

(33 ((S-1)). o), replacing L by L and (Ls >

(it is recalled that [L°s, Z]=0) . We find on setting « Ar _ 1 =0, « w = l: AT-2

r JV

*-0

La-0

P-1

This implies the two identities AT-2

J" N

ff-l

ft-0

La-0

p-l

La-0

9-JV

1 J

J

(the second follows from the fact that the order of the commutator on the right is < — 1 ) The commutator in the first identity is equal to [(! Wv^{s)-{-^)v_^S)u^]^-r.

2

(7)

a0

We sustitute in (7) the result of Proposition 3.10: B-l

We wish to represent the formula obtained in the form (cf. Sec. 8, Chap. I):

J.I

with appropriate

B.

kl

.

In order to compare (7) and ( 9 ) , we make i n (7) and (8) the change of

i n d i c e s : a — p — ~{ = k, f4-8 = y, P —1—8 = / .

We then find t h a t the sum of the f i r s t terms of

(7) leads t o the f i r s t terms of (9) i f we s e t

2(-»)6(*+/-./+1)n>^'-(/i;*)«/^ B

l.*i

=

do)

for j + k + l + \
for J + k + l + \>N

(we have used one of the identities for the binomial coefficients proved in Sec. 8, Chap. I), Similarly, the part of the second sum of (7) corresponding to given

J, k,I,

can be

written in the form (for k-\-l + j - \ - \
-£<-«v(!li)('V)*4r^-*^^M'n<-*(^£^> This agrees with (9) and (10) and completes the proof of Theorem 3.2 a ) . The fact t h a t the operator J£tBjdJ — ( — d)ioB,j i s Hamiltonian i s proved in Sec. 8 , Chap. I .

51

3.12.

Proof of Theorem 3.2 b ) . We f i x r

due to t h e system r e p l a c e d by

dx 1

dxL = [ < L > , L] .

a c t i o n duftiir ))-*®^ )),

5 and c o n s i d e r t h e e v o l u t i o n of

v_x(s),

Repeating word-for-word t h e proof of Lemma 3.8 w i t h 8

( i t i s only important t h a t 1

r and

[du d] = [du d g ] = 0 ) w i t h t h e

coefficient-wise

we find ) mod Im d.

Further, = \{Lr)

7Z]oZ°= < Lr > ~oZ°*—l° < U >~o2°<*-'> =

= < If )~oZ°*-[Lo < Lr > TZ 0 ^- 1 '! +Z°*= < U >~ = [ < U ) ~ L°*\-[Zo < 2/ >~ S i n c e t h e r e s i d u e s of t h e commutators l i e i n

Imd

L°i-%

by Lemma 3 . 3 , we f i n a l l y o b t a i n _i(s)6
According t o t h e d e f i n i t i o n s of S e c . 7, Chap. I , t h i s means t h a t

XLi(r) and *Li(s) commute.

We shall now clarify the algebraic significance of commutativity. )

let B be some Hamiltonian operator, and let Q, PG^=fc[ay ].

Let H,=(a0, .. . ,tt/v-2)>

Returning to the conventions •.

of Chap. I, we denote by XQ the evolution, field XQU=B-= 6u

3.13. Proposition. If Q, P commute in the Hamiltonian structure with operator B , then XQ takes the minimal d -closed ideal JP, generated by the components of B-= into itself.

ou I n o t h e r words, t h e XQ flow i s t a n g e n t t o the f i n i t e - d i m e n s i o n a l manifold of 6P

solutions of the system of ordinary differential equations 5 - = = 0 . 6u Proof. Since [
But XQ^ = D (B^) 6a

\

B^i,

6« / 6u

and by the Hamiltonian criterion for B the latter expression is equal to B-=(-=: B~)-\6a \6a' Ou / D[B-=i-)B— . The first term is equal to zero, since the commutativity of P, Q implies \ bu' )

bu

that ~=-B-=G\md ou'

bu

cKer-=. bu

The second term (more precisely, its components) lies in Jp ,

since the Frechet Jacobian belongs to

M„+l(53[d]).

This proposition motivates the search for solution (in function rings) of the equations « < =5-=, which remain for all time on the manifolds B ——0, where P is an integral. 6u 6P 67> Actually, instead of the equations —6KerB the equations -= = 0 — the extremals of the bu

6a

Lagrangiaa P— and the flows induced on them are usually investigated (cf. Gel'fand and Dikii [3]). Both formulations of the problem are equivalent if KerB = k^-1 and in place of the equation — = c&kN-x it is possible to consider the equation -=(P — c'u) = 0, noting that 8u

P-~c'u is in this case an integral together with merits special investigation.

ou

P .

In the general case the question

Regarding the Hamiltonian property for the induced flows see the work of Bogoyavlenskii and Novikov [1] and Gel'fand and Dikii [3]. 3.14. We shall apply these considerations to the Lax equations Li = \zJci

< Lsi ) , Z.J and

their integrals "S d,v" (r't ''^Q . By Theorem 3.2 the condition 5 - ^ = 0, implies that *• * r / + « ou 52

\"^idl{Lri

>, Z.] = 0. Thus, the search for solutions of the Lax equation lying on the "quasi-

extremals" of their integrals

|5-==0 in place of -==0|, reduces to the joint solution \ ou

of the system of equations Lt=[P,

oa

/

L], [Q, Z.]=0 . The condition

[Q, I]=0 is called the

auxiliary stationary problem (for the nonstationary problem Lt=[P, mainly devoted to the description of such solutions. 3.15.

L\).

Chapter III is

We set N = 2, L = d2-{-u . The equation

Example: The Korteweg-nde Vries Equation.

Lt = u,t-—\ ( L32 ) , L] is called the Korteweg-tle Vries equation, while the equations | 2 {L'i^Ct,

L\

axe. its higher analogues. As s^Q2

Lt =

it suffices to take half integers, since

< Ls ) =LS for integral s (this remark applies to general We observe that Corollary 3.9 assumes the form

N) . *=sv_\(s — 1), so that increasing s

V

by 1 corresponds to "variational integration." 3.16.

The Matrix Case. We shall now briefly describe the changes to be made of the

definitions and results of 3.1-3.12 in order to extend them to the matrix case. We set N

L=£iUkdk

, where t/Ar = diag(c1

ct) is a semisimple matrix of

Mt(k), which is diagonal for

simplicity and UQ,..., f/.v-i are matrices with independent variables as elements: f/i = («i,os); «jv-i,ap = 0 of

for

a, p with c a = Cp .

Having chosen an admissible root of an iV-th power W

UN , we can construct the fractional powers

further investigate equations of the form exhaust a l l Lax equations over the ring We set L^=y^lV_fi(s,W)l-^.

Z£, as in Sec. 2 for

dxL= 2

Ci W

- ( Lw ) •

r ff

*=o

L<*-o

We shall

recall that they now do not

&\d\, $ = Af,($30)> $o = *l a i%l-

A representation analogous to (7) holds as before but only

up to a commutator in the term dxUQ, which dropped out for /v-i

We

S^QN .

1 = 1:

1

AT-1

p-i i

J

L

'

The contribution from this commutator will vanish if we go over to equations for the matrix traces of the Uk in place of the

Uk themselves.

This becomes even more necessary if we wish

to carry over to the matrix case at least a part of the results regarding the Hamiltonian property. UV(m+n+i)

Indeed, the key Lemma 3.3 ceases to hold, since the matrix no

longer need be a total derivative.

W/<m+B+1)—(— l)B,+»+1.

However, this expression differs from a

m n 1

total derivative by the commutator [U, V< + + >] , and therefore after going over to traces we again obtain Tr res [a,'b] 6?33 .

In the considerations of 3.4 and thereafter in place of Ql(3S)

i t is now necessary to take the $[d]-module Af,(21 (S3o)); B^-^M^Q 1 ^))

is defined

wise, and the subsequent constructions are modified in an obvious manner. must represent

coefficient-

In Lemma 3.5 we

5P in the form 2 Qi.*$U-i,*t + du>, where Qi,c£p=g^— (coefficient-wise).

The

proof of the analogue of Lemma 3.8 leads to the result Tr8V_1(s, W) = STT res8Z 0 L OU_1) mod IIIK?.

53

Analogues of Corollary 3.9 and Proposition 3.10 now connect =

TrV^is, W)

with Tr V,(s— 1, W).

ou

l,afl

'

Substitution of expressions analogous to (4) into the equations for Tr
then (under weak additional conditions) L and P are

connected by a polynomial relation with constant coefficients. This reduces the problem of solving the stationary Lax equations to the problem of imbedding one-dimensional commutative rings (i.e., rings of functions on affine algebraic curves) in rings of differential operators. The fruitfulness of such a reduction will be demonstrated in Chap. III. 4.2. We shall actually prove a more general theorem pertaining to rings of operators generated by several differentiations
dn .

Let S3 be a not necessarily commutative

Q -algebra which is free of finite rank as a

module over its center 33a- Let du .. .,dn:33-+33

be differentiations taking 33o into itself

and commuting pairwise. We write i=>(h, • • •. *«). \i 1 = 2 *y d'=dil

• • • dn,

33 [d] = { 2 M ' i *<653}

with the usual rules of multiplication. Let Z,i,..., £r6$[d] be a finite family of operators. We call it independent if the sum of 33 -modules 2 33LPl... if/ is direct. We write LP=LPt...

L"/

and |/>|=2 pj .

4.3. LEMMA. We assume that in the category of free 53-modules the concept of rank is well defined (i.e.,

3?/=3?,S=$T=S)

the module. If Llt...,Lr

, and the rank of a submodule does not exceed the rank of

is an independent family of operators, then r < « . L\,...,Lr,

Proof. There exists a constant /, depending on the degrees of the operators such that for all m > 0 there is the following imbedding of free

^-modules: £i 3hlPa \p]<m

2 33d' .

But the 33 -rank of the module on the left grows asymptotically like c-jn.' and

\l[
that of the module on the r i g h t l i k e c2(lm)n.

Therefore,

r<«.

4 . 4 . We s h a l l c a l l a family of operators Lu ..., Ln£33[d\ maximal if i t i s independent and there e x i s t s a free 33 -module of f i n i t e rank Mcz33[d], such that 33^] = ^ML', and the t

sum on the r i g h t i s d i r e c t . Example;

in the case

n=\

i s maximal if the coefficient

the family consisting of a single operator bNdN-{- ...

bN i s i n v e r t i b l e in 33 .

33bpNdpN+l = 33dpNJrl, i t follows e a s i l y t h a t 33[d\=^ THEOREM. Let

Lu...,Ln

(iu...,i„),

54

q>0,

Since

/

be a maximal family of operators in

be an operator such that [P, Z.i]= . . . = [ / > , £„I = 0 .

33dl .

2 33d> \ D> and the sum on the r i g h t i s d i r e c t .

p_0\i=0

4.5.

Indeed, we s e t M=2

+b0,

Then there e x i s t

33\d] and l e t Pe33[d]

d-constants i q

lying i n the center of 33 and not a l l zero such that 2 aitqL P =0

aiiq, i = .

We f i r s t prove t h e f o l l o w i n g a u x i l i a r y a s s e r t i o n . over 33\d] which commute w i t h one a n o t h e r and w i t h 4 . 4 for t h e family

Lu...,Ln,

LEMMA.

33\d].

We choose a module

Mc33[d],

as i n

and s e t M [h

4.6.

Let X,, . ..,X„ be independent v a r i a b l e s

X„] = { 2 mtV i m&Ai} cz33 [d] [X,

X„J.

There i s t h e d i r e c t sum decomposition n

where the last expression on the right is the left 33[d] [>.l7. ..,/.„] -ideal generated by LL — X;. Proof.

a) We show first that the left side is the sum on the right.

Since

33[d]=^MV, y

i t suffices

to v e r i f y t h a t for any

the r i g h t .

But mV = nOJ -\-m{JJ — W), so t h a t i t remains to check t h a t V — \!

left

i d e a l g e n e r a t e d by L, — X£ .

Jk=£0 .

We s e t j'=(ju

m£M Let

and j

j = Uu..

•••.A-i.O, ...,0) .

t h e element .,/„) and l e t

The f i r s t term on t h e r i g h t i s d i v i s i b l e by

Lk—lk

k be t h e l a r g e s t index for which

L1'>.J* - X; Vt* =

j .

T h e r e f o r e , i n d u c t i o n on t h e number

results.

b) We now show t h a t t h e sum on the r i g h t of (11) i s d i r e c t . v e r i f y t h a t if

^lmlk^^^[d\[).\(Ll—,kl),

then

^=0

t h e n a c c o r d i n g t o t h e f i r s t p a r t of t h e proof a l s o t h e homomorphism of l e f t

l

X i ' . . X " i n t o L\ . . . ! „ " .

l i e s i n t h e k e r n e l of t h i s homomorphism.

4.7.

for a l l

t.

For t h i s i t s u f f i c e s

to

But i f ^,m,lie'^l^ld][k](Li

^imtL'e^^[d\m(Ll

— 'kl).

— h),

We c o n s i d e r

S3[d] -modules $[d][X]->-53[d], which i s t h e i d e n t i t y on

takes the free generator zero i n t e r s e c t i o n w i t h

l i e s in the

, w h i l e t h e second i s analogous to LJ — W,

y" has fewer nonzero components t h a n

of components g i v e s t h e r e q u i r e d

i s c o n t a i n e d i n t h e sum on

Then

Li - V = V'L'k* - l'"Vkk=L' Kk-V'KkJr

b u t t h e index

mV

33[d] and

The submodule 2 ^1^1 IM(^« — h)> c l e a r l y

T h e r e f o r e , i t a s w e l l a s t h e e n t i r e k e r n e l has

$[
Proof of Theorem 4 . 5 .

We c o n s i d e r the

$[X]-module M = 33[d][X] /^~®[d][X](L, — X f ).

According to Lemma 4 . 6 , i t i s f r e e of f i n i t e rank over 33[l], s i n c e i t i s isomorphic to 7W[X] . I t i s t h e r e f o r e f r e e of f i n i t e rank over $?o[X], where S30 i s t h e c e n t e r of t h e r i n g

33.

Since t h e o p e r a t o r P(*33\d] commutes w i t h a l l t h e

£,• , m u l t i p l i c a t i o n by P on t h e r i g h t

induces a

Therefore, the ring

S 0 lX]-endomorphism of t h e module

a c t s n a t u r a l l y on

A!.

M.

On r e p r e s e n t a t i v e s t h i s a c t i o n can be w r i t t e n as f o l l o w s : mobXiP'^bml'PJ,

Since X„,/>]

M

530[Xi> • • •. X„, P\ cz33[d]

b£BQ, m£M.

i s f r e e over 33Q\KU . . . , X„] t h e r e i s a nonzero polynomial Ffa,..

such t h a t

M°F = 0*

.,X„, P)G$?olXi.- ••>

According to Lemma 4 . 6 , t h i s i m p l i e s t h a t 33[d][X\oF(Xu..

.,XniP)ci

55

2 53 [d] [>•] (^ — \.)

.

From the calculations in the f i r s t part of the proof of t h i s lemma i t

i

follows that then 33[d\ [i.]oF (Llt...,

L„, P ) = 0 where the s u b s t i t u t i o n of

be done i n the reduced notation for

F .

Lt in place of

I,

must

Applying F(LU .. .,Ln, P) to l6$]d][X], we find that

F(LU . . . , ! „ , P)G33[d] n 2 ® I d H M ( £ | — h) = {% by the second part of the proof of Lemma 4 . 6 . i

We have thus obtained a polynomial r e l a t i o n between

Lt and P but with coefficients in

330 • We s h a l l now show that for s u i t a b l e F these coefficients are To t h i s end we choose a free b a s i s of the S0['-l-module a

t i a t i o n $b[>-I->-S8b[X] nd i t s l i f t

M and denote by D a differen-

M-*-M with the following p r o p e r t i e s :

a) On 33 the d i f f e r e n t i a t i o n b) DX,=0, i = \,

d-constants.

D coincides with

d.

...,n.

c) The chosen basis of M i s annihilated by

D.

We denote by A the matrix of the endomorphism of m u l t i p l i c a t i o n by P in t h i s basis and by F i t s c h a r a c t e r i s t i c polynomial. In addition to D,

there i s the d i f f e r e n t i a t i o n

the l e f t by d in 53[d][X].

I t also extends

d .

d:M-*-M, induced by m u l t i p l i c a t i o n on

Therefore, D—~d\M-+~M. i s a

330[l] - l i n e a r

operator: (d — D) (bm)=0—D) U—d —D

We set

and identify U

b • m+b (d — D) m = * (d—D) m.

with its matrix in the chosen basis. Since right multi-

plication by P commutes with left multiplication by

d , on calculating the action on this

basis of the composition of d and P in two different ways, we obtain dA-{-ALf=UA

dA=[U,A\

.

This implies that

t r a c e of


i s zero for a l l

dAn = 2 A;(?AAB-'-I = 2 re>0

.

or

n-l

n-l

A

' [ ^ ' A l A a_M = IC/, A"].

Therefore, the

This means that the coefficients of the character-

i s t i c polynomial of A are d -constant by Newton's formula. 4.8.

Integrals of the Stationary Lax Equations. We apply the preceding considerations _

JV-2

N

,

c

i

/

to the case 3S=k [a<>'\.. .,«(/l 2 iy>0] i L=*d +2 «« * /-o factor ring of 33 by the d -closed' ideal generated by [£, P].

We d e n o t e

b

?

_

=

® ®tct)

the

Let d:33-*-33 be the induced

differentiation where L, P(<33[d\ are the images of the operators L and P. It is obvious that L, P is a "general solution" of the stationary Lax equations with operators of the given degrees N,M, if the constants c0,...,cM are taken to be free variables and Si = i/N. Since the leading coefficient of L is equal to 1, Z L,

P

are connected by the relation

is a canonical relation: ^dtjV

l J

^dtJL P =

0, where

generates a maximal family, and dtj

are d -constants in 33. There

are the coefficients of the characteristic polynomial

/ of the endomorphism of m u l t i p l i c a t i o n by P for the 33 [>.]-module 33[d][ky33[d\[l\(L — X). l i f t s dtJ of the elements 7?,, to 33 are i n t e g r a l s of the equation [L, P]=0. 56

The

We note now that by Proposition 3.13 we have another c o l l e c t i o n of i n t e g r a l s of t h i s equation. dTr=0

Namely, for any r£QN,

in

P

r>0 we have

~l f (r)

B

^ j 21 ci ""'J+t * = !£®

so

that

93 .

The r e l a t i o n between the i n t e g r a l s

dtj

and

Ir

i s known for the higher Korteweg—de

Vries equations; in p a r t i c u l a r , the ones are expressed in terms of the o t h e r s . 5.

The Zakharov—Shabat Formalism 5.1.

[13].

In t h i s section we present an i n i t i a l version of the method of Zakharov—Shabat

Our objective here i s to c l a r i f y the algebraic side of t h e i r construction — the

s t r u c t u r e of "dressed" d i f f e r e n t i a l operators — while omitting functional-analytic considerations.

We begin, however, by describing the basic functionals.

5.2.

We consider the space of columns
functions of the v a r i a b l e x and possibly of the additional parameters t K(x,y;

t,z)

and F(x,y;t,z)

be two (NxN)

and

z .

Let

matrix-valued functions. We assign to them the

i n t e g r a l operators K, F: oo

oo

(Sty ( * ) = $ * (x, y) <»j (y) dy, (F| (y) dy. —oo.

X

Similarly, for any operator L we understand by (LK)~ the operator with kernel LK and i n t e g r a t i o n from x to oo, and by (LF)* the analogous operator with i n t e g r a t i o n from — oo to oo .

The dependence on

t, z

i s not indicated here e x p l i c i t l y .

The functions

K, F, and

<> ! are assumed such that a l l c l a s s i c a l formulas used below for d i f f e r e n t i a t i o n of i n t e g r a l s with respect to a parameter are valid (and, of course, the i n t e g r a l s themselves e x i s t ) .

The

following sequence of lemmas on the commutation of various operators i s preparatory to the formulation of the main theorem. 5.3

LEMMA. [dt, F\ = (dtFy,

[d„K\=(dtK)",

and similarly for

dz.

The proof i s obvious. Let FL+

Lx = Zli(x; t, z)dlx . where the U are (NxN)

matrices.

We denote by the symbol

the kernel 2 ( — dy)'(F(x, y)li(x)) and similarly for KL+ . i

5.4. Proof.

LEMMA.

[Lx,F\ =

{LxF-FL+}\

Integrating by p a r t s , we have oo

oo

(l^.^H) (•*) = £, J F(x,y)^(y)dy—oo

oo

J F(x,y)L^{y) = —oo

'oo

= J LXF (x, y) t (y) dy - $ FL+ (x, y) <1< (y) dy= —oo

5.5. LEMMA.

If Lx

{LxF-FLpl

—oo

satisfies the conditions of the preceding lemma, then

57

LfK-(LJO~=oK(f.x), where the d i f f e r e n t i a l operator

i s additive in Lx and for l = l(x,t,

°K(LX)

z) i s defined by

n-i

°K(ld?M= ~ 2 ld'x[{d--^K) (x, *)]. i=0

Proof. of

I t i s clear that the l e f t and r i g h t sides are l i n e a r on matrix-valued functions

x, t, z.

i t therefore suffices to compute

From n to n-\-\

a

ie(dx)-

the computation i s as follows: cK(djfi) d.=«?j+iJ K(x, y)^(y)dy-

(

-**H *

For re=0 the formula i s obvious.

K)(x,y)*Wdy

[ (d^K)(x,

y) *(y) dy=

+ aK(d*ft - J (d»+lK) (x, y) * (sr)rft/= - (
n-1

= - (djjAD (x, x) <•(*) - 2 <^+1 K*r : -'*o <*• •*) * <•*)]• The l a s t expression coincides with the formula for °K(dx+l)ty, indicated in the lemma. 5.6.

LEMMA. We assume t h a t the coefficients of Lx do not depend on x . koLx-(KLpA

Then

= ^(Lx),

where *K i s a d d i t i v e i n Lx and ii—I

** (W3=2 ( - ir* (dj-'-'/o <*. •*) wxi-=0

Proof. As above, for ra=0 the assertion is obvious. We carry out the inductive step from n tora-f-1. Splitting the second integrals into parts, we have: op

lk4dnx+1-K(idnu+1r]*(x)

=$#(*.

u)idnv+1*(y)dy-

X

00

- 5 (-1)"+1 (dl+1K) (x, y) /* (y) dy= OS

00

.=$*-(*, 9)«;+I$(0rfj,+(-l)«+« <<^0(*. x)ty(x) + $(-iy*HdlKH*, X

Here we have used the fact t h a t dy(lty) = ldyV since l = l{t, z) .

The sum of the two i n t e g r a l s

here i s equal to [Koidx - K (ldlr\ dxif = t , (WJ) d/f by the inductive hypothesis.

The e n t i r e expression i s therefore

( - I f " (
58

y)ld^(y)dy.

X

5.7.

We now consider the ring

33 [dx] of matrix-valued operators in dx, the coefficients

of which are polynomials of the r e s t r i c t i o n s to the diagonal of the p a r t i a l derivatives d*K, d$K and the derivatives of these r e s t r i c t i o n s along the diagonal. ( k c o n s i s t s of the constants, i . e . , R or C in the analytic c a s e ) . the same order as Lx, and i t s coefficients depend only on s i b l e to apply

x, t, z .

as i s evident from Lemma 5.5, reduces the order.

LX£MN(k)[dx]

Then {l-\-zK)Lx

has

I t i s therefore permis-

<sK and i t s powers to i t while remaining in 33 [dx] .

°K-33[dx]-^33[dx\,

Let

But the operator

Therefore, the following

expression i s meaningful: i ; = 0 +OK)-^KLX=(1 where M>ordLx

..+(-\)Mo*)zKLx63S\dx],

-OK+*K-.

.

The operator

L*x i s called the "dressed" operator

Lx , and the mapping Lx~*-(l +<3k)~hKLx

i s called the "raiment" of Lx . The f i r s t part of the next theorem shows that L*x i s roughly speaking the d i f f e r e n t i a l p a r t of the operator obtained from Lx by means of conjugation with the operator \-\-K : 5.8. K, F

THEOREM, a) L*x{\ +K) — (l+K)Lx

= {LxK — KL+}* . b) We assume that the functions

s a t i s f y the equation oo

K (x, y) + F (x, y) + J K (x, s) F (s, y) ds=0 X

o r , more b r i e f l y ,

K+F-f

K*F=0

.

There i s then the i d e n t i t y

{L'xK-KL+y) +

(LxF-FLl)+{LxK-KL+y)*F+K*(LxF-FL+g)^=0.

c) Under t h i s same condition we have dtK + and s i m i l a r l y for 5.9.

dz

dfF+dtK*F+K*dtF=0

.

Application of Theorem 5.8.

If K+F+K*F=0

some operators R, 5 , we s h a l l say that the pair +

and RK+SF + RK*F+K*SF = 0 for

(R, S) " d i f f e r e n t i a t e s " +

(AT, F) .

to Theorem 5.8, the pair (Z.*— L u + a.d, + $dz, Lx—L y + *dt + $dz) d i f f e r e n t i a t e s

According

(K, F), i f

a, p

are any c o n s t a n t s . We now suppose that the kernel F s a t i s f i e s the system of l i n e a r d i f f e r e n t i a l equations LxF-FL;+zdtF=0, LS-FLl+Wf^O. We find by Theorem 5.8 b) and c) that Vx K - KL*+* dtK + {L'JC - KL\ + *dtK) *F=0, LxK-KLl + ?dzK + {LxK-KL+„ + VdzK)>F=0.

59

We suppose, moreover, that from t h i s i t can be concluded that

LxK-KL*+?dzK=0. Using Theorem 5.8 a) andLemma 5 . 3 , we obtain, on the other hand, (L"x + adt)^+^)~^+f()(Lx+oidt)=lCK-KL+y + oLdtKr=0, {L'x + B
and

l+K

i s i n v e r t i b l e , we find that the d i f f e r e n t i a t i o n operators

I ' - f B ^ are adjoint to Lx+a.d, and Lx+$dz,

We now take the i n i t i a l matrix operators Llx

respectively.

and L2 x to have constant c o e f f i c i e n t s .

We obtain the following r e s u l t . 5.10.

THEOREM.

If [I lx ,Z. 2 ^]=0 and F, K s a t i s f y the conditions of 5.9, then L'x, L*^

s a t i s f y the equations of Zakharov-Shabat [L'lx+adt, L2% + P«?1=0, or

We proceed to the proof of Theorem 5.8. 5.11.

Proof of Theorem 5.8 a ) . According to Lemmas 5.5 and 5.6 L'xoK=(LxKV+cK(Lx), KOLMKLW

+xK{Lx).

Therefore, L

U^+^)-^+k)Lx={LxK--KL+r+Lx-Lx+ok(Lx)-rx(Lx).

But

Lx=(l+c„p(\+-z)Lx. Lx-Lx-oKLx 5.12.

identity

Hence + tKLx H ( l + 8 *)" 1 (1 + V ) - 1 + M 1 + ^ ) _ 1 (1 + t j r ) - v l 4 r « 0 .

Proof of Theorem 5.8 b ) . Applying the operators of Lemmas 5.5 and 5.6 to the K+F+K*F*=0;

we obtain I j r /Sr+I^ 7 + (i x Ar)*F+o J r (^)F=0,

(12)

KL++FL++K*FL+=0. By Lemma 5.6 applied to the columns of

F , we have

K*FL+= -K*(LxF-FL+)+K*LxF=*

60

(13)

-K*(LxF-FL+)+KL+*F

+ xK(Lx)F.

Substituting t h i s expression into (13) and subtracting the r e s u l t from (12), we find LXK-KL+

+ LXF-FL++{LXK-KL+)*F

We s e t LW = LX, LW=a'-i(aK — -zK)Lx for

i>\

+K*(LxF-FL+)

+ (aK-^){Lx)F=0.

(14)

and apply (12) to Z in place of

Lf . We

obtain ^)K+L^F+(HJ)K)*F

+ <,K(LW)F=0.

We take the a l t e r n a t i n g sum of Eqs. (14). with signs

(14) ±

( — 1)' over i = 0,..., M,

M>ordLx.

Noting t h a t M

we obtain L*KK-KL++LxF-FL+

+

(L*xK-KL+)*F+K*(LxF-FL+)=0,

which completes the proof of the theorem. 5.13.

Formal Analogues. Under appropriate analytic assumptions we have

K(x, y)= 2n-<^*H*' *)Q/-*Y and

kr- 2 (d'yK) (*, *) J {1=^ f (y) dy. X

(if

x\l

'.

It is possible to eliminate the factor i

times. Therefore, setting

under the integral by integrating by parts

(d'yK)(x, x) = ut, it is natural to assign to the Volterra

operator R its symbol oo

Here the ut may be considered a r b i t r a r y elements of some d i f f e r e n t i a l ring

3S, or matrices

with independent coefficients as in 3.16. In order to write formulas for the operators <»*•, *#, i t i s useful to introduce further the generator vh corresponding to (d'xK)(x, x). I t i s not hard to see that they are l i n e a r l y expressed in terms of the ttj and t h e i r derivatives (using the formula ((dx-\-dyyK)(x, x) = dx(K(x,x))) :

61

Comparison with formulas (3) and (4) indicates that conversely the u,t are expressed in terms of the

vl .

In these generators we obtain n-l

n-i

'•K-.S3 [d}->33 [d\:ld"- 2 ( - l r 1 " " * * - ! - ^ ' i=0 00

Conjugation by means of K — l + £(—

l)'ttjE~' takes commuting operators in

(53((;_1)), °)

into commuting operators: [I„ i 2 ]=0^" 1 Z:„ /?, ^»Z,/?] = 0

(we here identify

L\, and L2 with their symbols).

Generally speaking, the operators K^LjK

Considering only t h e i r d i f f e r e n t i a l parts ( K~lL[K ) ,

are not purely d i f f e r e n t i a l operators. we find

ord[ ( K^UK > , < K~lL2K > 1 < max ord Lt -1. Thus, t h i s method leads to the construction of c e r t a i n "general" solutions of the equations of Zakharov—Shabat. In conclusion, we s h a l l c l a r i f y which d i f f e r e n t i a l operators are conjugate by way of some K . N

Now l e t that

N

jKufi1, £w£lG$!im

be two symbols

{tlh wt

are any elements).

dUff=0 , «JV i s semisimple in the sense of Sec. 2, and that

We s h a l l assume

d:53+->33+ i s s u r j e c t i v e .

oo

5.14.

THEOREM.

For the existence of a symbol / o o

ff

u

l

2 M +2 i t i s necessary and s u f f i c i e n t that Proof.

\

v

k

/

1+2'°-*^"* \

oo

l

v

with

the

P ro P ert y

»

H

-£~ H + 2 ^~ °2w'6'

tttf = WN and Uw-i—<WN-\
We rewrite the l a s t equation i n the form

ct>(U>l L i=0 JV

We c a l l the k-th

equation (for

'

V

'

/-"

i> ) the equality for the coefficients of

E* on the l e f t

and r i g h t s i d e s . The N-th equation gives Uff = wN ; the (N— 1 ) - s t equation has the form [uN, v_x\ = TOAT-I — tifir-i . This shows the necessity of the condition of the theorem and uniquely determines vzl . The r e s t of the argument i s p a r a l l e l to the proof of Theorem 2.10. We suppose that from the equations with indices N N_k+l the q u a n t i t i e s v_u ..., *CA+1 have been 62

determined.

Then the (N — k)-th

equation uniquely determines the element

UN-lV_k +i — 1 U +i«lAT-l + NUNV'_k+l + [UN, V_k] = W.

Since

JCGS - ,

WN-I—UN-I-{-X,

xv_n+i

.

For d e t e r m i n i n g

t h e f i r s t two terms can be r e w r i t t e n i n t h e form we

"Q—n+i

[ttjv-i, i»^ +1 ] —

obtain the equation

NaN («J:, + 1 )'-(^I* + J )=o' + . which i s s o l v a b l e i f 6.

d:33+-*SS* i s

surjective.

The Benney E q u a t i o n s ; Main R e s u l t s 6.1.

We r e c a l l t h a t t h e system of Benney e q u a t i o n s h a s t h e form y

it + uux-u^ux\y^d^ u.

ht+Kudy)

+ hx=Q,

=0.

The meaning of the notation is explained in Sec. 1 of this chapter.

The formal investiga-

tion of the equations is based on the following lemma of Benney. h

LEMMA. We define the moments An(x, t) = }u(x, y, tfdy, re>Q. From system (15) there o then follows an infinite system of equations for the moments: 6.2.

A„,t + An+Ux+nAn_1A0,x=0, Proof. without

«>0.

M u l t i p l y i n g t h e f i r s t e q u a t i o n of (15) by nnn~i

(16)

and r e g r o u p i n g t e r m s , we o b t a i n

difficulty

(u")t+(W*l)x—\uniuxdvi)

We now i n t e g r a t e t h i s r e l a t i o n on y from

0 to

+nun~1kx=0.

h .

The f o u r t h term becomes

t h e t h i r d [because of t h e second e q u a t i o n of ( 1 5 ) ] becomes W(A, + uhx)\y-h A

f i r s t term of t h e l a s t r e l a t i o n to and

A„+1,x

\(u")tdy,

.

Adding t h e

h

and t h e second to

^(W^) x dy

, we o b t a i n

An,t

, respectively.

I n Benney's work i t i s shown t h a t t h e r e e x i s t two sequences of polynomials [J4 0 , . . . , An]

nAn„xAo,x , and

and F„£Q \A0, ..., Ak+l\

//„6Q

such t h a t (16) i m p l i e s l o c a l c o n s e r v a t i o n laws for

the

system (15) of t h e form Hn.t + Fn.s-O,

«>0.

(17)

In Miura's work [14] it is shown that there exist two further sequences of polynomials Hn, Ft£Q [u, A0,...,

An_i]

such that (16) implies local conservation laws of the form

63

Hn,i + Fn,x + {JRnv)y = 0; n>\,

? = — \uKdn. o

(18)

(In Miura's table the coefficients of v do not coincide with H„ because of misprints and omissions.)

As usual, under the assumptions of rapid decay of solutions^ a t i n f i n i t y , from

t h i s i t i s possible to obtain q u a n t i t i e s conserved in time: these are

\ Hndx,

rt>0.

—00

The next two sections are devoted to a new construction of relations (17) and (18). The description of the generating function of the system of polynomials H„ as solutions of a certain integral equation with parameter plays a central role in the construction.

6.3.

THEOREM.

We set


solution

Then there exists a unique

0

i-0

n(X) of the equation l*<X)+0(p<X)) = X

(ig)

a) in the class of formal s e r i e s of the form K+Q[A0, Ait.. .][[k~1]]', b) in the class of functions analytic i n X of the form h(x,t))

X+o(X""') in a neighborhood of oo (depending on

u{x,y,t),

. 6.4.

THEOREM.

System (15) implies equations for

J*(X) of the form

(1/-(^/2+A)x=0,

(20)

[&*+#i-[*%<*+^l-[$<*+#i\**v= 0 . Theorems 6.3 and 6.4 are proved in Sees. 7-8.

(21)

The Benney conservation laws (17) (up to CO

constant factors) a r e obtained from (20) by s e t t i n g 5jj(—Ty/riX~('+I)

and equating coefficients of

X~(B+I).

ji=X — ^ ( — 1)'//A~ ( ' +I) , \t.2/2+A0=* X2/2 + Miura's conservation laws are deduced

s i m i l a r l y from the r e l a t i o n s (21) i n which (^+«) - 1 i s to he interpreted as

^(—lyu^'^&^lu,

km. 6.5.

The Reduced Equations.

Eqs. (15) by adding the condition depend on height.

As already mentioned, the reduced system i s obtained from «y—0 : the horizontal component of the velocity does not

The reduced system for the functions u(x, t), h(x, t) has the form (the

c l a s s i c a l equations for long waves): u

>=-[!T

+ hlx~'5Z\~drh

This system i s Hamiltonian with Hamiltonian operator '#—(<)

64

(22)

o /'

and

Hami

ltonian

H-

The integrals Hn of the full system (15) are converted into integrals H\ n

0

of the

a

reduced system (22) after substituting An*+hu =A n:

H ndQ.\a, h\ . We explicitly compute these

integrals in Sec. 9, we show there are no others, and we establish their commutativity. [«/21

6.6.

THEOREM,

a) Up to constant multiples we have # J = 2 tnJlun-2k!ikn, A-o i

=

n.k

and any polynomial b

>

H

of

u, A, for which

where

(23^

(n—2k)\k\(k+l)l

HtGdxQ [a, A] i s a l i n e a r combination of t h e

H^.

[ffl,ff°m]6dxQ[u,h].

Theorem 6.6 e n a b l e s us t o i n t r o d u c e t h e " h i g h e r reduced e q u a t i o n s " [*H\

( « , =

u

AT

-ST-

YdHV ff=^cnHl,

c„eR.

(24)

From what has been s a i d , they admit c o n s e r v a t i o n laws of t h e form / / ' , / + P"a,x=0, where the Fn

depend on

c0,

...,cN.

I n S e c . 10 i t i s shown t h a t t h e c o n s t r u c t i o n of commuting H a m i l t o n i a n s {Hn°}

can be

considerably generalized. In S e c . 11 we pass to t h e i n v e s t i g a t i o n of t h e f u l l system (15) o r , more p r e c i s e l y , of t h e system of e q u a t i o n s for t h e moments ( 1 6 ) .

We b e g i n by showing t h a t they a r e Hamiltonian

a s a system of e v o l u t i o n for i n f i n i t e l y many unknown f u n c t i o n s e s t a b l i s h i n g the following We s e t

B=Bid+d°Bif

I , the operator 6.7.

B

—

of two v a r i a b l e s by

facts.

, where

2?i, y=k4, + j_i, i, / ^ 0 .

Then, as was shown i n Sec. 8 of Chap.

i s Hamiltonian.

THEOREM.

and H a m i l t o n i a n

An (x, t)

The Benney e q u a t i o n s f o r t h e moments a r e Hamiltonian w i t h o p e r a t o r

B

y / ^ 2 = _"2"(-^2 + ^o)-

(The H a m i l t o n i a n formalism i s h e r e a p p l i e d to t h e r i n g JL = k\Ai'

J, i, / > 0 , A\!) b e i n g t h e

00

independent v a r i a b l e s .

I t i s a l s o p o s s i b l e to work i n the r i n g

\J C*°(x, A\\i-\-j<&k) *—o

or i n

various intermediate rings.) 6.8.

THEOREM. Let H&A

be any element.

Then a system of equations for u, h of the

form

\/-0

where

^ii)==~rr~<

Ix

0 \,/-0

i m p l i e s t h e system of e q u a t i o n s

Jx >"«

At = B—=-

w i t h Hamiltonian

(25)

H .

65

This theorem is proved in Sec. 11. Theorem 6.7 is a special case of it for

H=—-~H2,

when the system (25) becomes the original system of equations (15) of part 1 and Theorem 6.7 becomes the Benney lemma.

Theorem 6.8 indicates the rather surprising situation: in-

spite of the loss of information on passing from

u, h to A , the equations of evaluation

for A with any Hamiltonian can be lifted to equations of evolution for u, h .

Of course,

the question of lifting solutions requires separate investigation. 6.9.

THEOREM.

Let the elements H&.J, be defined from Benney's conservation laws (17).

Then they commute relative to the Hamiltonian structure with operator B

.

This theorem is proved in Sec. 12. It makes natural the consideration of systems of the form (25) with Hamiltonian H = ^£;//,-, which we call higher Benney equations in analogy with the higher Korteweg-nde Vries equations. By the general formalism the higher equations possess conservation laws of Benney type Hl t-\-Fi x=0,

where the Ft

depend on H . Con-

servation laws of Miura type hold for one of the higher equations. 6.10.

THEOREM.

There exist Miura conservation laws for the system (25) with Hamilton-

ian H=cH3. Unfortunately, we have been unable to determine if this fact holds for the other higher equations.

Theorem 6.10 is proved in Sec. 13.

We began by considering the reduced higher Benney equations (with the additional condition uy=0),

for which Ai = hul . As was mentioned, they are Hamiltonian related to the

operator f J J) in the ring Jt°=Q[uO), 6.11.

h^\j>0\.

The Hamiltonian structures described in the rings Jl and d>° are

THEOREM.

compatible. A precise formulation and proof of this theorem is given in Sec. 14. That section also contains an explanation of the reasons for which the system (25) contains the operators

7.

The Function u(X) In this section Theorem 6.3 is proved, and further information regarding the function

u(X)

and its coefficients needed below is presented. 00

7.1.

|>W as a Formal Series. We set |i(X) = X — 2(~~l)Wi*-('+1) and seek — !«_! from Eq. (19) which we rewrite in the form

The Function

the coefficients Ht oo

»

|~

oo

"|-(i+D

2(_iy// ( X-('+"-2x-('+')(-l)^ i l_2(-iy// / >.-(/+ 2 ) This o b v i o u s l y i m p l i e s t h a t [A0,...,Aa_2,N0,...,Hn_2] 66

f

or

/ / _ i = 0 , H„ = A0, Hi = Ai and f u r t h e r n>2

.

=0.

ffn — An-\-P„,

(26) where

I n d u c t i o n on n immediately shows t h a t t h e

PndZ Hn£An-\-

Z [Aa, ..., An_2]

e x i s t and a r e u n i q u e l y d e t e r m i n e d .

3 4 , Ai. Ai+4A0A2+2A\+2A30 7.2.

The F u n c t i o n

h(x,

t)

t h e moments An

the

\Hn(x,t)\

The f i r s t

Hn a r e : AQ, Au A2 + A20, A3 +

. (A(X) a s an A n a l y t i c F u n c t i o n . become f u n c t i o n s of

For given d i f f e r e n t i a b l e

X, t ; i t would be p o s s i b l e to a t t e m p t to e s t i m a t e Cn(x, t) for a s u i t a b l e f u n c t i o n C(x, t),

and show t h a t they grow no f a s t e r than

and t h i s would e s t a b l i s h t h e a n a l y t i c i t y of

u{X,y,t),

i*(X) i n

X .

However, i t i s i n c o n v e n i e n t t o

o b t a i n such an e s t i m a t e d i r e c t l y , and i n s t e a d of t h i s we apply t h e method of i t e r a t i o n s t o the integral equation (19):

|i(X) + W(X) + a)-1flty = 0 .

of a f u n c t i o n

X w i t h t h e p r o p e r t y (19) and t h e e s t i m a t e

p. a n a l y t i c i n

We s e t

'rhe

K*.) = X-fs(X).

uniqueness

e=0(X _ 1 )

follows h

from the uniqueness of the formal series.

*-N = — jj(« +

To prove existence we set £o = 0.

0

\ + eN-i)~ldy

and show t h a t

e = lime^ e x i s t s ( f o r g i v e n

so l a r g e t h a t t h e f o l l o w i n g i n e q u a l i t i e s a r e

('.*, 0) uniformly i n X, when |X|

satisfied:

|A(jX|-i7)-'(lr-4A(lX|-L0-2r1<(IM-^)/2, \ | X | - f / > 2 V r A ; t/ = s u p { « | 0 < # < A } . In p a r t i c u l a r , if

u, A

is

(27)

a r e bounded t h e r e i s a domain of a n a l y t i c i t y p, n o t depending on

To t h i s end we e s t a b l i s h t h e f o l l o w i n g

x,t.

« i n e q u a l i t i e s by i n d u c t i o n :

• n - i I •*<* u I 5„_i — e „ _ 2 j A

dy

I n d e e d , | ^l | =

\%\—U

< A s u p | a + X|-'
^\%\—U

<'

'

" .

F u r t h e r , for any W > 1 we

0

have for

|SJV+I

— ^ K A s u p K i i + ^ + ^KM + ^ + S A r - O r M ^ - e i v - i l .

i V = l , we f i n d by (27) 2

(» + Jt |

2

2A(|X| — U)~ \ti |<4A(|XI— U)~ \ £i I .

_1

Setting here

N=\

1

< ( | X \ -U)~\

| « + X + > 1 r < 2 ( | ^\-U)~\

This p r o v i d e s t h e b a s i s for i n d u c t i o n .

t h e i n e q u a l i t i e s of (28) a r e s a t i s f i e d for a l l

ft
.

and u s i n g (28) whence

|«2-ti|<

Suppose now t h a t

Then

je/sr+i—ejv^AsupKtf + X + e^Xa + X + SAr-OI-1! e^ — e^r-i |-<8 | &N — e„_i \, by (28) f o r

n=N—\,N.

Hence JV+I B-l

by (27). This completes the proof of Theorem 6.3. Below we shall mainly use

n(k)

as a formal series. The validity of the corresponding

computations in the analytic version can be trivially verified.

We shall establish several

curious algebraic properties of the series for p.(A,) and its coefficients

Hi.

An

the weight of the variable n + 2 , and we

introduce the corresponding graded ring Z[i4B].

In this gradation the polynomials Hn are

7.3.

Homogeneity.

We call the number

67

homogeneous of weight

n+2 . This fact is obtained by an obvious induction on n from the

identity (26). 7.4. The Symmetry At++— //,-, X-*-^.

a

We define a homomorphism

into itself by the conditions a{A,)=—Hh

of the ring Z[-A,]|[X-1j]

o(X) = [i.. it is obviously an automorphism. We 2

shall show that it is an involution: o = id. ' (!*) — 2 "< (° (P))""''4-0 = I 1 •

Applying- a to (19), we obtain

0 n tne

other hand, by the

definition of (J., X — 2 ( — 1)'//(A—f'+1> = fi . But from the first equation, as in 7.2, the element 1=0

8

1

,

(t )Gt*+Z[.AJ]|[ji.- ]] = X-f-Z[.A,][[X-1]l

Hence a{v) = \ . Applying a again

is uniquely determined.

to the .series for ji , we find i»—2(—1)'°(#;)(JL~(i+1, = x; since H + 2 ( — 1)'AtJl~('+1>=* by (19), t^o

<-o

in view of the uniqueness of the expression for h in terms of |t, we obtain o(// i )=— A t , i.e., o(—fft) = At.

7.5. Summation of the Benney—Miura Series. In the work of Benney [27] and Miura [46] the generating series for the conservation laws were constructed in a different way. oo

oo

•O

Benney

oo

V

V

/ o
+

,+1

\T" 'Z'

started from the series T(z)=^Anz't and showed that y>Hnzn = £i\zij^) -(n , ^ ; similar formulas are given in [27, 46] for other series from (20), (21). After the substitution z = — X - 1 and passage to 4>(X) = X_ir( —X - 1 ) all the series of Benney and Miura can be rep00

resented in the form S i l p ^ u ' V ' M I ' M I . where < ? = — 0, ty£Z[At]((X"1)). Our basic integral equation (19) was obtained by applying to these series the following identity which is of independent interest. 7.6. Proposition. Let k be some

Q -algebra, and let ?GA[[*-1]]. ^(Qr1))

.

Then

00

^ l j j r ( g r ) [
'

p& + k[[X-1]]

Proof.

We n o t e f i r s t of a l l t h a t t h e s e r i e s f o r

terms X_2B+const , so t h a t t h e sum converges able

i s t h e unique r o o t of t h e e q u a t i o n

JJ. ==
'

C and d e n o t e by

^

1^-j \yn$i\ b e g i n s a t l e a s t w i t h t h e

*._1 - a d i c a l l y .

t h e r o o t of t h e e q u a t i o n

We i n t r o d u c e t h e a u x i l i a r y v a r i -

pc = c

e x i s t e n c e and u n i q u e n e s s f o l l o w immediately from Theorem 6 . 3 a ) .

Its

We s h a l l show t h a t

oo

Ssrf^r^l-'S?*^

in the ring

For

*MH^'H •

c==1

this i m

P l i e s the result of the

a—0

proposition. This i d e n t i t y i s e q u i v a l e n t t o t h e sequence of i d e n t i t i e s which we e s t a b l i s h by i n d u c t i o n on Sfj

=1 .

n ,

For

Suppose t h a t i t i s t r u e for

n
s t e p , we u s e t h e f o l l o w i n g remark. to

c and

X , we find

[
the r e s u l t i s obvious, since

and a l l


pc\c-o'=^*

I n o r d e r to make the i n d u c t i v e

D i f f e r e n t i a t i n g t h e r e l a t i o n ft. = c?(ft,) + *. w i t h r e s p e c t

d\>.c/dc = [ 1 — ccp'OOl -1 ?^).

and f u r t h e r d%fdldc=d*v.tldl*-
68

«=0

\^\

.

d\xc/d\ = [l— c?'(ft.)]_1, whence

dpjdc

=
^H^]-i[(ir^-ti)]-i((iTBP-(rt)wiJ-(^nS!,(rt)w+f^jVw*(h)+?wt'W]}|t--

This completes the proof. 8.

The Conservation Laws In this section we shall prove Theorem 6.4.

In order to establish a formal version of

it, we introduce the independent variables A\», Af^^A;,

I, J>0 , and we consider the follow-

-1

ing differentiations of the ring Z[A\»\[[X ]] into itself. At

into 1, and it takes AW l)

into A\J+

-1

for ki=i,

into zero; didt

and X

-1

[cf. (16)] and X

A\n for j>\

The differentiation

dldAt

takes

1

and X" into zero; d/dx takes A\'l

commutes with d/dx, and takes A t into — Al^ —

iA^A^

into zero. Moereover, all these differentiations are continuous in the

1

X" -adic topology. 8.1. LEMMA. Let D (|I))DJI = D X - « « ( H . ) ,

be one of the differentiations described above. Then (1+®'

where, by definition, *' ( . ) = 2 ( - l ) m ^ ( ' + !)(-)-(l+2>. * ° ( - ) = 2 (-l)^A(-)-(i+1)

COROLLARY, a) (I + Q'fr))dv./d\=>l ^

; b) (?IX/
; d)

= (_1)/+v_((+1)^

Proof. We apply D Lemma 8.1.

to the relation (19); we obtain D\i +
Setting here D = dldX, we obtain Corollary a ) . Finally, putting successively and

D = d/dx, didt, d/dA'i,

using Corollary a) and the fact that 0>d/<M'(•) = (— l)'(-)_(/+I), we

obtain Corollaries b ) , c ) , and d ) . 8.2.

Proof of Formula (20). We multiply relation (16) by ( —l)»{i-(«+«) and sum on tl We obtain d®/dt(p)-\>.{d!dx(v) — Ao.x\r1) + Aa.x®'{v)<=0, or d9ldt(v) — ^d<Sf/dx(v) +

from 0 to oo. A0,x(l+®'(v))=0

.

Multiplying the last equality by dpldl

and using Corollaries a ) , b ) , c ) ,

we obtain (20). 8.3. Proof of Formula (21). We carry out the proof by analogous formal computations in the extended ring Z[AO\ «<*•'>, -o^] [[X"1]] , where k,l,m,n>0;

«<°-~0>=tt; «<8}-« and cor-

v

responds to — \uxdr[ ; all the new variables are independent of one another and of the previous variables. The differentiation into AU+v,

d/dx takes «<*•" into

commutes with d/dx, l)

_I

— Ax , u into — aa* - ' — u<°- v — AW; and X X

didt

\

and X" into zero; the differentiation d/dl is trivial on the new variables; 1 0

-1

A ,)

1

the differentiation didt d/dx,

«<*+••'), «<«> into- ^™) +1) .

d/dl and takes v\% ' into v\"ln,

A0 into

into zero; the differentiation d/dy commutes with

and takes «<*.'> into «(*•'+», -v\% into —(
(the differentiations are chosen in correspondence with the first and second equations

of (15)). We now calculate (21) using the prime to denote differentiation with respect to X: 69

((*' G*+ti)~% = {(".+«) | i ' ( - 1 * ' (|»,+a,)J (i» + a)' 2 , —0*1*' 0*+")""')* = — [(!* + « ) O w ' + M\r) - «*' G** + K * )I (l* + a )~2. (?!*' (s*+")"')y=16* + ") (w,i»' + «|* j) - V (|ty+*>>)! (i*+«)~2S u b s t i t u t i n g i n t o the numerators of t h e s e e x p r e s s i o n s the r i g h t s i d e s of t h e r e l a t i o n s y. y =t*j=0, vy=—ux, respect-to

p./=p.[ijr + A),jr

[formula ( 2 0 ) ] , v-'t=p'px + w'x

X), ut = — uux — ayv~Ao,x

numerators i s e q u a l t o z e r o .

( t h e d e r i v a t i v e of (20) w i t h

and c o l l e c t i n g l i k e t e r m s , we f i n d t h a t t h e sum of a l l

This completes the proof of Theorem 6 . 4 .

There a r e a number of r e l a t i o n s between t h e " d e n s i t i e s " of conserved q u a n t i t i e s Hlt and t h e " l o c a l c u r r e n t s "

Fh Ft

tiations S J - j L + g y A ^

which we now d e s c r i b e .

and

differen-

D = | ^ ^ - .

8 . 4 . LEMMA, a) 25f* = dp/dl — 1; b) Proof.

We i n t r o d u c e two f u r t h e r

Hl

—d\x/d\d>.+«)_1.

£>;J.=

a) Using C o r o l l a r y 8 . 1 d ) , we find DO

( t h e l a s t e q u a l i t y follows from 8.1 a ) ) , b) S i m i l a r l y , D | i = 2 " ' ( - 1 ) , + 1 i*-('+1,di*/
8.5.

Proposition.

The following r e l a t i o n s hold:

a) « / / „ « / « / / « ; b) / / . - j * . ^ . ; d) £ > / / „ = - / / „ , DFn=-Fn; ^ "-"""(M. n + I» Proof. (21).

cM, n"—

^ H . - ^ r . ;

®Hn=nfin^, r

dAl+i

"-

R e l a t i o n a) follows immediately from Lemma 8.4 a ) ; d) i s o b t a i n e d from 8.4 b) and 2

R e l a t i o n e ) i s obtained s i m i l a r l y : t i o n s e ) , f ) , and g) follow and n o t e t h a t

Homogeneity.

w e i g h t s n and n+\,

Fn

d/d-Al+l(\x /2 + A0) =(—

+ l;

t h i s and (20) imply b ) .

1)'}*-«+I>
The r e l a -

.

S e t t i n g t h e weight of

a equal to

a r e homogeneous of weight

n + 3;Hn

1 , we find from 7 . 3 and Eqs. and

Fa a r e homogeneous of

respectively.

I n t e g r a l s of t h e Reduced System Theorem 6.6 i s proved i n t h i s 9.1.

i s equal to

section.

Proof of Theorem 6.6 a ) . The d e r i v a t i v e of k l

k l

(u h )t=—(k-\-l)a h ux

k l l

kn l l

— (ku " h -{-lu h ~ )hx.

ukhl,

by v i r t u e of the system (22)

T h e r e f o r e , for any polynomial P g

Q[U, h\, r e p r e s e n t e d as t h e sum of terms homogeneous i n u, ft, P = ^PL 70

— d\y.ldAt .

from a ) , b ) , and c) r e s p e c t i v e l y i f to the l a t t e r we apply D

[D, S)] = [D, didA[]=0

( 2 0 ) , (21) t h a t the

9.

s;

e)

C)

F u r t h e r , a c c o r d i n g to 8.1 d ) , d/dA0(\>.2!2 + A0)=—dp/dl

8.6.

(J*+«)->.

of degree d e g P , ,

we

have

Pt= — ^idegPiXPiUx—\u-0£-^"jj—\P-hx.

Pt&faQ[h,u]

The P o i n c a r e lemma i m p l i e s t h a t t h e c o n d i t i o n

i s equivalent to the requirement

that

A(PHd/dA(2degP,.P,)-^o(a(?/=0. In

Q[u, A] we i n t r o d u c e t h e g r a d a t i o n w i t h weight w, by s e t t i n g

Obviously,

A i s a homogeneous o p e r a t o r of weight —2.

w(u)=\y

«>(A) = 2 ( c f . 7 . 3 ) .

T h e r e f o r e , t h e k e r n e l of A i s t h e

d i r e c t sum of i t s homogeneous components, and i t s u f f i c e s t o compute i t on a component of weight

A l ^ c„,ku"r'2khk 1 = 0.

n , i . e . , to s o l v e t h e e q u a t i o n

t i o n s show t h a t _1

A = 0 on -1

(n — 2k—l)A (A+l) Cn,* k e r n e l of

Q + Qu, w h i l e for

for

k>\

.

«>1

we have

Somewhat complicated computac„,0 = 0 and

This i m p l i e s t h a t a l l homogeneous components of t h e

A a r e o n e - d i m e n s i o n a l and a r e g e n e r a t e d by polynomials

Remark.

The e x p l i c i t form of

of t h e f u n c t i o n

c„,*+i = (rt—2k) .

H\ of t h e form ( 2 3 ) .

Hn° could a l s o be o b t a i n e d by computing t h e c o e f f i c i e n t s

u°(X), which i n t h e reduced c a s e s a t i s f i e s t h e e q u a t i o n

+h(n°(%)+u)-l=k.

n°(X)

I t c a n n o t , however, be proved t h a t t h e i n t e g r a l s found form a complete system.

On t h e o t h e r

hand, by g e n e r a l i z i n g our arguments t o t h e f u l l system (15) i t i s p o s s i b l e t o show t h a t t h e Benney i n t e g r a l s a l s o exhaust t h e space of i n t e g r a l s which a r e polynomials i n 9.2.

Proof of Theorem 6.6 b ) . According t o t h e g e n e r a l formalism of Chap. I , t h e

Poisson bracket system

A{.

[Hm°, Hn°] i s t h e d e r i v a t i v e w i t h r e s p e c t t o

ut=(dHm°/dh)x,

h,

t of

Hn°, by v i r t u e of t h e

=(dHm°/du)x.

In o r d e r t o cover a l l p a i r s («, m) s i m u l t a n e o u s l y , we i n t r o d u c e t h e new formal g e n e r a t es ing f u n c t i o n >(X) = ^ / / n X B / r e ! and i n t h e r i n g Q [a, A] [[X, X^] we compute t h e d e r i v a t i v e with r e s p e c t to setting

t of

v(X) on t h e b a s i s of t h e system

v = v(X), v1 = v(X1),

we show t h a t

u.t = s>{\i)hx, A^ = v(X1)H
v0ldxQ[u,

AJ [[X, X,]].

More p r e c i s e l y ,

The c o n v e n t i o n s r e g a r d i n g t h e

d i f f e r e n t i a t i o n a r e analogous t o t h o s e d e s c r i b e d i n S e c . 8. We n o t e , f i r s t of a l l , t h a t because of (23) v_ v

y %\

M^"h»»Mttf-

tn kU

n

= Zt 2i -

K lm

y,

u 2

"' *

—^An—2k)\

n-0lf=0

(hXt)k

***" = h y fc\(k+l)\

n

«.*

y(^^--<•

*"=0

v

ua*\e^

ml — wVlk)e

Zd *!(* + l)| £i

,

' m=0

CO

where

io(-) = X ~ 2 ^ >•/! ; t>••

* t i s important below t h a t

[(i>(AX2)]AA = X2A_1co(AX2), which i s obvious

A«=0

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


Now t h e e v o l u t i o n of v

*t = «"< + ***/ = The i n c l u s i o n

VIA*

+

VI«

V

AA = =^ 2 A 'V

v, due t o t h e H a m i l t o n i a n =,

V

«( IM*X

.

Moreover,

vB = Xv.

N,, i s determined by t h e r e l a t i o n s

+ViA a aJ + vA(vloftAJ. + v laB u x ) = (v„v1Aft+vAvlaA)AJ(. + (vBv1AB +

^Gg^Q[«, A] [[X, X,]] i s e q u i v a l e n t t o t h e i d e n t i t y

wlttJux.

(vBviAA + vAvlBn)B = (vBv1AB + vAvlBB)A,

2

which i s checked d i r e c t l y by means of t h e r e l a t i o n s vAA = X A"'v, vB = Xv and analogous e q u a l i t i e s for

v

71

10.

Other Spaces of Commuting Hamiltonians The sequence of Hamiltonians

{Hn0} constructed i n Sec. 9 i s an example of a general

construction which makes i t possible to form infinite-dimensional spaces of commuting Hamiltonians in the ring 10.1.

Notation. 2

derivatives;

C (A)

R[u, A].

C 2 (a) and

We now develop t h i s construction .

denotes the ring of functions of «; with two continuous

2

C (a, ft) have an analogous meaning.

The l e t t e r s

independent formal parameters; the derivatives of them with respect to zero.

X, X, denote

u, A are equal to

We consider two formal Laurent s e r i e s with a f i n i t e number of negative powers

•f(X)6C2"(a)((X)), ij* and

ifi

T,*(X)6C2
.

Weset

We write if-n"<X), iff =^B(X,);

if ( X ) T J * M = 2 ^ («.*)*'•

have an analogous meaning.

We suppose that

if, iga> %*, ^

vanish only a t

isolated points. 10.2. operator

THEOREM. The Hamiltonians L

tf\

{^(a, A)} commute pairwise r e l a t i v e to the Hamiltonian

if and only if there e x i s t functions

outside a s e t of isolated points and a s e r i e s

V J B ) , V2(h) which are continuous

46R((X)), such that

if, V s a t i s f y the equations

Ktt=-a(l)Vdu)-f, fthh=a(\)V2(h)^. Example.

The Hamiltonians

{H\} are obtained by s e t t i n g

(29)

Vi = l, V2 = A_I, a(X)=X2

(cf.

9.2)." Proof.

As in 9,2 i t suffices to verify t h a t condition (29) i s equivalent to the condi2

tion

r)0/dxC (ut

A)((X)), where the derivative with respect to t i s taken on the basis of

equations with Hamiltonian

Hence the condition

i)i* Ut*=-riihx, ht^%„x .

yfidtdxC'fa

Computing as in 9.2, we find

A)((X)) i s equivalent to the i d e n t i t y

which a f t e r cancelling l i k e terms and a s u i t a b l e division acquires the form «„/')")hf U t t /^) _ I == (^hh^^ihhf^i)"1

-

Firom t h i s

ifc

follows t h a t both sides of the equation are independent of

It, A. and hence have the form a(X)a(Xif1, where ia(X)eR((X)) . f

•»)«<,/-»j«=0(X)V i(tt), i^/if=a(X)V 2 {A), where tf, if,

respectively.

Further, t h i s implies that

V\ and l^2 are continuous away from the zeros of

Obviously, the converse i s also t r u e , and t h i s completes the proof

of Theorem 9 . 2 . 11.

L i f t s of Equations of Evolution Theorem 6.8 i s proved i n t h i s s e c t i o n . 11.1.

B,

Writing out e x p l i c i t l y the equation for At with Hamiltonian

introduced in Theorem 6.7, we obtain Ai,,^iAt+!-xH{i),x+{jAl+j-,H{i))x,

72

H for the operator

Hm=WMr

(30)

I n o r d e r to d e r i v e t h i s system from ( 1 5 ) , we c o n s i d e r s e p a r a t e l y t h e e v o l u t i o n of b a s i s of

on t h e

(15).

11.2.

Using t h e e x p r e s s i o n s f o r h

tit

and

ht

of ( 1 5 ) , we f i n d

h

h

M

At,i-^u'dy~\tu u/ty+u'\hht h

= iluT-(2WH U ) \

dy-

y

-yu^UydyU^ju^Hu^

d-n+u/^rSjAj^Hu^

.

We first transform the second (double) integral in this expression.

6V>°

JX

Sy

( 31)

It has the form

\dy=u'\h\l"2ijuriHw\dy--[u.ity2ijui-Wu)\dy=

\\ B'((2;^«// y) WJ -u'(^ju^HU)\ b'lL

At

\J>°

)x>

0W°

Ix

0

\l>0

lx

=«' i*U^2 y» /_1 ^(^-2 /"'-'IA^A,) - M 2 M ^ U A
b/>°

;>o

J

b

h

\/>o

)

x

h

="' I* ( 2 MJ-IHW) - ji \2 y«'+y-w(/)rft/+C2 y«^//(y, («'),
A b i>°

since the HU)

\oi£i

do not depend on

J>0

lx

y . This coincides with (30).

The derivation of Theorem 6.7 from Theorem 6.8 coincides with that of the Benney lemma 6.2 if it is noted that Eqs. (30) for H— — -j(A2-\-Al)

coincide with the equations in the

Benney lemma, while Eqs. (25) coincide with the origin system (15). 12.

The Benney Integrals Commute 12.1.

XHfifidx^

Theorem 6.9 i s proved i n t h i s s e c t i o n . •

We r e c a l l t h a t i t means t h e f o l l o w i n g :

This a s s e r t i o n follows from a s t r o n g e r f a c t which we now f o r m u l a t e .

shown i n S e c . 7 s e t t i n g

l

n(X) = X — 2 (— l)'ff,\-i'+ \

, we have

As was

p.(X) + $((i(X)) = X, where $(X) =

i-0

CO

= 2 (—l)MjX_(i+1»; the formal series for

!*(*) is uniquely determined by this equation. We

i-0

choose a v a r i a b l e

X

n o t depending on X, and s e t

i t i s p o s s i b l e to d e f i n e t h e e x p r e s s i o n s

XHP

and

(i = [i(X), {M=|J.(X 1 ) . X^ \x

(on

Exactly as i n Sec. 8

X the differentiations

act

trivially). 12.2.

THEOREM,

a ) For any

H = ^ciHJ

we have

73

2

/J f-f

CjHj or H—y.x .

00

by

We

XHA{):

00

1

®, w = 2 (- )^-

(/+I)

i_o

Taking the summation on /

•^(/)=jTr=gx" #

As above, we s h a l l write

multiply formula (30) by (— l)'i*—('+*> and sum (replacing Aiit

f\hf

2(<^w/.*+; M^"«>u y—o

to the o u t s i d e , we obtain

00

00

°°

CO

/—O

f—0

/—°

<—0

We transform the two inner sums on the right:

2 (-1)' (*+y)A+/-u*-
i-0

J

=(- i)VI, [ $([»)-2 (-iM#-<*+4 00

00

\

i-0

<-0

)

"-W-i

K(i i )-2(- 1 ) M *.^- ( * +I) ]-

We now substitute these expressions into (32), multiply the formula obtained by dp/d\ , and use the following identities (Lemma 8.1): t(^.)d[i./dX = —pt = —XH^.,

®x(}>.)d\i/d\=—px


==l—d\>./d\,

. We obtain

-A-W=2^).'K-1)V(1-S)+2(-1)*+/(*+1)^^-*-2S1+ ]-o

L

*-o

J

+ 2 ffu) 1 (- ly (pOr+2 (- ir'Mt.ti^SJUThe first terms in the inner sums give a total derivative with respect to x of the form I 2^(/)(— 1)V I •

We s

^ a H prove that the remainder is zero. After this, part a) of

Theorem 6.9 is rapidly obtained, while part b) follows from the formula V-nj)5%=(-l)i+Vru+1)fe

(Lemma 8.1) which gives w+ li—l/T-'Ifl/ - 27=b PM,) (-iy^=2«sv35 «Ss=57=o

We prove, f i n a l l y , that the remainder vanishes. the coefficient of the remainder for remainder for of the

74

/ / = ( — \)}Hf;

(/+1>

?.f

I t suffices to do t h i s for ff=pi

, />0,

. Indeed,

then vanishes which corresponds to the

t h i s implies the required a s s e r t i o n for any l i n e a r combinations

Hj as well by l i n e a r i t y .

The remainder has the form

J=o

We fix

/-2

oo

7—2

*"« 0"

;'=0

t=0

s > 0 and show that expression coefficient of l a w Iin U this uixa C A J / I C O D M / U the UUC L uciiitxcub u i

du

,d|i

p.* f* £

vanishes.

This

coefficient is equal to

in,.).x(-ir,+ S (-iry(ft+i)A*jti„,., + 2 j—Jt—2—s

'

Substituting here !Aio)=(—l)y+Vr(/+1) dT ' w e

k^"!;)^

2

»—-""

obtain

(-i)* +1 (*+i)A*k w+,) Jf 1 ] x +(-i)* +1 M» lJ r l ir" +1) 3g =(i*r('+1,g£)jr+

(-i)»+I(*+i)i4»i»r(/+,)^!) +( s + ! ) 2

+( 2

(-i)*+^*.^,U).

(-i)*+,^.J«w-w+,,S!=

In passing to the last expression we have again used Lemma 8.1.

13.

Hiura's Conservation Laws 13.1.

-±-\-A^Au

The evolution of a on the basis of system (25) with Hamiltonian / / = - g - / / 3 = has the form a ui = \Al+AoU. + ^-'j —uy[(A0+u2)xdrt 6

=AitX + AQiXa + A0ux+a-ux—yUyAo,x+UyW,

(33)

y

where w = — \(a2)^-/; . (In place of -3-^3 i t i s possible to take c// 3 with any constant 6 1 c; -jwas chosen to simplify the c o e f f i c i e n t s . ) The evolution of p, under the same system has, according to Theorem 3.2 a ) , the form !*/=(—Ai +Au* + t 5-) j t =--Aix+ Ai««* +Ail**+ l*Vjf

(34)

The conservation laws of Miura type of t h i s system are obtained from the following r e l a t i o n s . 13.2

THEOREM.

I

t>= — \Ufd-q, |i' = d(i/dX

[l>> + B)1»-[(4+Ii 2 )l''(l^+Br i l, -[(w-A0,xy)^'(p.

+ « ) " • - ^ ' ] y = 0 , where

[cf. formula ( 2 1 ) ] .

6'

COROLLARY.

of

There exist constants

Rn, Pa, G„,0, G„,u G„,2, such that under the evolution

u, h according to system (25) with Hamiltonian

\_ -3-^3, 3

there are local conservation

laws of the form

Hn. t+F„, x+(Gn0y+Gnlv + Gn2w)y=0. 75

To derive the corollary from the theorem i t i s necessary to expand the expressions under the d e r i v a t i v e sign i n Theorem 13.2 i n powers of A-1, using the formula (p-{-u)'1— 2J (— 1)'M';JI-<'+1\ and equate to zero the coefficient of X~("+l) in the sum thus obtained. 13.3.

Proof of the Theorem.

For the proof i t i s necessary to d i f f e r e n t i a t e a l l three (n + «)2. to eliminate

terms of the expression by the usual r u l e s and multiply the r e s u l t by the denominator. (33),

fit

In the expression thus obtained the r i g h t sides of the formulas for

(34), fi/=2uji'nx+|iW+^o.vH/.

and

2

vy=—ux,wy=—(u )x

ut

are then s u b s t i t u t e d .

An

algebraic sum of f o r t y - s i x monomials i s obtained, a l l of which happily cancel if i t i s noted that 14.

Hy=\ly=0. . Compatibility of the Hamiltonian Structures 14.1.

operator

Theorem 6.7 contains a Hamiltonian s t r u c t u r e on the ring B .

B°=l^ Q!

In 6.5 a Hamiltonian s t r u c t u r e on the ring

was considered.

where A\=hul,

(/

A° = Q[u \

A<^} with operator

The rings A and A0 are connected by a homomorphism

A-^A^'.P-^P0,

which commutes with the s t r u c t u r a l d i f f e r e n t i a t i o n s in A and A0 ; these we

denote by the same l e t t e r d :A\>'>>-+A\>+1), hU)"-ku+1), u.v>>-u(>+i). With respect to i t the evolution d i f f e r e n t i a t i o n s defined.

A =Q[AW] with

Xn'-A-+A

Let G

HQA

and Xw:A ^>-A

0

be any Hamiltonian. can be

The precise formulation of the compatibility theorem 6.11 i s as follows.

14.2.

THEOREM. t o r any

P, QeA we have (XQP)» = XQ,P»

.

For the proof we need the following lemma. 14.3.

Proof. tively.

LEMMA. For any

Let

S1A, QlA°

Q&A we have

be the modules of d i f f e r e n t i a l s of the rings A and

They are freely generated over

8A<", htt^K 8A(/> .

The d i f f e r e n t i a t i o n

A0, respec-

A, A0 , r e s p e c t i v e l y , by the d i f f e r e n t i a l s

d extends to these modules, and the v a r i a t i o n a l

derivatives are uniquely determined by the following conditions:

f u r t h e r , the homomorphism d for which (8Q)o=8Qo .

A-+A° induces a module homomorphism Q1A-+£1A° commuting with in p a r t i c u l a r , (8A/=8i4°=/Aa''- 1 8a+«^A, whence (§Q)°=8Qo=2 iJX>:\°(jhuJ-lhii,+um) +,dufl.

Recalling that hu/~1=A0J^i and the c h a r a c t e r i z a t i o n of obtain the required r e s u l t .

76

8Q°/8a, 8Q°/8A indicated above, we

for

14.4.

Proof of Theorem 14.2.

P\, P2

in place of

If the identity (XQPf=-Xo>P°

holds for a given Q and

P , then it also holds for this Q and Pi+Po. ^1^2 > and

dPx

in

place of P . This follows from the basic properties of differentiations, the commutativity of XQ, X<3», and the fact that the homomorphism reduces the identity for P = Ah

d.

It therefore suffices to verify

t>0.

According to formula (30), we have

XQA = 2 10 + J) AUhX QQ/*Aj)x + jAi+]-UxhQilA}\. /x>

Therefore, according to Lemma 14.3,

/>o = f teM f 2 " ' («Q/Wy)0] + a 1 [ 2 / * " ' - 1 WlMp] On t h e o t h e r

= iAzt'-1 $Q°lU)x + ul (5Q°/Btt),.

hand, ATQ.A?=A'Q. (ha1) = ihul-lXQM

+ u'Xph = Ma'" 1 (8Q°/5A)^.+«' (?Q»/8a),.

This completes t h e p r o o f . CHAPTER I I I SOLUTIONS OF ALGEBRAIC TYPE 1.

Introduction

1.1.

This chapter i s devoted mainly to a description of algebraic structures a t the

basis of e x p l i c i t formulas for c e r t a i n classes of solutions of Lax equations.

These e x p l i c i t

formulas include both solutions of m u l t i s o l i t o n type as well as solutions of quasiperiodic character w r i t t e n in terms of theta functions and also solutions of mixed type.

An invariant

d e f i n i t i o n of t h i s c l a s s of solutions was given in 3.14 of Chap. I I : for nonstationary solutions of Lax equations and the equations of Zakharov—Shabat these are solutions of a comp a t i b l e system obtained by adding to the i n i t i a l equation a u x i l i a r y stationary equation

[Q, L]=0.

Lt=[P, L] or Lt +Py=[P,

L]

the

We s h a l l c a l l them solutions of algebraic type.

1.2. With t h i s i n mind the f i r s t object of study are the stationary equations [Q, L]=0. According to Sec. 4 of Chap. I I , commuting operators are connected by a polynomial r e l a t i o n with constant c o e f f i c i e n t s . This r e l a t i o n defines an affine algebraic curve CoI t s most important c h a r a c t e r i s t i c s are the following: the genus g of i t s nonsingular projective model C ; points of C, lying at i n f i n i t y ( r e l a t i v e to C0 ) and singular points of the curve C0 . The singular points of Co , a t l e a s t when they have the simplest form, i . e . , are double points with separated tangents, are responsible for the m u l t i s o l i t o n component of the solutions of the corresponding nonstationary equations: to each double point there corresponds one s o l i t o n . In the case of genus g = 0 t h i s i s the e n t i r e solution; in the case g>0 it

77

has a quasiperiodic component related to a motion on the complex torus — the Jacobian variety of the curve of functions O on

C . Points of C lying at infinity control the imbedding of the ring

C 0 into the ring of differential operators (in the simplest case the

image of this imbedding is generated by Q and L, where [Q, L]=0) . In particular, they make it possible to relate

the order of operators to the orders of poles of functions at

infinity. One further fundamental invariant of a solution, its rank, is determined in terms of the imbedding of the ring O in S3[d\ . The simplest description of rank is as follows: under weak assumptions regarding O and

5$[d] the order of operators in the image of O,

greater than some constant consists of all multiples of some integer r > l ; this r is the rank.

Solutions of rank

ranks.

1 have now been described much more completely than solutions of higher

For the Korteweg—de Vries equation all algebraic solutions have rank

it Z. = (?2 + ". and Q has a representation as a linear combination £ici{L?i

1, since for ) , where

st

are

odd integers, so that the degree of Q is odd, and the monomials LmQ" beginning from some place onward can have any integral order. 1.3.

In order to clarify the mechanism of the difficulties related to the rank, we

consider the curve C0: ©(x, X)=0(<E»(Q, I)=0 is the relation coupling Q and point cdC0 we write the system of linear differential equations If 53 consists of (/, /) matrices of functions, then height



is a

L) and for each

Q4» =x(c) +, £<Ji = X (c) ty. column of functions of

I . It is found that for almost all points cQC0 the solutions of this system form

a free (right) module Fc

over the constants in Si of the same rank which coincides with

the number r introduced above.

The system of linear spaces {Fc} for each value x can be

equipped with the structure of a vector bundle over C0, which actually extends to infinity. This is proved (for the case /=1 and a nonsingular curve [7].

CQ ) in the work of Drinfel'd

It is probably true in general. In any event, all solutions considered in the litera-

ture possess this property.

The functions $ with respect to c can then be interpreted

as sections of this bundle; the dependence on x is determined by the variation of the bundle. For

r=l

the generating module of solutions described in a suitable trivialization

of the bundle is called the Akhiezer function (see, e.g., the work of Krichever [16] and Matveev [44]).

If a trivialization is not fixed following Drinfel'd [7], then the action

of dx on sections is determined by an appropriate connection V* on the bundle described which extends

g-

and is trivial along C.

Since the space of modules of one-dimensional vibrations over the curve C essentially coincides with its Jacobian

Jc » finding the connection

V* reduces to constructing a

suitable vector field on this Jacobian and verifying its integrability. of the field are found to be rectilinear coverings of the torus

The integral curves

Jc . On the other hand, the

sections of the corresponding bundle lifted to the universal covering of

Jc, are represented

by the classical theta functions. This explains their occurrence in the explicit formulas for the unknown in Lax equations (see the derivation of these formulas in the paper of Matveev [44]) . 78

From t h i s the d i f f i c u l t i e s r e l a t e d to rank r ^ 2 bundles with rank ^ 2

become c l e a r : spaces of modules of

over algebraic curves as well as t h e i r a n a l y t i c uniformization have

been much l e s s studied. 1.4.

In t h i s paper we have t r i e d to get by with a minimum of machinery from algebraic

geometry and a n a l y s i s .

Therefore, for us the basic objects w i l l be special algebraic s t r u c -

t u r e s : the bimodules of Krichever-Drinfel'd which are described i n Sec. 2 and their standard r e a l i z a t i o n i n Sec. 3.

The bimodule technique makes i t possible to reduce the problem of

solving Lax equations to a problem of the v a r i a t i o n with respect to x, t, y

of c e r t a i n

algebraic functions on a curve C, the number of which i s exactly the rank.

For

r=Ithis

problem, j u s t as in other versions, i s solved almost to the very end, and the basic conclusion regarding the r e c t i l i n e a r i t y of the motion on the Jacobian i s attained in a very economical manner.

This i s the topic of Sec. 4 .

In the case of rank r ^ 2 our technique makes i t possible to construct a t l e a s t some solutions of m u l t i s o l i t o n type which we c a l l matrix s o l i t o n s .

The j u s t i f i c a t i o n for t h i s

name i s that e x p l i c i t formulas for them contain exponents of the form where

K\, K2.

and

exp(K\X + K2y + Qt) ,

Q are matrices of rank r, rather than scalars as i n e a r l i e r known

formulas. From the bundle point of view our construction i s motivated from the fact that if a curve of genus zero with s i n g u l a r i t i e s i s considered as

C0 , then a f t e r l i f t i n g the bundle

to a smooth model of C0 i t becomes invariable (a not very complicated theorem of Grothendieck). Therefore, v a r i a t i o n s of the bundles over C a r i s e only due to the joining of fibers a t those points of the smooth model C, which coalesce on

Co .

The space of modules of bundles

e s s e n t i a l reduces to a product of l i n e a r groups over k , and the problem of describing s u i t a b l e vector f i e l d s on them and connections can be handled. I t has been observed repeatedly in the l i t e r a t u r e that s o l i t o n solutions correspond to degenerations of conditionally periodic s o l u t i o n s .

Nevertheless, a detailed algebrogeometric

i n v e s t i g a t i o n of such solutions can provide useful information even in the case

r = l , since

the invariants of complex degeneration are well revealed in the language of the s t r u c t u r e of s i n g u l a r i t i e s of curves but r a t h e r poorly i n the language of theta functions (see the work of Matveev [44] where the simplest degenerations of h y p e r e l l i p t i c curves are treated analytically) .

This i s even more applicable in the case r^2,

where the corresponding "matrix

theta functions" are unknown. In Sec. 5 we construct bimodules o f . a r b i t r a r y rank, and i n Sec. 6 we investigate multis o l i t o n solutions of rank 2. The ring

We s h a l l here describe the simplest case of a single s o l i t o n .

33 consists of meromorphic functions of x, t.

1.5. The simplest Lax equation having s o l i t o n solutions of rank two has the form dtL = [P, L] , where L = dx-\-vd2x +wdx + z, P = udx-{-cdx + u; here u>, egR are constants, and tt, v, w, and z are unknown functions of

x, t .

Writing out the coefficients and eliminating

u , we obtain the equivalent system of equations 79

— ->r' vt = 2-Hj.^ — 2w r — em-1 vx, — " )_1 ^/ = — " « + 2 c^.v + *>»* — o r ' a r , — 2 ^ , - *r% = -j ^Wr + -o- '»-fx - « « + 1 v*w -

c z

^ x-

According to Sec. 3 of Chap. I I , t h i s system can be represented i n Hamiltonian form. Beginning with an operator

L of j u s t fourth order an i n t e r e s t i n g feature of the corresponding Hamilton-

ian of the d i f f e r e n t i a l operator appears for the f i r s t time: i t s c o e f f i c i e n t s depend e x p l i c i t l y on the unknown functions (in the present case only on v ) . t h i s system has the following form. o€R, a&Q .

We s e t

We introduce the a u x i l i a r y

l=x+ct

The s i n g l e - s o l i t o n s o l u t i o n of

and choose an a r b i t r a r y r e a l constant

functions

., ch 2ag —cos 2a| + 2sin 4a2mt ;x, — 4fl'- h 2 a | + c o s 2ai + 2cos ia'sot' „ sh2al-si n 2a| 0 ,-? — <"* Ch 2a| + cos 2a\ + 2cos ia'vt' Then

Z = — 4p<>,„ — Qpixx + 8 &lx)2 + 4 (\HP2)x + fotihtcx —

*f\&\

A special feature of the behavior of this soliton consists in the following: it moves as a whole with speed —c, but changes form with period n(2a2co)~' . At infinity (with respect to |)

the amplitude of these variations decays rapidly, but at times when cqs4a2at=—1,

ejection (in the functions

infinite

Hi, w, z ) periodically occurs at its center g = 0 . Thus, our

solution displays a geyserlike behavior and deserves the name "geyseron" or "shooting soliton" We remark that the vibration of the solution with frequency 4a?co formally occurs due to the same mechanism as the vibration of a free relativistic electron in the SchrBdinger solution of the Dirac equation. Another interesting feature of the behavior of the soliton is that the speed —c is completely determined by the equations while the amplitude (measured by the factor a) may vary arbitrarily.

In Sec. 6 where this example is considered on the basis of the general theory

it is shown that — c is also the common speed of the multisolitqn solutions wh^Lch thus represent a "coupled system" of solitons in contrast, e.g., to the multisolitpn solutions of the KortewegTie Vries equations. 1.6.

Finally, the last section of this chapter is devoted to the description o?f a

special class of solutions of the Benney equations and its analogues.

In the notation of

1.3 of Chap. II these solutions are obtained by adding to the Benney equations the conditions Uy—0, A =^ii£L .where c is any constant after which the problem reduces tp a well known problem.

The character of these invariant manifolds is not altogether clear in contrast to

the auxiliary stationary problems for Lax equations. elimination of the condition

80

This hinders, in particular, the

% = 0 . We remark that since the conservation laws H of

Benney do not depend on the derivatives of the moments An, i t i s here meaningless to apply the technique of r e s t r i c t i n g to extremals of conservation laws, since by the r e s u l t s of Chap. I I the s e t

-==0 6A

i s defined a l g e b r a i c a l l y r a t h e r than by d i f f e r e n t i a l equations.

Questions hereto r e l a t e d merit further 2.

investigation.

The Bimodules of Krichever and Drinfel'd 2.1.

Notation.

In t h i s section

k i s any field of c h a r a c t e r i s t i c

0, 53 i s a central

k -algebra which i s not necessarily commutative, d, dt, d2:33->-33 are three pairwise commuting k - d i f f e r e n t i a t i o n s , and 33[d] i s the ring of in 53, and the commutation r u l e

d-differential

d°b — bd = db .

operators with coefficients

The d i f f e r e n t i a t i o n s

dt and d2 a c t

coef-

f i c i e n t - w i s e on 33[d\ . We fix some ring

O of an affine curve over

of ^ - a l g e b r a i:0-*-33[d\, /(?)=/,„

for any

k .

tpG<9 •

Suppose there i s given an imbedding

Obviously, [Lf, Lq\=0 for a l l cp, <J>g<9 so that

prescribing I i s equivalent to giving an e n t i r e class of solutions of the stationary Lax equations. We s t a r t by associating with the imbedding i a c e r t a i n additional s t r u c t u r e s : the {33, O)-bimodules the imbedding i

fe-linear

space M having a s e r i e s of

of Krichever and D r i n f e l ' d .

We then show that

is recovered on the basis of the given bimodule, and solutions of the Lax

equations and the equation of Zakharov—Shabat are also constructed.

In the next sections

we i n v e s t i g a t e and construct the bimodules themselves. 2.2. Construction of a Bimodule on the Basis of the Imbedding and consider the following s t r u c t u r e s on M . a) $ right.

a c t s on Jl by m u l t i p l i c a t i o n on the l e f t and

b) In M there i s a distinguished element c) In M

i.

We s e t J£ = 33[d]

0 = i(0) by m u l t i p l i c a t i o n on the

1 — the i d e n t i t y operator.

there i s an increasing f i l t r a t i o n

d) The fe-linear operator ^J-.M-^-M, ^7m — d°m acts on Jf

. These s t r u c t u r e s s a t i s f y the

following axioms which are t r i v i a l l y v e r i f i e d : e) The actions of

33 and O commute so that i

f) ^_i = {0}; 33MiczJfl for a l l g) For each free generator of

V

i > — 1 the factor

is a

(33,0) -bimodule.

i. Jti+ilJtl

i s a free

53-module of rank

1; 1 i s the

M0.

h) For a l l b£33, —1,

\7J(tcz-Mi+\ and

V induces an isomorphism of

In other words,

33 -modules

81

A connection of the action of O with the f i l t r a t i o n

{Mt} i s not postulated, but i t c a n

sometimes be described e x p l i c i t l y on the basis of the following Lemma due to D r i n f e l ' d .

2 . 3 . LEMMA. We assume that O has no zero d i v i s o r s , i{p)t-k and that for any 960 the coefficient of the leading power of d of the operator Lv i s not a zero d i v i s i o r in 33. We denote by C the smooth projective e x i s t s a unique closed k -point that for a l l r>\


k-model of the affine curve

Spec0 .

Then on C there

oo, the image of which does not l i e in Spec©, and such

ord,?^ of the operator Lv i s equal to

rordMcp .

The integer

does not depend on 9 and i s called the rank of the imbedding !•• COROLLARY. Under the hypotheses of the lemma MiLy
If, moreover, the lead-

ing coefficient of L/«**i+r 0rd q>—l«

Proof.

The mapping V.O-+Z, v(
v(G)=£{0}, ti(A) = {0}; v(»<{') = '»(?)'+«(min(v(9), v(if)) .

I t i s easy to see that v extends

-1

to the quotient f i e l d of O by the formula «(9<|> ) = z> (9) —tf(),and a l l properties described are preserved.

Therefore,

vdefines

a k-valuation

on the f i e l d of functions on

00 be the k -point corresponding to t h i s valuation. for some
I t does not l i e on Spec©

The group of values of ordoo coincides with

of v i s rZ for s u i t a b l e i n t e g r a l

r ; t h i s i s the rank of

This lemma i s usually employed in the following manner.

Z,

9 with

oogC(A)

requirement means that

since z>(
i. Suppose that we are interested L = d2-\-u

In order to ensure the existence in O of a function

orddZ.(j, = 2 , we must have on C a function with i t s only pole at

In the case

Let

while the group of values

in solutions of Lax equations [Q, I ] = 0 with an operator L of low order, e . g . , for the Korteweg-nie Vries equations.

C .

and the closure of

Spec©

00 of second order.

has no s i n g u l a r i t i e s a t i n f i n i t y t h i s

C i s a h y p e r e l l i p t i c curve, possibly degenerate, and

r=l.

We s h a l l now indicate how solutions of Lax equations are constructed on the basis of the bimodule Jf . 2.4. 2.2e)-i).

THEOREM. Suppose that a bimodule M

i s given with s t r u c t u r e 2.2 a)-d) and axioms

Then there e x i s t s an imbedding i:G-+33[d] and an isomorphism of bimodules 33[d]^J(,

which preserves a l l these s t r u c t u r e s . Proof. We denote by S[V] the ring of k -endomorphisms of the space M generated by m u l t i p l i c a t i o n s by elements S3 and the operator V . The canonical mapping S3-*-33[V] i s an N

imbedding by 2.2 g ) . Any element of S [ V ]

can be represented in the form

^ * ( V ' using the

commutation r u l e V* — bS7=db [2.2 h ) ] . This representation i s unique by 2.2 i ) . There i s therefore a unique ring isomorphism 33[V\—33[d], V"-d, which i s the i d e n t i t y on 53. I t follows from 2.2i) that M is a free 53 [V] -module of rank 1, generated by 1, and that the isomorphism 53[Vl—S[d] extends to an isomorphism of the modules J£=53[Vl 1— 53[d]. We point

82

out that in t h i s argument the

O -module s t r u c t u r e on M was nowhere used.

This is important

for Theorem 2.6 below. For each

cpgO we now define an operator

action of O on Jt .

By the foregoing,

Ly&B [d] by the formula

Z.

l

The mapping f--Lv

is

k -

l i n e a r , has t r i v i a l k e r n e l , and i s m u l t i p l i c a t i v e by 2.2 e) which completes the proof. The next two theorems concern nonstationary equations. 2.5.

THEOREM. Suppose that a bimodule M i s given with the following additional

structures: a differentiation (V, V i ] = 0

di'.S3-^3i which extends to a

and (V, l ) ? = V i ( l f ) for a l l

in 2.4, and we define Proof.

For

P£33[d\ from the equality

We have (£„ — Id-?)1 = 0 a n d

[Vi, Id-?11=0

cpg© .

and [P, Id-9]1=0, while

di -connection ' Vi'-Jt-+Jl,


(V!— P ) 1 = 0 .

|Vi, £
Then
Hence since

[Vi— P, Ly — Id-?]1=0. But [Vi, V] = 0 .

(dilg,— \P, l
Suppose that a

1 is the free

53-module Jt i s given with the structures 2.2 a ) - i )

in the d e s c r i p t i o n of which a l l mention of

O is omitted.

additional s t r u c t u r e s are defined on . Jt :

53 -connections

where V, Vi, V2 commute pairwise. V11=H, V21=P1 .

t i o n by noting that

We define operators

Suppose, moreover, the following Vf.Jt-^-M,

extending dJt j = \, 2,

L, P£53[d] from the conditions

Then (?1P + d2Z. = [P, L].

From the r e l a t i o n

Remark.

Therefore,

over 531G>], by the argument of the preceding s e c t i o n .

THEOREM.

Proof.

with

[Vi— L, V2 — P] 1 = 0 we obtain, as above, the required a s s e r -

[Vi, V 2 ]=0, [Vi, P]=tfiP, [V2. L] = d2L.

Although as i s evident from the formulation of the theorem, the

O -module

s t r u c t u r e on Jt is not e s s e n t i a l , those modules which are constructed by the method of Krichever and Drinfel'd carry t h i s s t r u c t u r e with some O by the very construction. 2.7.

We remark in conclusion t h a t in the ring of symbols

53((?"')) i t i s possible to

obtain additional information on commutative subrings using Theorem 5.14 of Chap. I I .

Let

N

€>,vcS ((;"')) be such a subring and suppose i t contains an element is a

d -constant, i s semisimple, and

tiN-id\iiN, 53]=S~.

L = 2^ufii,

for which ttjv

Theorem 5.14, assuming s u r j e c t i v i t y

OO

of d:5i*-+33*, implies the existence of the symbol 1 + 2 ™-**~* = Q w i t h t h e P r o P e r t y Q~l°L»Q= *=i UN\N . Therefore, the e n t i r e ring Q l°0°Q consists of symbols which commute with UN\N. Now i t i s evident from the r e s u l t s of Sec. 1 of Chap. I I that the space of symbols commuting with uN\N,

c o n s i s t s precisely of symbols of the form ^ vfif, where v£Ker df)33+,

since a l l such symbols commute with terms in the given i n t e r v a l

UN\N , and the subfactors of t h i s space, considering

of o r d e r s , have the required dimension.

This implies several conclusions.

83

a) If the leading term KJV-IGS

-

, and

+

53

UN of the operator L i s a d -constant and is semisimple, if

i s commutative, then any two operators commuting with

L, commute with

one another. Indeed, then the ring (Ker
3S=Mt(%),

au =diag(Ci, .. .,ct), ct=£Cj for

i^=j-

1

In t h i s case the e n t i r e ring of symbols commuting with u^ *, consists of diagonal matrices of symbols with constant c o e f f i c i e n t s .

Therefore,

/ " p a r t i a l " order functions are defined

on i t : the orders of the symbols a t the s i t e ( i i ) ,

1<£
The argument of Lemma 2.3 can

be applied to each of these separately, so we obtain the following r e s u l t . b) If in the commutative subring Ocz3S[d\ with ^ == Af4(.^0X there i s an element L with invertible there are

«Af=diag(c 1 ,.. -,ct),
Ci=f=Cj, UAT-IG®-, then on the smooth projective model of Spec©

i n f i n i t e l y d i s t a n t points the orders of the poles a t which are determined by

the p a r t i a l order functions of the symbols of the corresponding o p e r a t o r s . In the c e n t r a l i z e r

L in 53((?-1)) the p a r t i a l orders do not depend on one another, but

in O they may be r e l a t e d and even completely determined by one of them. the points a t i n f i n i t y there a r e not necessarily 3.

Therefore, among

/ distinct points.

The Standard Realization of a Bimodule Over a Field 3.1.

In t h i s section we assume t h a t the basic d i f f e r e n t i a l ring

53 i s a f i e l d .

This

does not reduce appreciably the generality of the r e s u l t s , since for almost a l l i n i t i a l cond i t i o n s the solutions of the s t a t i o n a r y and nonstationary Lax equations are l o c a l l y a n a l y t i c , and we can seek them i n the f i e l d of germs of meromorphic functions. applications i s R or with the

C ; i t i s assumed to be a l g e b r a i c a l l y closed in

d -constants in

The base field

k in

$3 and to coincide

S3 .

Let 0(Z33[d\ be a commutative subring containing

k .

I t contains no zero d i v i s o r s .

The orders of i t s elements form a semigroup; hence, there e x i s t s an integer «o> such that a l l those orders greater than «o> form an arithmetic progression with difference the proof of Lemma 2.3 i t i s c l e a r that Jf=ffi[d].

The semigroup of orders of

r . From

r coincides with the rank of the (S3, O)-bimodule O is f i n i t e l y generated; choosing i t s generators and

operators of the corresponding orders we find that the ring 6 i s f i n i t e l y generated over k .

I t i s thus the ring of an absolutely irreducible curve.

Let

oo be the point of i t s

smooth model defining the orders of operators of O as in Lemma 2 . 3 .

We s h a l l assume that

the leading coefficient of a t l e a s t one element of O of nonzero order i s a constant.. From formula (1) of Chap. I I i t then follows that a l l leading coefficients of elements of O are constants. In t h i s section we consider i

as a l e f t

53®0=530-module with action

(bf)m = bmf,

k

bgB, m£<M, /GO, and we r e a l i z e M as a submodule of vectors i n the r

ordinate space K

over the quotient field

the standard r e a l i z a t i o n . 84

K of the ring

330 .

r —dimensional co-

This r e a l i z a t i o n i s called

In order to describe i t and carry over to i t s language the basic s t r u c t u r e s r e l a t e d to M , i t i s convenient to use the elements of the theory of algebraic curves. is the field of functions on an algebraic curve over with f i e l d of constants extended to

53 .

The points of the field are l o c a l rings in K con-

53 .

Spec©,

are called constants and the remaining points are v a r i a b l e s .

Those f i e l d points which come from points of the complete smooth model of

constant p o i n t .

The principal part of an element

/ modulo the maximal ideal corresponding to Any k - d i f f e r e n t i a t i o n of

dO={0} .

K

53— the smooth complete model of Spec©

taining

of

The field

The same l e t t e r

fdK

a t a point

For example,

oo i s a

P i s called the class

P.

53, in p a r t i c u l a r

d, extends uniquely to K by the condition

d i s used to denote the coordinate-wise action on

K''.

We can now formulate a theorem on the standard r e a l i z a t i o n . 3.2. r

McK

THEOREM.

The bimodule Jl = 3&\d\ i s canonically isomorphic to the 530 -submodule

with the following p r o p e r t i e s : a) (530)rcJ£; the factor space b) The element

Ml{?BO)r i s finite-dimensional over 53.

\^Jt i s represented by the vector 1=(1,0,.. .,0)'g.^.

c) Let .W(i) denote the subset of elements of M, the j - t h coordinates of which have poles a t

l

oo of order not exceeding

^(i) =(«^i)~ for a l l d)

i>i0,

where

The connection ^:M->M

\£Mr(K)

~'

with leading d .

form a

53-basis of

330 -module M has no t o r s i o n .

(^!b n fn)m = ZJbntnfn of that

[j^bnfnjtn

.

O .

If 2&„/„=£0

is

any

Choosing the g r e a t e s t

i s equal to the leading term

M has no n o n t r i v i a l t o r s i o n .

V =
e emen

l

t

«o with

Indeed, suppose

For each n(
Clearly, the elements {/„} form a k-basis

£0.

I and

r>2)

T i s the semigroup of the orders of elements of fn^P

all

i s induced by a connection on KT of the form

We note f i r s t of a l l that the n

Then M(i)
t0 i s a s u i t a b l e constant.

i s a matrix of the form (for

Proof.

, y'=0, .. . , r — 1 .

of

of

O .

53(5 and m£J(,

b^^O,

They therefore

m^O,

then

we find t h a t the leading term

*„„"*/„„> i . e . , i s different from zero.

This means

This implies t h a t the canonical mapping JI-+K ® M:

mo m^*\®m.

i s an imbedding. K®-M = Kr .

We e s t a b l i s h the isomorphism elements of

Y are d i v i s i b l e by

r,

We s e t M' = 5 3 1 0 + . . . + 53V~11©.

the orders of the elements of /

to ymodr , and therefore the sum of the 53V 1© i s d i r e c t . of M' include a l l i n t e g e r s greater than some space Mn

i s finite-dimensional over

Since a l l

83S7'\0 are a l l congruent

Further, the orders of elements

/to. and therefore

M = M' -\-Jini.

Since the

53 , the 53© -module Ji/JC i s a t o r s i o n module. Indeed, 85

if

tion of the form ^!>&/'£.#',

i.e., Jl'-+Jt

Thus, the imbedding K®M'

^bj1

{ V y l | 0 < / < r — 1} (more precisely, {l^V^l}) • We identify

with the space Kr

and K®M

mmodJC.

is annihilated by

induces an isomorphism K®M'' -*K®M . In view of the above,

has the canonical basis

K®JC'

of columns of height r with coordinates in K by

This defines the canonical imbedding Jt->Kr.

means of this basis. Jt.

m£Jl, then there is always a nontrivial rela-

/G
We denote its image by

r

The inclusion (33Q)
already been proved: (330) space Jl^M'^M^.

is the image of Jt',

r

while Jll(330)

is isomorphic to the factor

The element IQJC is represented by the vector (1,0,...,0)' .

The filtration with respect to order is easily described on (*o/o> • • •»^r-i/r-i)'' (bfi33, ffcO) if and only if

not exceed — —

bj=£Q.

for

we may remove the condition Thus Jfwcz(Jfi)" tary to (630y .

j — rordM/;
to the vector

by Lemma 2.3.

(b0/0,...,

Therefore,

, i.e., the order of the pole of f}

at oo does

Taking the order of the pole of the zero element to be — oo, bj^O.

. In M we now choose a finite-dimensional subspace over 93 complemen-

The orders of the poles at oo of all the coordinates of elements of this They are therefore contained in Jt . for the sufficient-

subspace are uniformly bounded. ly large i.

(330)r:

there corresponds the operator 2 */V'l//» the order of which

is equal to max(y'+ordZ./.l ^/^O) =max(y— rotdoJj\bj=^Q) br-.1fr-iyG(Jti)~,

over 33 have actually

This implies that J£(/) = (^i)* for sufficiently large i.

We have now verified asser-

tions a)-c) of Theorem 3.2. In order to establish d) we note first of all that the connection V extends uniquely to a connection \7:K®Jt-+K®Jt

with V (f®m) = df®m-\-f®S7m

for all f&K,

m£Jt (this is

a standard fact regarding the extension of connections on a localization). With the identification of V — d

difference Since

/

is a

K®Jl

with K'

we obtain a connection T

/C-linear mapping K'-+K .

S7'-Kr-*Kr.

The

Let A be the matrix of this mapping.

+1

V(V 1) = V' 1 , we have on setting e, = (0.. .0 1 0.. .())< a representative of

VJl

in Kr,

i (tj+i

for j
— 2,

2 V A W ) f°r J = r-\.

V-o

Therefore, A has the form indicated in the theorem. 3.3. Remark.

The standard realization of Jt

This completes the proof.

shows that the matrix A, or its last

row (A0, ..., Ar_j) is essentially the unique invariant of Jt

(for given S3 and O ) : Jt

is

OO

recovered from it as the connection tions that M

V-

^33(d+Ay(\,Q,.

..,0)', together with the filtration, the action of 33 and

However, (A0, ..., A,_,) cannot be chosen arbitrarily, because the condi-

be invariant with respect to multiplication by

and that the filtration of Jt

O , have finite type over O ,

be described in terms of the behavior at oo, as in Theorem

3.2, impose strong and nontrivial restrictions on A . We shall occupy ourselves with them

86

in the next s e c t i o n s .

Here we note only t h a t part of these r e s t r i c t i o n s have the character

of d i f f e r e n t i a l equations: of

A i s to be considered as a c o l l e c t i o n of functions on the model

Spec©, depending on parameters x, (x, t), (x,u,t)

and satisfying c e r t a i n d i f f e r e n t i a l

equations in these parameters. 4.

Bimodules of Rank 1 4.1.

In this section we give a classification of bimodules of rank 1 which satisfy two

additional conditions. We can formulate the first one immediately: it is that the ring 530 consist precisely of all functions of the field K, having a pole only at the point oo . This condition is equivalent to the condition that complete model by excising the point at

oo.

Spec© be obtained from its smooth

The second condition will be formulated later.

Xv

We fix Jl and i t s standard r e a l i z a t i o n

Jt(*K.

Let Afc/X" be defined as i n Theorem 3 . 2 . We c a l l a pole of

M any point

f(*Jl, having a pole of order order of the pole of 4.2. field

THEOREM,

P of the field

> 1 at

f(*Jl a t

P.

K, for which there e x i s t s an element

We c a l l the order of the pole P

P (oo i f there i s no g r e a t e s t o r d e r ) .

a) There e x i s t s a f i n i t e number of nonconstant points Px,...,Pt

K and positive integers ax,..., as such that

having poles of order < a ;

at

P;(l
b) The degree of the divisor

and a pole of any order at

D = £alPt

a l l the points Px, ..., Ps, oo and such t h a t

i s equal to the genus

Ph oo, r e s p e c t i v e l y .

respective forms

Proof.

Pit oo.

z—2(co)

oo.

g

of the field

K-

O , which i s f i n i t e a t

z— z(oo) are local parameters a t the

Then the principal part of

-[J \p' y +£;•

classes of the points

z — z(Pt),

of the

M consists of a l l functions of the field

c) Let z be an element of the field of fractions of the ring points

the greatest

A a t the points Pi,<x>, have the

+ c °°' where cu c«= l i n e in the fields of the residue

The divisor of the poles of A i s precisely

Let P be any point of the f i e l d

K,

and l e t

Op

oo +

2 ^

be i t s local r i n g .

It is

oo

known that d takes Op poles of M

into itself (cf. [20, Lemma 2]).

Since •M'='2i
must be contained among the poles of A, so that M

We shall first show that at oc M

has a finite number of poles.

has a pole of exactly first order. If A had no pole

at oo then neither would Jl, and this would contradict the inclusion S30cz.M. of

A

at oo were of order

precisely

ja

a > 2 , then the order of the pole of

(an easy induction on j using the fact that

of a pole at a constant point).

But

(d-fA)-'l

d

must coincide with

would be

does not increase the order

corresponds to the operator di^[d\=JC,

by Theorem 3.2 c) the order of the pole at oo of the function large j

(d-\-L)i\

If the pole

/ . Therefore, the case a > 2

(d^-A^l

and

for sufficiently

is impossible.

87

Suppose now t h a t A and hence of shows t h a t

P=£<x> i s a constant p o i n t .

Jl .

We s h a l l show t h a t i t cannot be a pole of

Indeed, otherwise, the same argument as in the preceding paragraph

JC contains functions with a pole a t

i s f i n i t e l y generated over

P

of a r b i t r a r i l y higher o r d e r .

530, while elements of SBO have no poles a t

P

But M

at a l l ;

this

i s thus impossible. We now consider a nonconstant pole Pt of the module otherwise, as above,

M could not be f i n i t e l y generated over

3BO have no poles a t

Pi .

the f i e l d of f r a c t i o n s a t feM

.

Since

Suppose t h a t at>l

I t s order i s f i n i t e ,

for

SBO , since the elements of

i s the order of t h i s pole.

We choose z

in

O, as in part b) of the formulation of Theorem 4.2 and an element

with a pole of order

((z — z(Pi))~"i+1)

,M .

at

at

Pi .

I t may be assumed t h a t

(d + k)ff<.M must a l s o have a pole a t

Pt

—z(Pt))~aur

f=(z

0

of order no greater than

o-i, and aidz(Pi

0 + A)/-A/+

we find t h a t the expansion of A a t

\l+l

0{{z-z(Pi))-a^,

+

Pi must begin with —

gifg\pr

s

We set D^^a^i

and denote by X(D-\-joo)

in the linear space over SB of all func-

tions of K, the divisor of the poles of which does not exceed

D4-Joo.

According to

Theorem 3.2 a) and c), for sufficiently large /' we have OC(joo)czJijC%(D+Joo). .A(j=Z(D + joo)

for / > 0 , since in Jt}

Therefore,

there are elements with principal parts at P, of

order exactly at, and taking linear combinations of them with suitable coefficients in 530, we can make these principal parts whatever we wish. Now the dimension of Mj

over SB is equal to y' + l, while the dimension of

for y > 0 by the Riemann-Roch theorem is equal to degZ) — g + l+jdeg a point of first degree and AegD=g.

oc.

Therefore, oo is

This completes the proof.

4.3. We now formulate the second condition imposed on oar bimodule divisor Pi+...+P,

£(D-\-Joo)

•$>. It is that the

defined in the preceding section be nonspecial, i.e., there exists no

differential of first kind that vanishes at Pi+'...+Ps» nonconstant function

But then from the existence of a

A with a divisor of poles Pi +...+Ps+co

g, ft,=l; the function

it follows that

deg(SPj) =

A is uniquely defined up to an additive constant.

The divisor Pi + ...+Ps

x=x0,

is clearly nonspecial if this is so at the initial point

and the set of such initial conditions is dense in the g -fold symmetric product of spec 0. We shall now show that the conditions on A of the Jacobian coordinates of Px-\-..

.^P,

become a condition on the linear variation

with respect to X in the functional case. We

shall assume that SB is the ring of germs of meromorphic functions of X Pt

as germs of holomorphic paths on the Riemanri surface of Spec©, parametrized by

may be assumed that they are distinct, i.e.,

88

and represent the

s=g.

x.

It

We choose a basis of d i f f e r e n t i a l s of f i r s t kind O ,' a function 2 as in Theorem 4.2 c ) , and set points Pi, ...,Pg.

We set

/j = ./* (JC) = 2

\ mt.

">,, ...,10^ in the f i e l d of fractions of

«!>i = Uidz.

The functions

We define the element

ut are f i n i t e a t the

b&i from the condition

7—1 oo

A=

r- Z (oc) + Q < 1 ) 4.4.

near

THEOREM,

the divisor

°°-

a) dxJi = bU:(oc) for a l l

i = l,...,g.

P\-]r • •••\-Pe move in a constant d i r e c t i o n

Hence the Jacobian coordinates of (Ui(oo)) with speed

b(x) a t the point

x . b) Conversely, suppose D = P i + . . . + P e is a nonspecial divisor of the field K, Jacobian coordinates of which vary with x as in part a) with some function

the

b£33 . Then

there exists a unique (up to a constant of 33 ) function A with principal part

z_z,00\

oo

at

oo and divisor of poles

D+oo,

and £ 33 (d-\-&) dl i s a

(53, ©)-bimodule.

7=0

Proof,

a) For any d i f f e r e n t i a l

be poles of «;A

only a t the points

u> of the field Pu...,Pg,

K we have ^res e (joA)=0 . •

oo, so that

*

There can

Q

,

]£,• res p (i»jA) = —res^^A) . 7=1

°°

'

Therefore, £

£

dxJ'i^^iMP'j)oxz(P;)=2

S

res p .

y-i

7-1

Ui

f_f{P)J

'

7

7=1

b) Conversely, suppose that Pi + . ..-\-Pg dji

dz = - ^ resp (^A) = resoo(oJiA) = tei(oo). varies as in part a).

read in the reverse order shows that at the points Pj ^ ^

with

Then the calculation of

the principal part of A

, since the matrix («;(P/)) is nondegenerate because Pj+.. -+Pg

begins

is nonspecial.

Arguments analogous to those given in the UX(PX 7>o

proof of Theorem 4.2 show that 2 33(o +... +P e -\-J<x>), and all the axioms of a bimodule for this space are verified without

difficulty. 4.5. field.

Theorem 4.4 gives almost a complete c l a s s i f i c a t i o n of bimodules of rank 1 over a

In order to apply i t to find solutions of nonstationary equations, i t is useful to

have in mind the following s i t u a t i o n s . a) Let us assume that 33 i s a ring of germs of functions of ary Lax equations we must extend

dt to M

of V, must have the form dt + kf where kfijt

.

by the condition

&t€K .

The commutation condition has the form

Since

x, t .

[V*, V/]=0.

To solve nonstationThis extension

(dt + bt)\eJl , we necessarily have

dxkt= — dt&.

uniquely determined by i t s principal part a t oo (because

Since each element of Jl

Pi-\-... +Pg

is

is nonspecial), A,

X

can only have the form — )\ dtkdx + (an element with a

^-constant principal part at

On the other hand, since the connection V< must take A

oo ) .

into itself, the behavior of A,

near P;- is determined by the same sort of conditions as in Theorem 4.2 c). The argument used in the proof of Theorem 4.4 shows that the motion of Pi+.-.+P* with respect to t also becomes rectilinear in Jacobian coordinates; its direction and speed are determined by the principal part of A, at oo.

The same is true for the motion in y for solution of the

Zakharov—Shabat equations.

89

jArfx

b) We s e t

x

$=e '

.

Then for any m^M we have \(dx + K)m]-b = dx(rti<§ ,

can also be r e a l i z e d as the space

Therefore, M

Jtif, to which a l l s t r u c t u r e s carry over i n an obvious

way, while the connection Vx goes over in to ordinary d i f f e r e n t i a t i o n with respect to x . The function

f i s called the (stationary) Akhiezer function.

Prescribing t h i s function i s

equivalent to prescribing A and hence the bimodule Jt i n the standard r e a l i z a t i o n .

In the

papers of Krichever [16] and Matveev [44] t h i s function i s taken as the i n i t i a l object.

If

A does not depend on t nor A, on x , then i n the nonstationary case the function ty(x, t) = expi \ Arfjc + 5 bjAt J .

On M§ the connections 'Vx and V* become dx and dt,

respectively.

c) Since A, A* a r e determined by t h e i r divisors and principal p a r t s a t oo, they can be written out e x p l i c i t l y i n terms of the c l a s s i c a l Riemann theta functions.

The coefficients

of a l l operators which enter i n the s o l u t i o n of Lax equations a r e expressed i n terms of A, At and t h e i r d e r i v a t i v e s .

This leads to e x p l i c i t formulas for the s o l u t i o n s .

For further

d e t a i l s we refer the reader to the papers [16], [ 4 4 ] , and Sec. 6 of Chap. IV. .5.

Bimodules of Higher Rank dver a Rational Curve with Double Points In t h i s section a c l a s s of bimodules i s constructed which may have a r b i t r a r y rank r ^ l .

For the motivation see 1.4 and 2 . 2 . The r e a l i z a t i o n s given here a r e close to the standard r e a l i z a t i o n s but do not coincide with them. 5.1.

The I n i t i a l Objects.

We s e t & = R or C, and we l e t 3$o be a r i n g of germs of

k - a n a l y t i c or i n f i n i t e l y d i f f e r e n t i a b l e functions of the variable x (for s t a t i o n a r y Lax equations), of (x, t) (for nonstationary Lax equations), or of (x, t, y) (for the equations of Zakharov-Shabat); $ = Af,(5?n) (the matrix algebra of order / over 530); d = dx = dldx,
d2=dy=didy. To construct O we choose 2N d i s t i n c t numbers ah f^eC, i — \,...,

a,=JFj

i f k=U

N with the condition

and we s e t 0={/,(X)6A[M|v; = l, ...,N,

(1

/(«>)=/«

>

I t i s obvious t h a t O i s a r i n g of functions on the affine l i n e with N pairs of identifiedpoints realized as a subring of the functions on the l i n e i t s e l f , i . e . , &[>•]• Similarly, we construct M as a submodule of the t r i v i a l

(53, *M)-bimodule

JV=3$\\\r

of rank r with i d e n t i f i c a t i o n conditions a t the points (a*, P,) . We s h a l l write elements of 53r and jr = §imr as columns of height in

53, 53 [X], respectively ( X commutes with 53 ) .

by the formula

90

r with coordinates

We introduce on Jf the l e f t action of 53

where t

S = Af/(,<S0)-

is the transpose in

We define the right action of

ordinate-wise multiplication on the right by GL(r, 53) .

k\k\

on JV as co-

£[M^r » where Er i s the identity matrix in

Finally, we introduce on JV the natural left action of the group GL(r, 53).

Obviously, the actions of 53 and k\k\ on JV are effective and commute with one another and with the action of GL(r, 53). The identification rules defining M

are described by a collection of N

g;gQL(r, C®53), on which additional conditions will later be imposed.

matrices

Having chosen this col-

lection, we set M = {m(\)£
k=R

N, m(ai)=gim(^)}-

pV-=a<, this implies the condition

and

(2)

gi=gf ! .

The remainder of this section i s devoted to describing those conditions on (*/• p\)>

(gt) and

which enable us to introduce on JC the bimodule structures with the axioms of 2.2.

5.2. on JV.

The Actions of

5? and

O.

We have described above the actions of k\K\ and 53

I t i s obvious from definitions (1) and (2) that

of 53 and

O on M commute.

The F i l t r a t i o n .

and the actions

That these actions are effective will become clear below

following construction of the element 5.3.

SSJtttzJl, JtGaJt

l£M.

We f i r s t describe an auxiliary f i l t r a t i o n on JV by setting

^f_1 = {0}, and for any a>0, 0 < £ < r ; l bi

( \ JG53[X]', b ...,b u k

* /,/ "ar^ = )j "

are polynomials in

X of degree


; (3)

^*+i, • • •, br

are polynomials in \ of degree < a— 1}. 00

(A polynomial

of degree

It is obvious that Ji"c:JVui

< — 1 is zero.)

and Ji"=\JJl", . 1=0

/,

r

Further, ^ ,Ti = " (©53eJ+i, where el+1 has and zeros elsewhere.

Therefore,

JVul/jVl

a

X Er at the s i t e

k + l for l = ar + k, a>0, 0<&
is free of rank 1.

The f i l t r a t i o n we need on M will be induced by this f i l t r a t i o n on JV and the followoo

ing s h i f t .

We temporarily set

!

Mw=J\ 'k(\dl

.

I t is obvious that J(=[J^ik)

We impose the following conditions on the collections 5.4. i
The Nondegeneracy Condition.

and 3HM{k)aJ(w.

(gi), (a,-, Pj) .

The block matrix in MrN(33): G = {a.[Er — ^[gi), 1<

0
i
Lemma.

If the condition of nondegeneracy 5.4 is satisfied, then Jtw = {0} for

and the natural mapping <MlJr\\Mi-+J\'*lJ:XiJVl is an isomorphism of

l>rN + l

53 -modules for

.

The f i l t r a t i o n g where any

J^=J£(*+rAr+i)

for k> — 1 therefore satisfies conditions 2.2e, f, and

53-generator of the module Jf(r,v+i) may be taken as

1 .

91

Proof.

According to ( 3 ) , Ji"rN = { 2

=

m k

i ' Imfi®>T\'

Therefore, by (2)

m lJ

2

lXJV-1

J I V/ = 1, . . ., iV, 2 W*r~ P/Sl) % = 0 • y=0

J

The condition i n braces may be w r i t t e n

Since G i s nondegenerate by hypothesis, i t follows that Jt^N) = {^} (and hence jt(t)—0

for

*
is a

33-isomorphism for

l>rN,

£,+i of the form et+i =s £/ mod Jft,

it

suffices

where e^i

is

described i n 5 . 3 . To find

ei+i = 2 O T A ;

we

™"st write down the system of l i n e a r equations for the nij, cor-

responding to the conditions ( 2 ) .

After displaying them over the commutative ring

$?o we

find i n the system a nondegenerate minor of maximum possible rank flN corresponding to the matrix

G.

The system i s therefore solvable.

5.6.

The Explicit Form of

1.

From the proof of the preceding lemma it is clear that

we may set _y-i

AN

l=\.m;\J+l° where the

ttij form a solution of the system

5.7.

The Connection

V = VX .

(i = \,

J, m&P,

W

...,N):

The following r e s u l t s with obvious modifications w i l l

also be applied to the construction of the connections Vi = V«, V 2 = V y . We f i r s t continue dx:3?,->- 63 coordinate-wise to t

Inasmuch as dx(blb 2) = dx(bi)b'2+b1(dxb2y Any other connection extending r

EndS[A.]

.

( T r i v i a l i t y on

for

63r and then to 8tf\l\

bu b2(<33 , t h i s action i s a

dx and t r i v i a l on

X, has the form

X gives the condition Vx(m'j)={S7xm)y

so that

dx\=0.

dx -connection on J" .

dx + dx, where dxGM,(53\)<]) = for

meJ",
We use the freedom in the choice of d.{ to ensure the condition Vx~MaJ(. 5.8.

LEMMA. Let

x

7x = dx + dx(l).

Then S?xM
V/ = 1,...,A', d^-g^AM-dA'dSi-

92

(6)

Proof. According to (2), m(\)£Jf,

if and only if m(ni)=-g,mQi)

N ,

for all i = l

whence (djn) (a,) = dxgim (&) + gidxm (,3,.)Tlierefore, \(dx+dx)

m\ (a,) = dxgirn (fc) + gtdxm (fc)+rfx (a;) g,/» (p,).

On t h e o t h e r hand,

Si Wx + dx) m] (M = g t d,m (f>,) + g
S o l u t i o n s of E q s . ( 6 ) .

t i v e l y with

[dx{at,x),

commute r e s p e c dx{a.i,y)\

*=[dx$„x),

(Jf)=exp ( - Jrf^a,, E)rf6j gl (0) exp ( j <*,(&, ?)rf?J.

I n t h e examples we s h a l l t a k e

5.10.

(x) = e x p ( - JM*, (a,)) gi (0) exp (xrf,(p,)).

Tlie Connection of Then SJMiCM^

Vx with the F i l t r a t i o n i n

for a l l

if the following conditions are

/>—1

where d0x£Mr(5?)

Ji.

(8)

LEMMA.

Let

and V* i n d u c e s an isomorphism

S/xJKczJ(,

Vx=

JillMl_\'^.MlJ!xlMl,

satisfied:

for r = l : dx = d0x+dlx\, for

(7)

dv(X)6Mr(M,(fe[X])), and t h e n gi

.

and dx($t,x)

Then an e x p l i c i t s o l u t i o n of (6) can be w r i t t e n i n t h e form

gi

^x rdx

dx(a.it x)

e x p K r f ^ a , , S)rfM m d e x p K d x ( M ) d S K t h i B I s BO I f

dx($i,y)]=0).

J

We assume t h a t

rflre$*=GL(l,33);

r > 2 : ^==rf 0t . + rfi.v>.,

h a s z e r o s on the main d i a g o n a l and above and i n v e r t i b l e elements ( i n 53*)

a l o n g t h e d i a g o n a l s below the main d i a g o n a l , w h i l e

dlx^Mr(3S)

h a s an i n v e r t i b l e element i n

t h e r i g h t upper c o r n e r and z e r o s e l s e w h e r e . Proof.

Since

Jfl = JCf\^"i+rN+i

and Jll+llJll^J''i+rN-\-2l'd/'i+rN+i

v e r i f y t h a t under the hypotheses of t h e lemma V x ^ ' i C ^ + i and JV^I^^^t+ol^i+i since

.

for

L>rN,

i t suffices

to

V x induces an isomorphism

But t h e a c t i o n of VA- on t h e l a s t f a c t o r c o i n c i d e s w i t h t h e a c t i o n

dx,

d ^ ' j C - ^ j , and t h e r e q u i r e d r e s u l t i s checked by d i r e c t c o m p u t a t i o n . We have now completed t h e c o n s t r u c t i o n of t h e c l a s s of bimodules

t h e s e c t i o n i s devoted to some f u r t h e r 5.11.

The Rank of

M.

n , then

Jt-ficMi+w,

The r e s u l t of

remarks.

I t i s e q u a l to

r

i n the sense described i n the introduction

and a l s o i n t h e sense of Lemma 2 . 3 and i t s c o r o l l a r y . degree

M .

and m u l t i p l i c a t i o n by

Indeed, i f


= induces t h e isomorphism

Jt,/Jt,_i~ 93

•MUrnl'Mi+Tn-\

•

I t suffices to verify t h i s on the corresponding f i l t r a t i o n submodules of

J where a l l i s obvious from the d e f i n i t i o n s . 5.12.

The Connections

Vt, V y .

We s e t S7t = d(+dt,

V , = d , , - f rfr For solving nonsta-

tionary equations these connections must s a t i s f y conditions analogous to (6) but not Lemma 5.10.

In place of i t we need the

(cf. Theorem 2 . 5 , 2 . 6 ) .

commutation

conditions

IVX, Vt] =[V JC . Vy] = [V<, Vy]=0

They can be w r i t t e n i n the form dxdt-dtdx+[dx,dt]=0

and s i m i l a r l y for the remaining p a i r s . [^JC>^I=0

etc.

(9)

In the case dx, dtf>Mr (Mt{k[k])) they become simply

If these conditions are s a t i s f i e d ,

then the matrices

gi(x, t) for the Lax

equations have the form gi(x,t)=exp(-xdx(at)-tdt(lxi))xgi{0,0)exp(xdjctfi)

+ tdt$i)).

(10)

Conditions (9) with constant dx, dt, dy are most simply satisfied by taking dt, dy to be polynomials in 5.13.

dx . The Order

of the Operators. According to 2.3 and 5.11, the order of Lv, is

equal to rdeg-f . It is not hard to see that the order of P, found from the condition V/1=/>1, does not exceed

rdegdt

(degree in X ) and similarly for

Vy . These considera-

tions determine the choice of O, V/ in the next section. The case where O contains polynomials of degree 2 and r=l. leads to multisoliton solutions of the Korteweg-nde Vries equation and its higher analogues and has been treated in the literature in various ways (by the method of inverse scattering, the method of Zakharov-Shabat, Hirota's method).

The next most difficult case is the case r = 2.

6. Example: Solitons of Rank 2 6.1.

Parameters of an Ar-Soliton Solution. We shall construct an Af-soliton solu-

tion of a nonstationary Lax equation of the form Lt<=\P, L] , where the order of L is equal to four and the order of P is two.

The parameters of the N -soliton solution are con-

stants which go into the construction of the appropriate bimodule M. We take £ = R, SS — Sio be the germs of meromorphic functions of x, t of solitons N

R ; the number

is the number of pairs (*> PA) of dual points.

In order that O

contain a polynomial of degree 2, it is necessary and sufficient

that <*ft+P* not depend on for all k;

over

a

k . Replacing

X by

/.-[-const , we may assume that

a

ft+P* = 0

2

then X g<9 . Real a* lead to nonlocalized solutions. We therefore take a*

to be pure imaginary.

For convenience of subsequent computations we set o.k=2ia2kb, $k—-

— 2ia\b, ak, b&R, where b is chosen as follows. According to Lemma 5.10 and (8), the connection V ^ has the form u Q), a, *6R* • Replacing X by may assume that 94

dx-\-dx,

where tfx =

ar'X and modifying the value of ak, correspondingly, we

a—1; b remains free.

We s u b j e c t the c h o i c e of have o r d e r 2 .

V , to t h e c o n d i t i o n t h a t t h e o p e r a t o r

According t o 5 . 1 2 , i t s u f f i c e s t h a t for Vf = dt-\-dt t h e m a t r i x

X l i n e a r l y ; moreover, f o r c o n s t a n t c o e f f i c i e n t s of V<

commute l e a d s to t h e c o n d i t i o n

term

P, f o r which V , 1 = P 1 ,

dt=
dx

and

dt

dt depend on

the condition that

where (»GR, CGR*

( t h e a priori

V* and

admissible

0E2, «>0GR, does n o t l e a d to a l t e r a t i o n of t h e s o l u t i o n ) . To c o n s t r u c t t h e i d e n t i f i c a t i o n m a t r i x gk(X, t), which f i n a l l y d e f i n e s t h e bimodule

i t i s n e c e s s a r y t o f u r t h e r choose i n i t i a l c o n d i t i o n s gk(0, 0)GGL(2, C) . 1

must have g*(0, 0)=g f t (0, 0)" . We f u r t h e r s e t

We r e p r e s e n t

$ = b(x + cf),

T=
gk(0, 0) i n t h e form

M,

According t o 5 . 1 , we

gk(0, 0)=TT1-T*. T*eQL(2, C) .

Then formula (10) becomes

gk(x,t)=Gt(x,trGk(x,t),

(11)

Gk(x,t)=e<\exp(°l2'f'iy-

<">

where

Thus,

— c has the interpretation of a certain "group velocity" of the

N-soliton

solution; co is the common frequency of oscillation, and b is a scale factor;

The Element

and V>

k -th soliton.

determine the shape of the 6.2.

ah

1 . The coefficients m3-

of the element

1, defined according to

formula (4) are found from the system of equations (5) by transforming with the use of relations (11) and (12)

(/ — I,.:., AT):

2l(2^2/*)*Gy-(-2w^)*G;)]m/i =[{2ia)brG1-(-2ia)by'G]{

6.3.

The O p e r a t o r s

we s o l v e i n

M

L and

P.

The c o e f f i c i e n t s of and

/KAr_2 = ^ 3 J

= -

4d

+ wx+wVx

«, v, w, z

+ z)\ = ll2b2.

(1*) coefficients

by t h e formulas v = — 4dx\>.2, w = — 6d2xV.2—Abd&i + ^ d . ^ ;

xf12 - 6 A ^ 2 - 4MX|*, + 8 (dxV.2f + 4&M*ft! + 6^2d2x^+4fy2
To o b t a i n them i t s u f f i c e s to e q u a t e t h e c o e f f i c i e n t s of r i g h t s i d e s of

To find

L a r e u n i q u e l y e x p r e s s e d i n terms of t h e components of t h e

u=0; z

L = d*x + ud3 +vd2 +wdx + z.

(13)

the equation (Vx+uS7x

mff-i = r1\

We s e t

? J.

(15 )

Xw+2, Xw+', Xw on t h e l e f t and

(14).

S i m i l a r l y , for t h e e q u a t i o n

V / 1 = / > 1 , we f i n d P=mb~ldl + cdx—2
(16)

95

The condition

Lt — [P, L\ has the form of a system of equations for

v, w, z:

—bu>~lvt=2vxx—2wx—bcsTxvx\ —b«rlwt = —wxx+2vxxx + vvx - bc^~lwx—2zx;

(17 )

l

- b«r zt = ^ vxxxx + 2 vvxx - zxx + j vxw - bc
ba~l and the "speed" of the solution

equations; the remaining constants of 6 . 4 . A Single Soliton. acquires the form

c determine the form of the

N - s o l i t o n solutions can vary.

We solve Eq. (13) for the case N = l, ai = a, yi=iE2 .

The system

(iGQ + iG0) (*) = (2a2G0 - 2a2G0) (£), whence

(jj)=2d»(ReO^r'ImO0(J),

(18)

G0=e«"^xp§2ioa2)l.

(19)

where

/02ia2\2 (2ia2 0 \ i(l+i)a 0 \2 Since ^i Q ) —y 0 2ia2j \ 0 (1 -\-i)a) ' exp we obtain without d i f f i c u l t y

bt y

se

P 3 r a t l n g even and odd powers in the s e r i e s for

n _ . j i . - « ( c h ( 1 + < ) f l S . 0 + i)«sh(l + i)al \ «o-« '\(1 + i)-ia-i Sh(1 + i)al, ch (1 + i)&)•

(20) u u ;

To simplify the notation we set ch = cha£, sh = sha;, c = cpsac, s = sina;. cos = cos2a 2 t, sin=sin2a 2 t. Computing (20) e x p l i c i t l y , we obtain 0

/ch c cos — sh s sin, a (sh e cos—ch s cos — sh c sin — ch s sin)\ ~\ 2^(shccos+chscos — chssin+shcsin), chccos — shssinj'

,„.,..

ImG _/ch£sin-f-shscos, a(shcsin+shccos+shscos—chssin) \ I ^ ( s h c s i n + c h s sin+chscos —shccos), chcsin-fshscos] # After rather lengthy but straightforward transformations, we find from this

det Re G0 = j (ch 2aS + cos 2a\+2 cos 4a2t) and then, using (18), (21), and (22), _ n 2 ch 2a£-^cos 2ag +. 2 sin 4a2r (x, — zffla c h 2(j g + CQS 2 a g + 2 ^ os 4ffl!i ,, ji 2 ==— 2a

96

sh 2ag—sin 2ag eh 2a£ + cos 2ag + 2 cos 4a 2 T'

(22)

The qualitative behavior of the corresponding (v, w, z)

was described in the introduction

where, for simplicity, we set b = \ . 7.

Solutions of the Reduced Benney Equations and Their Analogues

fc—

•••—

—,.i

7.1.

i i —

i

...

-

•••—

- — — —

' ••——

The reduced Benney equations are Eqs. (22) of Chap. I I . 2

solutions of them for which ft = (p(u), we find that h=(u+c) /4, compatibility condition. 7.2.

cGR i s a constant from the

We have, moreover, the following r e s u l t .

THEOREM. Let c6R, fc=(«+c)74» and l e t u be a s o l u t i o n of the equation

2

(3u /4+cu/2)x Proof.

In attempting to find

.

Then the pair

ut = —

(«, h) i s a s o l u t i o n of system (22) of Chap. I I .

Under the hypotheses of the theorem we have ht = ("+c) «,/2 = - (a+c) (3a + c) ux/4 = - (u (u + cfi4)x = - («A),, ut= -(3u2l4+uc/2)x

= -(u2l2+{u+c)2/4)x=

We now consider the equation ut + (3u2/4+cu/2)x. 7.3.

Proposition.

~(a2/2 + h)x.

The following method i s c l a s s i c a l .

Let if> be d i f f e r e n t i a b l e , l e t

u=u(x, t) be a smooth s o l u t i o n of the functional equation u = ^{x—
{x, t) .

Then in t h i s range .[tt* +
COROLLARY. Under the hypotheses of the proposition u(x, t)

i s a solution of the equa-

t i o n ut +
Differentiating the r e l a t i o n

u=ty(x—
obtain ux = ty'(x~
t , we

We multiply the f i r s t

equation by q/("), replace q>'(u)ux by f(w)x, twice, add the r e l a t i o n s obtained, and take a l l terms to the l e f t .

We obtain a s s e r t i o n 7 . 3 .

If the Cauchy problem i s posed for the equation «(+0. f*("o)¥=0 »
both sides of the tangent a t the point u0 . sect <}> twice. Remarks, solution

A s u i t a b l e compression and t r a n s l a t i o n takes t h i s secant into the diagonal. a) The i n i t i a l condition

u=

A small t r a n s l a t i o n of t h i s tangent w i l l i n t e r -

2 + ct

y ^ = (2 +cii"

wllic

${x) = x on some segment leads to the c l a s s i c a l

h i s called the "ruptured dike" s o l u t i o n .

b) If the velocity p r o f i l e if a t the i n i t i a l time has an i n f l e c t i o n point with a horizont a l tangent, then by an a r b i t r a r i l y small deformation i t i s possible to obtain from i t a p r o f i l e with a maximum and minimum which leads to nbnuniqueness of the solution i n f i n i t e time. 7.5.

We now present a method for finding invariant manifolds of the form h= 9(a) for

the Hamiltonian system described i n Sec. 10.of Chap. I I .

Our c a l c u l a t i o n s w i l l show that

the manifold A=—^— i s invariant also for the higher reduced Behney equations.

I t would

be of i n t e r e s t to find an analogue of i t for the unreduced equations. In the notation of Sec. '10 of Chap. I I we seek solutions of the equations tlt — t\llix, ht = filuj[ subject to a r e l a t i o n of the form S(ti, A)=cGR, where tion.

S i s a s u i t a b l e dif ferentiable func-

We must ensure the consistency of the system

0=Sx=Sttux+Shkx.

(24)

We multiply (24) by — T J " ^ and add to (23); into the r e s u l t we S u b s t i t u t e the r e l a t i o n s C y l M ^

and ^iUl=aCk1)V2i^ and divide by a(Xj)Vfy? .

We obtain the equation

S A V>,+S tt V 2 A,=0.

(25)

In order that the system of equations (24) and (25) have nontriVial solutions necessary t h a t 0

5 J ^ , - ^ V 2 = = 6 , i . e . , SalV~Vx(«)±ShlVV2(A)=0.

(ux, hx) i t i s

This i s s a t i s f i e d if 5 i s a

h

function of K \rVxdii ± \ VV2dh possible to s e t simply

.

Since we are interested in the r e l a t i o n

Sts^W^du

± [VVjh

=const.

For the higher Benney equations the conditions (26) reduce to A = (a+c) 2 /4, as above. We express A i n terms of u from Eq. (26): u in terms of A and argue analogously).

S=conSt , i t i s

(26) u±2Vh

= const, i . e . ,

A = A(tt) (when t h i s i s impossible we express

Under condition (26) the equation ut = rnhx becomes an equation u-t = (ylih\h=inu))x °f t n e type considered in 7 . 3 . The equation ht = i\lax i s automatically s a t i s f i e d . Indeed, from Eq. (23) and then (25) we find (always with A = A(w)): 98

k

< = ~~Sft

U =

'

_

Sj, ^ *

=

_

Sft ft'**** + VlhuUx) =

From l i n e a r i t y c o n s i d e r a t i o n s i t i s c l e a r t h a t t h e same formalism remains i n f o r c e i f formal s e r i e s ian) .

"Hi i s r e p l a c e d by any l i n e a r combination of i t s c o e f f i c i e n t s

This shows t h a t the manifold

\YV^du

the

(as t h e Hamilton-

± \VV?dh = const i s i n v a r i a n t w i t h r e s p e c t t o

e q u a t i o n s w i t h any H a m i l t o n i a n from t h e c o r r e s p o n d i n g s p a c e . CHAPTER IV INDIVIDUAL RESULTS 1«

The H i r o t a Formalism 1.1.

I n t h i s s e c t i o n we d e s c r i b e t h e s e m i h e u r i s t i c method of H i r o t a for s o l v i n g non-

linear equations.

The method c o n s i s t s of two s t e p s : r e d u c t i o n of t h e e q u a t i o n t o s o - c a l l e d

b i l i n e a r form by f i n d i n g a s u i t a b l e s u b s t i t u t i o n and t h e n s o l v i n g t h e b i l i n e a r e q u a t i o n by means of some v e r s i o n of p e r t u r b a t i o n t h e o r y o r a lucky g u e s s . g r e a t d e a l i n common with t h e Zakharov-Habat

The H i r o t a formalism has a

formalism d e s c r i b e d i n Sec. 5 of Chap. I I , but

t h e i r e x a c t r e l a t i o n s h i p h a s n o t been c l a r i f i e d .

Our p r e s e n t a t i o n i s based on t h e papers

[ 3 8 , 3 9 ] ; s e e a l s o t h e l i t e r a t u r e c i t e d i n t h e s e works. 1.2.

The H i r o t a O p e r a t o r s .

L e t f(x, t), g(x, t) be two f u n c t i o n s of two v a r i a b l e s which

a r e d i f f e r e n t i a b l e an a p p r o p r i a t e number of t i m e s .

For a p a i r of i n t e g e r s

m, nwe

define

the expression

A system of e q u a t i o n s i n b i l i n e a r form i s o b t a i n e d by e q u a t i n g t o zero some system of l i n e a r combinations of such e x p r e s s i o n s f o r t h e unknown f u n c t i o n s

f

and

g ( t h e c o e f f i c i e n t s of

t h e l i n e a r c o m b i n a t i o n s i n a l l examples a r e c o n s t a n t s ) . 1.3.

The H i r o t a S u b s t i t u t i o n s .

We s h a l l p r e s e n t a sequence of s u b s t i t u t i o n s which

r e d u c e well-known n o n l i n e a r e q u a t i o n s t o b i l i n e a r a) The Korteweg-de V r i e s e q u a t i o n

form.

ut+6uux + uxxx — 0. S u b s t i t u t i o n :

u=2(\ogf)xx

.

Bilinear

form: Dx(Dt + b) The e q u a t i o n

D»x)f.f=0.

ut + 45a2tix + \5(uxuxx + uxxxu) + uxXXXX=0.

S u b s t i t u t i o n : - « = 2(logf) x x .

Bilinear

form: Dx{Dt+Dx)/./=0. This e q u a t i o n i s s i m i l a r to t h e second of t h e h i g h e r Korteweg-de V r i e s e q u a t i o n s (corresponding

99

to the operator

< {dx+uf2

> i n the notation of Chap. 2) which has the form

10(2uxuxx+uuxxx)+tixxxxx=0.

ut+3Qu2ux +

However, as Hirota observes, the l a t t e r does not reduce to 1.3 b)

by l i n e a r transformations. c) The equation for waves on shallow water: 00

u

3

3

"/ — xxt — ""< + "JC S atd^ + Ux = 0. X

Substitution: «=2(log/),.t.

Bilinear form:

Dx{Dt-DxDt+Dx)f.f=0. d) The Boussinesq equation: utt — Uxx—3(tt2)^—Uxxxx—0.

Substitution: « = 2 ( I o g / ) ^ .

Bilinear form

(Z>5-Di-Dl)/./-0. e) The two-dimensional Kbrteweg—de Vries equation: utx±uyy t i o n : u=2(logf)xx . Bilinear form:

+6(uux)x+uxxxx=0.

Substitu-

(D,£>,±D2+£M)/./=0. vt+6v2vx-{-vxxx=0.

f) The modified Korteweg-de Vries equation:

\

f—ig

Substitution:

)x

Bilinear form: (Dt+Dx)(f+ig).{f-ig)=0, Dx(f+ig).(f-ig)=0. g) The sine-Gordon equation: wt/ = sin« .

Substitution: v= —2i log(f+ig)l(f—ig)

.

Bilinear form: DxDtg-f=gf,

DxDt(f.f-g.g)=0.

h) The two-dimensional sine-Gordon equation: vxx + vyy —
Substitution:

(Dx+D*y-D*)(f.f-g.g)=0.

i ) The equation ity, + 3ta |<|»|2<|>x +?<[>„ + n|2<J) = 0.

(DX-\)F>F=-2\G\*. Substitution ty = G/F , F a r e a l function. Bilinear

form (f£>( + D 2 — X)G./ 7 =0, (Dx — \)FF=— 100

Substitution:

2|G| 2 .

The c o n s t a n t

X i s determined by t h e b e h a v i o r of

<> ] as

|JC|-»-OO.

k) The e q u a t i o n s of two waves

Substitution: (fl=Gl/F . Bilinear form: (D < +© i D J )O r F=0, i = l,2. 1) The e q u a t i o n s of t h r e e waves: fit+«!?i* -=^i?yT*. cpi = G i // ? .

Substitution:

Bilinear

{'./.*} = 0 . 2 , 3 } ;

form:

{Dt + vfi^OrF^qft/lt. 1.4.

Sample S o l u t i o n s .

fc=±l.

i = l,2,3.

We p r e s e n t two examples of s o l u t i o n s of t h e b i l i n e a r

equations

and r e f e r t h e r e a d e r t o t h e p a p e r s c i t e d of H i r o t a f o r o t h e r s o l u t i o n s . a) The i n t e r a c t i o n of t h r e e waves ( E q s . 1 . 3 1 ) . a r y s m a l l parameter

We expand

F, G,{ i n powers of a n a u x i l i -

e:

F = l+2* 2fl /2«,

Gi=gio+2 s2 "£i»'

G y =2^ +1 g/,2«+i,

y==i,2.

We s u b s t i t u t e t h e s e expansions i n t o t h e b i l i n e a r e q u a t i o n s 1.31 and e q u a t e c o e f f i c i e n t s of l i k e powers of

e.

W r i t i n g o u t t h e c o n d i t i o n s for t h e v a n i s h i n g of t h e l e a d i n g

coefficients,

we find two t y p e s of s o l u t i o n s : The s o l u t i o n w i t h

gio=0 .

Here *l ?2

__ =

™

I + «„ exp 2tj2 + 60, exp 2t|a » a,, exp t)8 l+«,2exp2Tia +ft0,exp2T|j' a.i exp t|, 1 + «0j exp 2r|s + 6o2 exp 2T|, '

where ri2=*Pt(x — Vtt),

v\3=P3(x — V3t),

fl

22~<7l 2 ( 0 , - 0 ft'

The l e t t e r s

a2]

and «3i

denote a r b i t r a r y c o n s t a n t s , while

Pi and p3 obey t h e c o n d i t i o n s

(«i—v2)P2=(vi—v3)p3-

101

This s o l u t i o n was obtained by V. E. Zakharov and S. V. Manakov by methods of i n v e r s e s c a t t e r ing t h e o r y . The s o l u t i o n with

gio^O .

Here li—gio T2—

«

1— exp2t|, i + exp2n, q,expt|, l+exp2t|, ' *iexpii

* * - I + ex P 2t ll ' where

(Sj - z>2A) (2, - fsA)=ftftffw a\=—giqy4: (2 t — vtfx) (fy — z ^ ) , *i = — q\q% • 4 (2,—«>,/>,) ( 2 , — ^ A ) c) The two-dimensional sine-Gordon equation (Eq. 1.3 h ) . s o l u t i o n s of t h r e e - s o l i t o n t y p e .

For t h i s equation Hirota found

I t i s i n t e r e s t i n g t h a t there are apparently no known

s o l u t i o n s w i t h a d i f f e r e n t number of s o l i t o n s .

/ + t e = H—0,1 2exP

2

The Hirota s o l u t i o n has the form

A

13>*>;'»1

«l*il,7+2(1lJi + ',c/2)fr|.

where ili=PtX+9iy

— &it,

and the c o n s t a n t s obey the f o l l o w i n g r e s t r i c t i o n s :

mA P

1.5.

li

(Pi-Pi)* + (gi—?/)'—(Pi—Q/)' (Pi + Pi)2 + (It + »y)»-(Oi + Qy)1 dij=PiPj+<]i
~~

The Hirota I d e n t i t i e s .

dU— 1 ~du+V

In seeking s u i t a b l e s u b s t i t u t i o n s r e l a t i n g the equations

i n the usual form to b i l i n e a r e q u a t i o n s , Hirota makes use of i d e n t i t i e s samples of which are given below.

The reader w i l l have no d i f f i c u l t y i n extending t h i s l i s t by i n d u c t i o n . Dxta-b = D^ab

=

(—iy*Dxtb-a; D«-1(ax-b—abx);

D™ exp fax) • exp {p2x)={p1

-p2)m

Dxab -c=^bc+a(Dxb D2xab.c=d^bc+2pxDxb.c+a(Dlb.c);

102

exp fa+p2) x\ • c);

exp (Wt) [exp (2zDx) a-b]-cd= = exp (eDJ [exp (eD, + W,) a • d] • [exp (tDx - Wt) c • b]; exp (id/dx) alb = [exp (zDx) a • 6]/cos h (sDx) b • b; da

Dxa-l\

dx b ~~ ~1F~' d' t a\ D^a-b a Dxb-b dx2\bj * ) • b2 3 d* a _D xa-b dx3 b b*

b b* ' p^.b^b-b b* b' '

2 cos h (ed/dx) log / = l o g cos h „ d* , Dxf.f

{iDx)f-f\

2^1og/=-ir-=tt;

D*J.fip~ihx+U-; 2.

P o l e s of t h e S o l u t i o n s 2.1.

I n [40] Kruskal s u g g e s t e d c o n s i d e r i n g t h e p r o c e s s of s o l i t o n i n t e r a c t i o n by t r a c -

ing t h e p o l e s of t h e a n a l y t i c c o n t i n u a t i o n of a m u l t i s o l i t o n s o l u t i o n i n t h e complex domain. For t h e e q u a t i o n

Vt + vxJr'Vxxx'

which i s c l o s e l y r e l a t e d to t h e Korteweg-rle V r i e s e q u a t i o n , ,—

t h e s i n g l e - s o l i t o n s o l u t i o n has t h e form

v(x, t) = 3y c t h - — ^2

i n a sum of p r i n c i p a l p a r t s near t h e p o l e s h a s t h e form

v(x,t) = 6^

l/^T-( x

(X-ct-tiKi/Vc

thz = ^

ct\

'-. I

The expansion of 2-

-?)

thz

* whence

)-'.

T h e r e f o r e , a c c o r d i n g t o K r u s k a l , t h e e v o l u t i o n of t h e s o l i t o n may b e c o n s i d e r e d a s a "parade of p o l e s " moving i n s t r i c t f i l e w i t h speed

c.

The t w o - s o l i t o n s o l u t i o n can a l s o be r e p r e s e n t e d i n t h e form of a sum of p r i n c i p a l p a r t s :

v(x,t)=62i(x-xW-^H)~^6^i(x-x^-^tr\ ««5l<2)

B=sl(2)

where xW, /=1,2 are constants, and the \W behave like c^t-\-nxilVc^i\

where

c

W)

are functions of time which for large

is the speed of the j-th soliton.

\t\

The speed of a

soliton is proportional to the square of the density of its poles. When a fast soliton overtakes one which is twice as slow, pairs of poles of the first soliton momentarily coalesce with the poles of the second.

If the speeds are close part of the poles of the fast soliton

spring into the gaps between poles of the slow soliton. 2.2.

In the joint workof Airault, McKean, and Moser [25] Kruskal's remark was developed

and led to the discovery that the poles of multisoliton solutions evolve in correspondence with known Hamiltonian equations of the type introduced earlier by Calogero and investigated by Moser. One of the results of their work which is most simply formulated is the following. We suppose that the function

103

*(*.*)-22(*-*,<<))-»

(i)

i-l

i s a solution of the equation vt+3vvx—f'0xxx=^.

Then the functions

Xj(t) s a t i s f y the

system •*/= 6 2 (x/—•**)"*' 7 = 1

re,

*#/ with the additional condition n

2(*;-**)~3=°. J=1

»•

«)

ft-1

The s e t (2) i s empty if i t i s r e s t r i c t e d to real values.

However, for

a=—^—.

the s e t of i t s complex points i s isomorphic to a Zariski-open, dense part of complex space (after symmetrization in 2.3.

d>\,

d -dimensional

(xu ..., xn)).

In [30], which was completed almost simultaneously with the work of A i r a u l t ,

KcKean, and Moser, D. V.Choodnovsky and G. V. Choodnovsky obtained analogous r e s u l t s for other equations.

Especially i n t e r e s t i n g i s t h e i r remark that the evolution of the poles of

the Burgers-Hopf equation ut=2uux+uxx i s free from r e s t r i c t i o n s of type ( 2 ) . 2 . 4 . Proposition. Function u(x, t) = £i(x — Xj(t))~1 i s a solution of the equation . ; ut=2uux-\-uxx, i f and only if the Xj{t) s a t i s f y the system

Xj(t)= - 2 2 (*,(<) -XtWr1n+i (For an i n f i n i t e set of indices j the following computations are formal; convergence r e quires a separate i n v e s t i g a t i o n . ) Proof.

If

u(x, *) = 2 ( J C — -MO)-1 » then i

2uax+axx=-2

2

(x-Xj(t)r(x-xk(t))-\

1, A; l+k

«,=2-M')(*--M0)-2. Comparing the poles of these expressions (assuming t h a t the Xj(t) do not pairwise coincide), we immediately obtained the required a s s e r t i o n . 2.5. For the modified Korteweg—de Vries equations algebraic r e s t r i c t i o n s on the motion of the poles again a r i s e : the function u(x, Q = 2g<(JC~JC/(*))~1» cf= ± 1 s a t i s f i e s the equation ut=6u?ux — uxxx, if and only if Xj^^iXj

— X,,)-2,

2M*y--**)",=0. k+i

104

2.6.

The results briefly considered here are close in spirit to classical expansions

in the theory of elliptic functions and also to the approach for obtaining solutions of Lax equations of algebraic type described in Chap. III. We recall that for the bimodule of rank 1 introduced there the invariant A also evolves in such a way that the poles of its derivative cancel in a particular manner with the poles of the function A that the poles considered there refer

itself.

It is true

to the auxiliary "spectral parameter" — the variable

point on the Riemann surface of the curve Spec O — rather than to * as a function of t. However, it appears likely that recalculation of the evolution of A in terms of the poles of solutions will lead to a generalization of the results of 2.2 and 2.6. 3. Pseudopotentials and Generalized Conservation Laws 3.1.

This section gives an introduction to the interesting formalism proposed by

Estabrook and Wahlquist [54, 32] and further investigated, in particular, in the work of Corones [31], Corones and Tests [26], and Morris [49]. The central feature of these papers is a certain generalization of the concept of a conservation law for evolution equations of the form Ut — K(u,tl',...)

where U = (ult.. .,«„) .

As we have repeatedly mentioned, an ordinary ("algebraic" in the terminology of Chap. I) conservation law for such an equation is a relation of the form Ft — Gx = 0, which is a formal consequence of the equation U( — K.

More precisely, let ^ = ^[a{;)] or

(J C°°(«F' i-o

!|y|<0< ^G<^> and l e t dt:J,-*Jl be an evolution d i f f e r e n t i a t i o n corresponding to F,G§.JL with Ft=Gx c o n s t i t u t e a conservation law. us

If

then

i s a solution of the system, then in terms of i t and a conservation law i t i s v : a solution of the system vx — F(as)
possible to construct a p o t e n t i a l

Suppose F, G are vector-valued functions of variables

ut=K;

v = (t»i,..., vN) .

u\!) of height

N and the auxiliary

We assume that the following system of equations i s consistent

(the r i g h t sides do not contain derivatives of vt ) : ut=K(u,~u',...); vx=F(u, u', ...;©); vt = G(«,«', ...;©). Then for any solution

(us, :vs) the r e l a t i o n ^-tF(us, .. .,'vs)=jiG(us,

...,VH)

• 3.2.

F and

satisfied.

I t is

Vs i s called the corresponding pseudopotential.

called a generalized conservation law, and In a l l examples considered

.. . ^ t s

G l i e in A®&N, where & = R, C and

3^ = Cx(vu

An algebraic model of t h i s s i t u a t i o n can be formed as follows. We consider the ring

dx:uy)®\*~u\J+u®l

and

A®$& .

dt:u®\*~K®l;

On the subring A®\

[dx, dt] = 0 .

there act the d i f f e r e n t i a t i o n s

The r i g h t sides of the two l a s t equations

of (3) can be considered as the extension of these d i f f e r e n t i a t i o n s to a l l of
d~t:\®v-+G.

105

The formal consistency condition of the system

vx=F

means simply t h a t the d i f f e r e n t i a t i o n s

dt commute.

dx and

and vt = G under the condition

ut=K

We may therefore pose the

following problem. 3.3. For some tion

Problem.

Given a

A-algebra

ft-algebra

A and two commuting

ft-differentiations

du d2: A->A-

33 describe the s e t of extensions du d~2-A<&33->-A®!® with the condi-

[di, d 2 ]=0. 3.4.

Without p r a c t i c a l loss of g e n e r a l i t y , we may assume that du d2£A®D(53) + D(A)®33.

In order to solve Problem 3 . 3 , under c e r t a i n r e s t r i c t i o n s on A we construct a Lie algebra ft over

X=%(j&, dl,'d2),

such that the s e t of a l l extensions

&>$ i s in b i j e c t i v e correspondence

with some subset of Lie algebra morphisms of Hom(X, D (33)) , where D(33) i s the algebra of k d i f f e r e n t a t i o n s of 3.5.

33 into i t s e l f .

We consider a basis

The r e s t r i c t i o n s on A

are as follows.

(£/,), j&J of the algebra A

as a l i n e a r space over ft, which

determines the s e t of s t r u c t u r e constants by

'&

'&

(4)

UjUk=2dmUt, where the r i g h t sides are f i n i t e l i n e a r combinations with coefficients in

ft.

We suppose, moreover, t h a t these coefficients possess thefollowing property: for each 1(<J there e x i s t s only a f i n i t e number of pairs (/, ft)g/XJ (respectively, elements /(*/), such t h a t tf^T^O (respectively, cft^O, cf>=£0). 3.6.

For the algebra

(4); moreover, djkl^0 we define the weight

(&[#<'>], dx,dt) the basis of monomials in

only if w(Uj)

Uj, Uk, divide

a^

obviously s a t i s f i e s

Ut, and these are of f i n i t e number.

by a d d i t i v i t y , s e t t i n g

w(uV)) — j -

Further,

Then dx i s homogeneous and

increases the weight by one, and the space of polynomials of a given weight i s f i n i t e dimensional; therefore,

Ut can enter i n only a f i n i t e number of the derivatives

Thus, only the condition on the c'jf (for d2 = dt) can be n o n t r i v i a l . to Korteweg-rie Vries (ut = uu' +u'") i t can be s a t i s f i e d as for < i)

weight wl(u ~ ) = i-{-2.

==

Then the field ^ 2^

and increases t h i s weight by 3, so t h a t

Ut

ttU

a

( )

'"*" '") ' r77r

is

d\Uj.

For evolution according

dx, by introducing the new homogeneous with respect to o>i

can enter the expansion of d2Uj only for

w1(Uj) = wi(Ul) — 3 ; there are a f i n i t e number of such j

.

Similar considerations are pro-

bably applicable to other Lax equations. 3.7. We now fix the algebra (A, di, d2) and basis (£//), j&J, satisfying the conditions of 3 . 5 . We consider the free Lie algebra £, generated by Xk, Yp j , k£j, with r e l a t i o n s

2 c ii ) J 'y-2 c ^y=S r f 7«l J , y. *J> &•

106

(5)

The condition of 3.5 ensures that a l l these l i n e a r combinations are f i n i t e . 3.8.

Proposition.

The s e t of commuting extensions

(di, d2) of the pair

(du d2) to

Jl®D(3is)-\-D(.A)®3$ corresponds b i j e c t i v e l y to the s e t of homomorphisms of the Lie algebra X to the Lie algebra D(53), which vanish on a l l but a f i n i t e number of the Xk, Y} . Proof.

Any extension has the form

d, =
where almost all the Zj, T)/ are equal to zero. The condition [du d2] — 0 means that

J

(6)

l.k

i

where the commutators are evaluated in the Lie algebra D(33).

Substituting here the r i g h t

sides of the i d e n t i t i e s ( 4 ) , we find

2 «$>®C/j®ty-2 c(fiUi®ll= 21 dmUi® hy **!• Equating coefficients of the £/,, we obtain the i d e n t i t i e s (5) with Xk, Yj in place of

»*. f\j

This completes the proof. 3.9.

In 3.7 an almost invariant d e f i n i t i o n of the Lie algebra

to the pair (-A, dlt d2) was given.

k

The dependence on the choice of basis

over £ with respect (Uj) with conditions

3.5 i s probably not e s s e n t i a l i n the sense that for a c e r t a i n class of bases a l l Lie algebras X obtained are canonically isomorphic. We c a l l

X the EW algebra for (
The search for pseudopotentials for the evolution equation represented by (..-#, du d2), naturally breaks into two s t e p s : a) description of the

EW Lie algebra X ; b) description

of i t s representations i n the Lie algebras of vector fields D(C'°(Vi, .. ,,vN)). Both problems are far from being completely solved even for the Korteweg-^de Vries equation.

Apparently, even a finite-dimensional

EW algebra for i t i s unknown.

In the papers

cited a t the beginning of the section a sequence of factors of EW algebras i s constructed for some i n t e r e s t i n g t r i p l e s (X ^2). which are generated by a f i n i t e number of generators and r e l a t i o n s , and some p a r t i c u l a r representations of them by vector fields are given. Factors of a f i n i t e - t y p e EW algebra if

Uj, Uh contain derivatives of

ly high" usually means

ut

St are most easily obtained by s e t t i n g Xj, Yh = 0,

of s u f f i c i e n t l y high order.

In examples " s u f f i c i e n t -

>3, and the additional wonder i s t h a t then by v i r t u e of the r e l a -

tions (5) a l l but a f i n i t e number of the Xh Yh automatically vanish. 107

If the factor obtained has a n o n t r i v i a l , finite-dimensional factor Lie algebra

S?o ,

then i t i s not hard to construct i t s representation by invariant vector f i e l d s on the corresponding Lie group.

We observe t h a t representations with an Abelian image, as i s evident

from ( 6 ) , lead to the usual conservation laws, so that non-Abelian representations and the corresponding pseudopotentials a r e of special i n t e r e s t . Another c l a s s of representations which can be constructed i s obtained i n the Lie algebra of fields over a one-dimensional base. p o t e n t i a l s simple.

Corones suggested c a l l i n g the corresponding pseudo-

This c l a s s i s nice i n that over a one-dimensional base

[£, t)] =0 implies

t h a t £=cn, c a constant, which makes i t possible to avoid the many r e l a t i o n s (5) and their consequences. With these rudiments of a systematic theory, we now present several samples of computations from t h e works cited above. 3.10.

The Korteweg-Tie Vries Equation:

Ui + uXXK+l2uux = 0.

i n the work of Wahlquist and

Estabrook [54] (see also Corones and Testa [26]) a factor EW-algebra £? i s constructed which i s given by the following generators and r e l a t i o n s : [Xu X3\ = [X2, Xa]~[Xlt X4\ = [X2, X6\=0, [XuX2\ + X,=0; [XuX7]-X5=0; [Xh X7]-X6=0; [Xu XS\ + [X2, X 4 1=0; [*3, X^ + lXu X6]+X7=0. The generalized'conservation law Ft — Gx

(7)

has the form

F=2Xl+3uX2 + Su2X3; G=-2(uxx +6a2) X2+3 (u2x - 8u3 - 2uuxx) X3 + +8Xi+8uX5+4u2X6+4uKX7. Any representation of (7) i n the algebra of vector fields on an N-dimensional fold gives

mani-

r

N concrete generalized conservation laws(A i , F, G are expanded in terms of

components). In order to obtain a n o n t r i v i a l factor of the algebra ( 7 ) , Wahlquist and Estabrook s e t . 8

Xg = [AT4, X3\ and impose the new r e l a t i o n [^1.^5] = ^ cmXm. They then verify that t h i s leads m=l

to the zero algebra in all cases except corresponding factor by

Ci-=... = e 6 = 0, c 7 = — c 8 = X, X^O. We denote the

Xt .

According to computations of S. I. Gel'f and, Xx has as a direct component the Lie algebra sl(2), engendered by the generators 2Xs + lXs, X6, X7 — Xs , and the representation of Wahlquist and Estabrook coincides (up to a formal diffeomorphism) with a representation of the leftinvariant fields on SL(2).

In their notation this is a representation in D(C°°(;/2, y3, ye)),

which is described as follows:

108

The generalized conservation law corresponding to -j—, has the form {2a + yl-l)t+4[(u If

+ X)(2u + yl-}.)+2uxx-2axysl=0.

(8)

a i s a solution of the Korteweg—de Vries equation and ya i s a solution of ( 8 ) , then

a = —a—y\-\-\ speed

>• .

i s a new solution which i s obtained by "adding" to

u

a single s o l i t o n with

The reasons for the appearance of the "Backlund transformation" i n such a con-

text remain somewhat puzzling. 3.11.

The Hirota Equation

function and a

ut = — 3auuux — $uxxx + i~(tixx + izu?u. Here

a

i s a complex

i s i t s conjugate, they are considered algebraically independent, and the

Hirota equation corresponds to a pair of equations for complex conjugation from the f i r s t .

u, u: the second i s obtained by formal

In [31] Corones constructs the following factor EW

algebra for i t : -<»*, + [»[*„ [ ^ . A - J I - O ; aX2 + HX2,[XuX2\\=0; [X3,[XuX2]]=0; / e ^ + iTl^i, [X2, AMJ-p [Xx, [X2, [X3, X,]]] - i P \X„ [Xlt [X3, X^-a -hX2-ii\X2,

[X2, Xft-WXi,

[X3, *,]=();

[X2, [X3, Xl]\] — j P[-^i. [X* [Xz, X2]]]-*[X3,

X2]=0,

—yP[Jf„[A' 1 ,[A'a,Jf ,111=0; -^[X2AX2,[X3,X2]]}=0; - * T [ * i . [Xu * 3 H - P l * 3 . [Xu [X3, ^,111-Pl^i, [Xz, [X3, X41H=0; -h[X2,

[X2, X3]]-HX2,

[X3, [X3, X2\\]-

-jnX3,[X2,[X3,X2\\]=0; -WXU [X2, X3]] + i-({X2, [X3, Xl]] + ^\XZ, \X,i X\]\- p [Xu [X3, [X3, JfJH-P [X2, [X3, [X3, X,]]]- p [X3, [X2, \X3, JMU + lXs, X<]=0; -il[X3, \XU X3]\-HX3, [X3, [X3, XJU + IX,, XA}=0; il[X3, [X2, X3]}-$ [Xz, [Xa, [X3, X2]\] + [X2, X 4 ]=0;

The corresponding conservation law does not depend on UU), u(i) for

j>3-

Corones observes that i t has non-Abelian representations by one-dimensional f i e l d s only i n the case

Be—av = 0.

I t i s i n t e r e s t i n g t h a t t h i s i s precisely the condition obtained by

Hirota for m u l t i s o l i t o n solutions of h i s equation. 3.12.

The Equation

the s u b s t i t u t i o n

3utt + uxxxx+6(uux)x = 0. This equation was investigated by Morris [49];

v = ut reduces i t to the usual system of evolution.

The EW factor algebra constructed by Morris i s defined by the r e l a t i o n s

109

[A-!, Xs\=A%;

[Xi, X10\ = j A^ -f -y pX3\

[Xu Xn] = ±*4-3^2; [ A 2,

A5] = — - j

[*2, * J - -AT U ;

A 8;

[X2, X 8 ] = — -g- X3\

[X2, A"ig] = — -£ X2;

[X2, Xn] = — -^ A"8;

[Ao, A 8 J = — -g- X3;

[X2, A n ]

=

— 4" X55

J A 3, A"i 1] = A^;

[ A J , A^IOJ = — 4- X 2 ;

1X3, A 4] = Tj A"10;

IX4, X 5 ] = — 3[iX8;

[X 4 , A'uJ = ~2 Xt—3X2; [X~5, Xs] = j X2;

[A 3 , A10] = — - j A"3;

[X 4 , A%] = — -j- [1X3—j Xi; |X 4 , Xn] = —311X5;

[X5, Xn] = — -jg XI — -pr X3;

[Xit Xl0] = — -j

[ A 8 , Xn] = — 4- A 10;

A8;

[A*j0l Xn] = — j X n . Morris c o n s t r u c t s i t s r e p r e s e n t a t i o n i n t h e a l g e b r a

£)(C°°(A:I,

JC2, J£3))

by s e t t i n g

(
^1^3;

A 3=

X4=x%dx+p (x^+x 2 d 3 ); X&=j

(Xidl +

X&=-^(Xid2 XlQ=-£

x3d3—2x2d2); — .*2
(Xidi — JC3O3);

Xn = -j- (JC3(?2 — JC2O1).

From t h e e q u a t i o n for t h e g e n e r a l i z e d c o n s e r v a t i o n law Morris deduces t h e r e l a t i o n s

x['" + -2ux[ + iju' + w)x'l = vxu Xi,t+3x'l + uxl=0, 3

where

TW i s a t r i v i a l p o t e n t i a l :

wx= —jUt,

1

(9J

3

wt= 4-«'"-+-j(uu')'.

Equations (9) make i t p o s -

s i b l e to apply t h e t e c h n i q u e of s c a t t e r i n g t h e o r y t o t h e o r i g i n a l problem. 3.13. conjugate.

The Nonlinear S c h r o d i n g e r E q u a t i o n ; The f o l l o w i n g f a c t o r

iut-\-uxx—2"Su«2=0,

5[A- 2l Z 2 ] + [K 1 .Z 1 ]-eZ 1 =.0;

[ X ^ R 2 1 ^ , ^ 1 = 0;

[A'1,K1l + [A'2,K2l = 2[Z 1 ,Z 2 ]-2[Z 1 ,i;i 110

i s

the formal complex

EW a l g e b r a for t h i s e q u a t i o n was found i n t h e work of

Estabrook and Wahlquist [ 3 2 ] : [Jf 1 ,Jfjl = [X 1 ,J' 2 ] = [X ! | i r i ]»=[A'„Z l ] = [ 2 1 , Z a i - 0 ; [X„Z,] = Z 2 , 2

u

\ZX,Z,\=\Y,\ (10)

(To these relations are to be added their formal complex conjugates.) Denoting complex variables by

tfu U-z and by *?i = 37-, dt = -=— te dyi

the standard

differentia-

tions, we obtain the following representation ( ft is any complex constant): X1 = 2(kyldl+kyldl); Vi=t[yld1—'yl'd1 —

Td2+-yd2); ;

___

(ID

Zx = — 2" {tfidi—E
fields.

Backlund Transformations 4.1.

E(u)=0

A Ba'cklund transformation

B, relating two systems of differential equations

and F(v) = Q, i s a system of differential equations B(u, v) — 0 such that E and B

formally imply F, and F and B formally imply

E .

In the language of

algebra, suppose that E corresponds to an ideal / E C ^ , F to an ideal ideal

IstzA®^'

Then IB^IE®^

+ 1®//=-.

differential 7pc53 and 53 to an

Thus, a Backlund transformation is an analogue of

a "correspondence" in algebraic geometry.

(In our case Jl, 33 are differential rings over

a common base differential ring K, over which a l l tensor products are taken; the ideals are assumed to be differentially closed and to be radical for analytic applications.) There is no systematic theory of Backlund transformations, and we limit ourselves to describing a portion of the experimental material at hand taken from [26]. 4.2.

The Backlund Transformation for the sine-Gordon Equation.

While investigating

surfaces of constant negative curvature, Backlund in 1880 found the following transformation relating the sine-Gordon equation uxv = sinu to i t s e l f :

Ba: where a is any constant.

2

(12>

a -a

This transformation makes it possible to obtain a sequence of

solutions of the equation in quadratures starting from the solution

U0—0.

Bianchi observed

that the pair of transformations B0l, Ba, commutes in the following sense: starting from the initial solution «0 , the compositions

Bai°Ba,

and Ba,°Bai

generate a certain common

solution tt3, for which

111

where «i

and

4.3.

« 2 a r e t h e s o l u t i o n s g e n e r a t e d by

A Remark of Rund.

suppose t h a t t h e e q u a t i o n

I n h i s work i n

Bai

and

Ba,

from

[26] Rund put f o r t h t h e f o l l o w i n g i d e a .

E i s t h e Euler-Lagrange e q u a t i o n w i t h Lagrangian

Chap. I ) , and we a r e i n t e r e s t e d i n Backlund t r a n s f o r m a t i o n s of t h e n seek i d e a l s Lagrangians

IB

ca(u) and

simultaneously.

a0 .

E

We


into i t s e l f .

such t h a t (o(«)— ©(u)Glmxd+ (/B) ; o n s o l u t i o n s of t h e system

One can B

a>(v) d i f f e r by a d i v e r g e n c e and t h e r e f o r e have zero f i r s t

Rund shows t h a t t h e t r a n s f o r m a t i o n (12) b e l o n g s t o t h i s c l a s s .

the variation A certain

m o d i f i c a t i o n of h i s i d e a i s a p p l i c a b l e t o t h e Korteweg-nde V r i e s e q u a t i o n . 4.4.

Backlund T r a n s f o r m a t i o n s f o r t h e Korteweg—de V r i e s E q u a t i o n .

We c o n s i d e r t h e

s p e c t r a l problem r e l a t e d to t h i s e q u a t i o n d e f i n e d by t h e Lax p a i r

\(dt+4d3x + 6udx + 6dxou)
w = <J»Jt/<|»,

we

see t h a t i t i s possible to eliminate

modified Korteweg-de V r i e s e q u a t i o n for

(13)

u from (13) .

The r e s u l t i s t h e

v.

vt-6v2vx+6\vx+vxxx=0, d i s c o v e r e d by Miura.

(14)

E q u a t i o n (14) h a s t h e t r i v i a l t r a n s f o r m a t i o n

i>>-»-— v; then a p p l y i n g t h e

t r a n s f o r m a t i o n i n v e r s e to (13) we o b t a i n a Backlund t r a n s f o r m a t i o n of t h e Korteweg-nde V r i e s e q u a t i o n i n t o i t s e l f which we d e n o t e by

B% .

Estabrook and Wahlquist found t h a t i f

ui=B),i(u0),

i = l,2,

then

B^B^tt^)

and £;!,,£>., (a0)

contain the s o l u t i o n

an analogoue of B i a n c h i ' s formula for t h e sine-Gordon e q u a t i o n . 4.5.

F l a s c h k a and MacLaughliri i n t h e i r paper c o n t a i n e d i n [26] s t u d i e d how t h e t r a n s -

f o r m a t i o n (12) a c t s on t h e spectrum of t h e Schrodinger o p e r a t o r .

Changing t h e n o r m a l i z a t i o n

s l i g h t l y , we w r i t e t h e e q u a t i o n i n t h e form ut—6uux + uxxx = 0 and t h e Schrodinger o p e r a t o r i n t h e form U=2WX

dx2 + u=l. , and

W

Setting

u=2wx,

we o b t a i n t h e new Schrodinger o p e r a t o r

dx2+U = L, where

i s found from t h e e q u a t i o n s

(Wx = - wx+(W - w)2+k \Wt==—wt+4[\Wx Here W—w=—i|>*/i|>, where If

wxx(W.—w)].

|x|-»-oo, t h e s c a t t e r i n g d a t a for / = —d x 2 +u a r e determined

For any complex number k w i t h lm&>0 t h e r e a r e s o l u t i o n s of t h e problem

d i s t i n g u i s h e d by t h e a s y m p t o t i c b e h a v i o r a t

112

+

/i(>=A,i|>. .

u i s r a p i d l y decreasing as

as follows.

+ wx + wx(W-w)2

infinity:

Lf=k2f,

elkx, x-+-\-o&; /i(x, k)^[b(&je-ikx+a^^ Mx,

x_+_00.

e 1X + a e k)^[zf^ " ^ ~"'X' e Utx, x ->- — oo.

x^ + oo;

On the imaginary axis there may be a finite number of points of the discrete spectrum of l:k2= —rf., J=l, .. .,N, which determine constants

Cj, j = \,. ..,N,

by conditions on their

oo

eigenfunctions: <}'/~]/c;exp( — rtjx),

if \y*(x)dx=l.

The collection (a(k), b(k), •»),, Cj) is called

—00

the scattering data for b. If

%(x, X0) i s any solution of /ij»0 = X0<|»0, and ty(x, X) i s a solution of lty = \ty , then

*P(x, \) = Y(x, X) —tjj(jc, X) ° *' x* i s a solution of the equation not vanish i t suffices that

JL*' = X'F .

In order that <M*> X0)

>-0 l i e to the l e f t of the spectrum of /.

We choose X0= — rf and denote by (A(k), B(k), 11r Cy)the scattering data for L . According to the analysis of Flaschka and MacLaughlin, there are the following facts. a) If %=fi(x, ty , then

A(k)=a(k), a<*)«3=£*<*).

Cj=^cP

and the discrete spectra of I and L coincide. b) If %=f2(x,

in) , then

A(k) = a(k), B{k)=^b(k),

C y =5±^c /r

and the discrete spectra of I and L coincide. c) If % = Dlfl(x,

if)-\-D2f2(x,ir),

Dx, D2 nonzero constants, then

A(k) = ^a(k), N

B(k)=-b(k),

Cj=^cj,

points of the discrete spectra for / and L coincide and, moreover, L has an additional

point — T]2, with normalization constant depending on Dx and D2 . This implies several curious conclusions. First of all, there are Backlund transformations which add no solitons (these correspond to points of the discrete spectrum). They only shift the phase of solitons present in the old solution (cases a) and b): Cj'&Cj) . Secondly, iteration of the transformations defined by h(x, ir\) and then D\Fi(x, ii\) + D2F2(x, tr\) : u-+U does not change

b(k)/a(k)

and Cj, but adds the eigenvalues — r,2.

For it

we have

U(x)=a(x)-2D/l(x,iti)ll

+ D J/»(2, hfldzj ,

113

where D is a certain constant. Finally, if tt(x, t) is a solution of the Korteweg-^de Vries equation, then, as is known t)=8ik3b(k,

jfb(k,

t), jfCj(t)=8^c/0

/i(x, if\\ t), changing

. This implies that Backlund transformation by means of

b(k, t) into *j~'.. b(k, f), takes the solution

a into the new solution U,

i . e . , i t commutes with theKorteweg-nde Vries flow. With this we conclude our brief discussion, referring the reader to the literature for further information. 5.

Lax's Method for Generating the Algebra of Korteweg-^de Vries Integrals 5.1.

We consider the algebra

.A=&[ap] and some operator

B which is Hamiltonian in

the sense of Sec. 7 of Chap. I .

I t generates a Lie algebra structure on -A/KerS —= . Ex-

plicit description of operators

B, for which in this Lie algebra there exists large, e . g . ,

infinite-dimensional, Abelian subalgebras is a very interesting question.

Equations

tlt=B—= , where F l i e s in such a subalgebra have infinitely many conservation laws. In ou Chap. I I we described such subalgebras for the Gel'fand—Dikii operators corresponding to Lax equations and for both the reduced and unreduced Benney operator. In this section we present the recurrence method of Lax for the operator B = dx, which leads to the Korteweg-nde Vries algebra.

The exposition i s based on notes of I . Ya. Dorfman

who has kindly permitted the author to use them. 5.2. by /

Let A = k\u.W \j>OJ,

the image of / i n

A.

d:u^>-*a^+1K

We set A = A/dA

We shall write

structure on A: {/, i"} = (d-g£---g|-] •

L e t

f~g , if

and for any f£A we denote

f=g-

There is a Lie algebra

H:A-*A be a linear operator which is formally anti-

symmetric in the sense that ///g-f g7//~0 for a l l / , g£A. We consider a sequence of elements /_i, /o, • •., fn£A, f_x^=cuy c=£Q. The following result is motivated by the remark in 3.15 of Chap. I I . 5.3.

Proposition.

If # - g j / i ^ - g j / i + i .

—l
a) {/;-//}=0 f<>r

Proof. H

~L~f>-v^f

—Ki,j
From the antisymmetry of H and d i t follows that for l < y < i<«: d4p--4p = ^Er,d^ET~d^W"^i^'

•

Iteratin

S

t h i s

argument, we obtain

<*Sfs 6/, 1° 6« * 6a '

Both expressions on the right l i e in Imd (the second by the antisymmetry of H) which proves assertion a ) . Further, H Ma-

~6u~

114

c v

- i - | ^ . . / / - ^ i = — c~J^--d -^f-~0 . 6a

6u

6tt " 6«

This completes the proof.

5.4.

Proposition 5.3 makes it possible to continue the commuting collection of elements

/_i, • ••,/ n 6^ to the commuting collection /_i, • • •, /n, fn+u if d~lH-^~/„glm-r— : as /n+1 we take any element of f-r— j d '//-£—/„. In order to guarantee the solvability of the equation for /n+i. we impose additional conditions on H. d'.

We recall that the Frechet operator D(g)

We define a mapping [D, //]:

A->-Z,(J4,

for g£A has the form D(g) =

.A), where L(A, A) is the space of linear

operators on A by the formula

\D,H\f=D(Hf)-HoD(f) (cf. Sec. 7 of Chap. I). If H=^htd'

» then

[D,H]f=

2 <*7-D(*i)-

We c a l l the antisymmetric operator / / Laxitive if for any pair / , d i t i o n Hf=dg 5.5

the operator

Proposition.

If

g(«A with the con-

\D, H\g»d — [D, H\f°H i s formally symmetric.

H i s a Laxitive operator and /_i,..., / B s a t i s f y the condition of

Proposition 5 . 3 , then the equation # §^/n = dg^/ n+ i i s solvable. Proof.

We s e t gt = -T- ft, i = — 1 , . . . , rt, and l e t

gnJrX be a solution of Hgn = dgn+i, which

e x i s t s by 5.3 b ) . According to the r e s u l t s of Sec. 7 of Chap. I , i t suffices to verify that the operator D(gn+1) i s symmetric: t h i s implies that g"n+i6g- A. to the symmetry of the operator

This, in turn, i s equivalent

d°D(gn^)°d.

We begin with the case « = — 1 . L a x i t i v i t y of H (for the pair f—0,

The operator

[D, H\cod

i s symmetric because of the

g = c i n the d e f i n i t i o n of t h i s property).

This means

that the following operator i s symmetric: doD (g0)od = D (dg0)od = D (Hc)°d = [D,H] cod. Now l e t « > — 1 . . Inasmuch as Hgn = dgn+U we have, D{Hgn) = d°D(gn+x}, i . e . , [D,H\gn + H°D(gn) = d°D(gn^). Similarly, from Hgn_^dgn we find [D, H\gn_x +HoD(#„_,) =d<>D(gn) . Multiplying the f i r s t equation on the r i g h t by d, the second by —H and adding, we obtain doD(gn+1)°d = [D, H\gn°d-[D,H\g^oH+HoD(gn)*d From the L a x i t i v i t y of H operators D(gn) Finally, 5.6.

+

d°D(ga)off-H°D(g^H.

i t follows that the sum of the f i r s t two terms i s symmetric. The

and D(g„_i) are symmetric, since gn and gn_i are v a r i a t i o n a l d e r i v a t i v e s .

d and H are antisymmetric.

This completes the proof.

Thus, for any Laxitive operator

H and

c(
115

d e f i n e s t h e g e n e r a t o r s of an A b e l i a n L i e s u b a l g e b r a i n 5.7.

Proposition.

Let

c0, c&k.

A.

Then t h e o p e r a t o r

H = d o 3 + 2 (cQu + Cj) da + c0u' is Laxitive. The proof i s o b t a i n e d by d i r e c t v e r i f i c a t i o n .

D i r e c t computations show t h a t t h i s i s t h e

g e n e r a l form of L a x i t i v e o p e r a t o r s of t h i r d o r d e r l y i n g i n Setting

c — -^, c0=— 2,

cl=0,

k [u, u'\ [
we o b t a i n by t h e formulas of 5.6 a sequence of

of t h e Korteweg-de V r i e s e q u a t i o n ( i n t h e form

u.f — §aa' — u'").

integrals

The f i r s t members have t h e

form a*

a

o

an"

•§-; —2-; a3—j-; 6.

5

. . 5

,, , 10 , „

1

—jtt^+ytta^-f j - a V - ^ s i t " " .

Solutions of Algebraic Type and Theta Functions 6.1.

This section is to be considered as an appendix to Chap. II: the formulas pre-

sented here can be obtained from the facts proved there regarding the structure of bimodules over a field if they are augmented to include the classical results from the analytic theory of Riemann surfaces and Jacobian varieties. Our exposition is based on the survey of Matveev [44]; for the proofs the reader should see this survey, the literature cited there, and also the paper of Krichever [16]. 6.2.

The Topology of Riemann Surfaces. To each field

and one-dimensional over

K , which is finitely generated

C, there corresponds a one-dimensional compact, complex variety

T , the Riemann surface of K . The field K is isomorphic to the field of meromorphic functions on

T . The genus of K and T is

a) half the first Betti number of

T;

b) the dimension of the space of holomorphic differential 1-forms on T : Abelian differentials of the first kind. A Riemann surface of genus zero is the Riemann sphere or the set of points of P 1 (C)of the projective line over C

. The genus of the curve

T:y2=]J_

ferentials of first kind on T have the form f(z)y-ldz curves are called hyperelliptic (elliptic for £ = 1).

(z—E.)

is

equal to g ; the dif-

, where /€C[z], deg/(2)
elliptic curve is given by either of the following two conditions (provided the genus > 1 ) . a) On T there is a function with a single pole of second order. b) The ratios of differentials of first kind generate a field of kind zero. On a compact Riemann surface T of genus g>\

it is possible to choose a basis of the

homology group #i(I\ Z) of the form {a„ bj}, i, j = 1 , .. .,g,

116

where (ah aj) = (bh bj)=0,

(ai,bJ) = Zlj.

2g+t

A typical choice of

{at, bj) on T: y2=i[

(z—£;); Et r e a l , i s as follows.

copies of the Riemann sphere with cuts along the intervals

(£\, E2), (E3, £4), . . .,(E2g+i, co) and

we glue them together crosswise along the edges of the cuts. the cut (E2j-i, E2j) is

Differentials and Periods.

Abelian.

A clockwise contour around

a;- ; a contour which on each of the sheet joins an interior point of

(£2;-i, E2J) with an interior point of (E2g+i, °°) is 6.3.

bj.

Any meromorphic 1-differential

to on T is called

The numbers A}(u>)= ^w, Bj(c«) = ^u> are called i t s periods (along a-

1 C

Cj = -^-y<>

We take two

a/t bj).

The numbers

b.

(over a circle around a logarithmic singularity) are its residues.

If all the

residues of *> are trivial <» is called a differential of second kind; all differentials of first kind are also differentials of second kind. A basis of differentials of first kind j¥=k,

(k)=^ik (0 for

and 1 for ; =fe) . Any Abelian differential is uniquely determined by its A -periods and principal parts

at singular points. For differentials of second kind it is possible to uniquely define by the conditions

Aj(a>)=0

©

and any value of the principal parts. For general differentials

the" same is true if the sum of the residues of the prescribed principal parts is equal to zero. 6.4.

The Riemann Theta Function. We consider g-dimensional complex space Qe and the

lattice Zg^Cg.

Let B be some complex

(gXg)

matrix for which there exists a constant

g

c>0,

such that

(\mBk, k)>cj?ik2

there i s a theta function

for a l l

k£Zs, k = {ku .. .,kg).

Corresponding to this matrix

6:0->C : 6

(P)= 2

exp{n(Bk, k)+2r.i(p, k)}, p£&.

If Bjk =B):{<.%) it is called the theta function of the Riemann surface F, referred to the {di,bj}

basis

of the group Hi(V,Z)

and the normalized basis of differentials of first kind

(ij);.) on T . The matrix B is called the matrix of periods of

T ; it is symmetric.

The following properties of the theta function are easily verified: a) 6,(_p)=:9(p). b) Q(p+v)=HP) c)

f

oT all vQZz.

%(pJirBi)=e''t'B'r2,lip'^{,p)

, where

We now assign to each point PQ

is a fixed point of

I\

Pff

B1

is the j-th column of B.

its Jacobian coordinates: P^
Here

The Jacobian coordinates of P are defined up to the lattice

of periods of V spanned by the columns of the matrix B , since different paths of integration from P0 to P differ by some cycle representing a homology class in H1(T, Z). The multivalued function r~>-C:P>-*8(u)(P) — e ) , where e(*C£, is called the Riemann theta function.

It is meromorphic and is either identically zero or has exactly g zeros on T.

117

The Jacobian coordinates of these zeros are

£—*, where the vector x is defined by the

relations p

i+m. aj

P,

(the Riemann c o n s t a n t s ) . If the Riemann theta function i s nonzero, then the divisor of i t s zeros i s nonspecial, and any nonspecial divisor of degree g i s a divisor of the zeros of a s u i t a b l e theta function. From properties b) and c) of the theta function i t follows that dlog6(w(/5)—e) i s a meromorphic d i f f e r e n t i a l on T, having as i t s divisor of poles the zeros of

9-

All poles

have f i r s t o r d e r . 6.5.

Analytic Description of the Akhiezer Function.

Riemann surface

V of genus

l o c a l parameter

ft-1

g , a point Po6l\

i n a neighborhood of

We fix the following data: a

a nonspecial divisor D = ^Pi

on T , a

P 0 , and two polynomials P, Q€C [ft] of degrees

n

and m , r e s p e c t i v e l y . 6.6.

THEOREM.

There e x i s t s a unique meromorphic function

<[>:C3X(r\P0)->-Cwith the

following p r o p e r t i e s : a) The d i v i s o r of the poles of b) At the point

P<> the function

f in

P

coincides with D for any X, y, t£C3.

<>| has an e s s e n t i a l s i n g u l a r i t y of the form

4 (x, y,t; P) = exp [ft (P) + P (ft (P)) + Q (ft (P))] • (1 + 0 (ft (P)"1). c)


This function has the form

4(*, jf, *, P) - e(o,o))e(-«(x,y.o)exp \ .H 01 i x + U, ^+ ( V)J> where u>i, 3 are normalized Abelian differentials of second kind with a pole solely at P0

and principal parts d(ft-f clogft), dR(k),

dQ(k),

respectively, and e

e{x, y . /)=s/(B(»,)j: +fi(»2)ff+flW ) + 2 « ( / > y ) + * . 5(co i ) = A ( o A ,

6.7.

x = (xffl)

is the Riemann vector.

The connection of this construction with the results of Sec. 3 of Chap. Ill is

as follows. Let 33 be a field of meromorphic functions in x,y,t, of meromorphic functions on T with pole at

and let O denote the ring

P 0 . Then on the space of functions

it is possible to introduce the structure of a (53,0)-bimodule with <J» as

118

33\dx\$

1 , in which the

filtration is induced by the filtration of

V*, V y , Vt

33 [
coincide with dx,dy,dt

,

respectively; multiplication by O is the natural multiplication. Further calculations along the lines of Chap. Ill but using the analytic information now at hand lead to the following explicit formulas. 6.8. The Korteweg-rde Vries Equation ut = 6uux — uxxx.

As was explained in Chap. Ill, in

order that under the imbedding . G-+3?l[dx] the image of O contain a Schrodinger operator of second order

— dx-\-u,

it is necessary that Y be hyperelliptic.

surface of the curve y1 — II(z— E}), Ej=£Ek.

Let T be the Riemann

We choose a normalized basis of differentials of

first kind

where the

cjk

as in 6.2.

are found from the system of equations

Then for

\o).=8,.

and the ak

P0= oo and a nonspecial divisor D=P\-\- . u(x,t)=-2dxlogB{xg

+

are defined

-\-Pg we have

tv+l)+c,

where g, V, /gO, c£C and gj=2icn;

i>, = 8/1 ^cnj

2«+l

^Em)

+ cJ2);

I

c=^Ej-2

2

$;•

f-1

(The nonuniqueness in the choice of path of i n t e g r a t i o n in the formulas for affect u(x,t)).

lj does not

The Akhiezer function 4>lx t P)-

BXaW + ^+.fa + Oe.Cfa+i)

i s a solution of the Schrodinger equation

{ — dx+u(x,t))y

normalized d i f f e r e n t i a l of second kind with i = dVz

/. f = y(P)ty.

3/2

+0(z~ dz)

as

) Here P e r

and

v is a

z->oo, and having no

other poles. If a l l or part of the pairs

(Em, Em+l) coalesce the l i m i t solutions w i l l be

g -solitons

or s o l i t o n s on an almost-periodic background. 6.9.

The Kadomtsev—Petviashvili Equations.

These correspond to the Akhiezer function


R(k,) = k2, Q(k) = k3.

In the ring

3S[dx] with the help of the bimodule S [ ^ H w e seek

operators 119

L = dx + u(x, y, t), P=d3x + vl(x, y, t)dx + v,(x,y, satisfying the Zakharov—Shabat equation u, Vu V2

t),

dyP—dtL =[L, P]. In terms of the coefficients

this system can be rewritten in the form ( v2y—ut + V2xx—uxx—zi,ax=0; «ux+2v2x—3ttxx+'Oiy=0; [2wijr='3ttx.

Solutions constructed on the basis of the bimodule actually even satisfy the condition 2t»i=3a .

Eliminating

V\ and

w2 from this system, we obtain the Kadomtsev—Petviashvili

equation 4 u*y + axt+T

¥xxx + 6" «A = 0,

which i s also called the two-dimensional Korteweg—de Vries equation. written in the form

I t s solution can be

i

2dx log8(xg+tv+yh+l). LITERATURE CITED

1.

2.

3. 4. 5. 6.

7. 8.

9. 10. 11. 12. 13. 14. 15. 16.

120

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