Nonlinear Partial Differential Equations for Scientists and Engineers

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Lokenath Debnath

Nonlinear Partial Differential Equations for Scientists and Engineers

Birkhauser Boston &Isel Bedin

Lokenah Dcbnah Depamnent of Mahemaues University of Central Plorida

P.O.Box 161364 Orlando, I% 32x16-I364 U.S.A.

Library 01 C o w C.Lslogiag-ln.Pnblieption Dst. Dcbnnfh, Lokensfh. Nonlinearpamaltbffemtid quaaons for saentl~aand enmnem I Lnkenath hbna!h. p. cm. lneiudss bibliographisaircfnmccs and index. ISBN 0-81763902-0 (Boamn : alk.pap=). .-ISBN 3-1643-3902-0 (Easel : alk. paper) I . mffcrcntiai equations. I~ i t k QA371.Ml 1997 530.153353--dc21 97-15110 CIP

mntcd on acid-het paper 0 IW7 Bi*UyScIBoston

BirkhiiUSer

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Copynght is not claimed for works of U.S Govmment employees Alirightsrucrved. NopanofUli~publicationmayknpodueed,smndinan~ievalsy~lon. oruansmiued, in any form a b y any means. clmmnic.mechanical.phMocopyinerrrording. or OlhewiSe. without pnor permissionof tbc copyright owxr. Permission to photosopy f a internal n penal use of spcifis clients is granted by Bi*hlxlBOStOnfalibranFIandafherusssrcgimrdwithIheCopyrigbtCIcaranecCents (CCC). provided tbat !he base fee of S6.Wp"copy. plus $0.20 per page is paid dinctly to CCC, 222 Rosswocd Drive. Danvm. MA 01923. U.S.A. Special w m t s should k Bddres~eddi~tlytOBi~~~~~B~~m",6lSM Cambridge.MA02139. ~~(1~~h~~u~A~m~e U.S.A. ISBN &8116.3902-0 ISBN 3-1643-39026 Typest by fbc author m M i m s R Wmd.

Pnnted and hound hy Made-Vail. Ymk. PA Pnntcd in the U S.A.

9 8 7 6 5 4 3 2 1

TO MY MOTHER with love, gratitude, and regards

*'__. the progress of physics will to a large extent depend an the progress of nodincar mathematics, of methodr to solve nonlinear equations ... and therefore we can Ibv comoarinedifferent nonlinear orobkms." WEFNER HEISENBERG

"Our present analytical methods seem unsuitable for the solution of the important problems arising in connection with nonlinear partial differential equations and. in fast, with virmally all types of nonlinear problems in pure mathematics. The Vuth of this statement ir psnicularly striking in the field of fluid dynamics ,,,," JOHN VON NEUMANN

"However varied may be the imagination of man,naNre is still a thousand times richer. .... Each of !he themes of physics ... presents (partial differential) equations under a new aspect ... without these theories. we should not know parual differentialeq"atio"s." HENRl POINCA&

'5ince a general solution must be judged impossible h m want of analysis, we must be content with the knowledge of some special cases. and that all the more, since the development of various c a e s seems 10 be the only way of bringing us at last to a more perfect knowledge."

LEONHARDEWLEX

"... as SirCyril Hinrhclwocd has observed...fluid dynamicisu were divided into hydraulic engineers who observed things that could not be explained and mathematicians who explained things that could not be observed." JAMES LIGHTHILL

Contents

lie

Refme

t

Linear Psrtisl Dafrrentinl Equations

1

1nvod"ctio" I 2 Basic Concepts and Definitions 1.3 The Linear Superposition Rinciplc 1.4 Some Important Classical Linear Model Equations 1.5 The Classification of Second-Order Lincar Equations and The Method ofCharseteristies 1.6 The Method of Separation of Variables 1.7 Fourier Transforms and Initial-Boundary-Value Fmblemr 1.8 Applications of Multiple Fourier Transforms to Panial Differential Equations Laplace Transforms and Initial-Boundary-Value Fmblems 1.9 1.10 Hankel Transforms and Initial-Boundary-Value Pmblcms 1.1 I Green's Functions and Boundary-Value Fmblems 1.12 Exerciser

i 2 4 7

1.1

2

10 20 32 44

49 58 67 78

NonlinearModel Equations and Variational FVineipla

93

2.1 22 2.3 2.4 25

93 93 94 99

26 2.7

lnuoduction Basic Concepts and Definitions Some Nonlinear Model Equaltons Variational Principles and the Euler-Lagrange Equations The Variational Principle for Nonlinear Klein-Gordon Equations The Vanational Pnneiple for Nonlinear Water Waves Exercises

104 105 107

dii

3

COntmlu,

First-Order, Quasi-Lmnr Equations and The Method of Characteristics 3.1 3.2 3.3 3.4 3.5 3.6

4

Fimt-Order N o d i n a r Equations and Their Appliestions 4.1 4.2 4.3 4.4 45 4.6

5.

6

Introduction The Classilicahon of First-Order Equations The Consmction of a First-Order Equation The Geometrical lnterprelationof a First-hder Equahon The Method of Characteristics and General Solutions Exercises

Introduction The Generalized Method of Charactenstics Complete Integrals of Certain Special Nonlinear Equations Examples of Applications to Analytical Dynamics Applications to Nonlinear Optics Exercises

111

111 112 116 118 130 135 135 135 139 145 151

156

ConstmationLaws and Shock W a v e

159

5.1 5.2 5.3 5.4 5.5

159 159 172 174 181

Introduction conservation Laws Discontinuous Solutions end Shoek Waves Weak or Generalized Solutions Exercises

Kinematic Waves and Spreine Red-World Nonlinear Pmblcm

185

lnvoduction Kinematic Waves Traffic Flow Problems Flood Waves in Long Rivers Chmatographic Madels and Sediment Transport in Rivers Glacier Flow Rail Waves and Their Stability Analysis Simple Waves and Riemann’sInvariants The Nonlinear Hyperbolic System and Riemann’r Invanants Generalized Simple Waves and Generalized Riemann’s Invanants 6.11 Exercises

185 185 189 202 204 210 213 219 239

Nonlinesr Dispersive Waves and Whithsm’s Equations

263

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

7.

111

7.1 7.2 7.3

Introduction Lincar Dispcrsivc Waves Initial-Value Roblcms and Asymptotic Soluuonr

250 254

263 263 267

CaVcnS

7.4 7.5 7.6 7.7 7.8

Nonlinear Dispersive Waves and Whilham's Equations Whitham's Theory of Nonlinear Dispersive Waves Whitham's Averaged Variational Principle The whitham Instability Analysis and 1ls Applications to water waves Exercises

Nonlinear D ~ b n - R e a d i o phenomena, n Burgem' and Fisher's Equations 8.1 82 8.3 8.4 85 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14

... .......

Burgerr' Equal,"" and lhc Plane Wave Solution Travclinp Wave Salulionr and Shock-Wave Stmc~urc The Cole.Hopl Transformation and Ihc Exacl S~lulion ofihe Uurgcn k y a t m The Asymplolic Behavior of the Exact Solulion of the B u r p , Equdlion 'lhc N wmc sol",,"" Burpen' Inmrl- m J Boundq-Value Problem Firher's Fqumon and Dilfuuun-RcaAon Pmcxcw Trs\clmg W v c Soluuonv and Swhility Analyiis Penurbalion Sululions of the Pnhcr Boundaly-Value Problem S t m h m ) hlcihodr and Similanl, Sulultd,n\ or Diffusion Equations Nonlinear Reaction-Diffusion Equations A Brief Summary of Recent Work with References Exercises

270 273 276 278 281 283

283 283 286 289 294 296 298 301 303 307 309 319 324 325

Solltotlsand The lovenc ksttering Transform

331

9.1

331 331 336 347

l"trOd"Cti0" 9.2 The History of the Soliton and Soliton Interactions 9.3 The Boussinerq and Korieweg.de Vries (KdV) Equations 9.4 Solutions of the KdV Equation, Solitons and Cnoidal Waves 9 5 The Lie Gmup Method and Similarity and Rational Solutions of the KdV Equation 96 Conservation Laws and Nonlinear Transformations 9.7 The Inverse ScatteringTransform (IST) Method 9.8 BIcklund Transformations and the Nonlinear Superposition Principle The Lax Formulation, Its KdV Hicmchy, and the 9.9 Zakharov and Shabat (ZS)Scheme 9.10 The AKNS Method 9.11 EIUClSeS

356 359 363 386 391 400

401

C0"lc"lS

10.

11.

The NonlinearSchriidingerEpuntion and SolitaryWPM

405

10.1 Intmduction 30.2 The One-Dimensional Linear Schdinger Equation 10.3 The Derivation of the Nonlinear Schrodinger (NLS) Emation and Solitarv Waves 10.4 Properties of the Solutions of the Nonlinear SchrbdingerEquation 10.5 Conservation Laws for the NLS Equation 10.6 The Inverse ScatteringMethod for Ihe Nonlinear Schrodinger Equation 10.7 Examples of Physical Applications in Fluid Dynamics and Plasma Phvsics 10.8 Applicalions to Nonlinear Optics 10.9 Exerciser

405 405

NonlinearKldn-Gordonsod Sine-GordonF,qustionr

453

Intmduction The One-Dimension Linear K l e i n - W o n Equation The Two-Dimensional Linear Klein-Gordon Equation The Three-Dimensional Linear Klein-Gordon Equation The Nonlinear Klcin-Gordon Equation and Averaging Techniques 11.6 The Klcin-Gordon Equation and thc whitham Averaged Variational Principle 11.7 The Sine-Gordon Equabon. Soliton and Anti-Solim Solutions 11.8 The Sulution olthc Sine-Gndon Equation by Separation of Variables 11.9 Backlund Tmsfomations f a the Sine-Gordon Equation 11.10 The Solution of the Sine-Gordon Equation by the Inverse Scattenng Method 11.11 Thc Similarity Method for the Sinc-orrdon Equation 11. I2 Nonlinear @tics and thc Sine-Gordon Equation 11.13 Exercises

453 453 456 458

.4symptotie Methodsand NonlinearEvolutionEquaHons

501

12.1 1ntmducti.m 12.2 The Reductive Perturbation Method and

501

11.1 11.2 11.3 11.4 11.5

12

407 412 419 422 424 438 449

459 467 470 475 484 487 491 492 4%

508

521

CO"l€llmD

12.6 The Permrbation Method of Osmvsky and Pelinosky 12.7 The Method of Multiple Scales 12.8 Method of Multiple Scales for the Case of the Long Wave Approximation

d 527 531 538

541 559

Ldex

581

Preface

Overvier Historically, p m a l differential equations originaled from UK SNdy of surfaces in geomemy and for solving a wide variety of problems in mechanics. During the second half of the nineteenth century, a large number of mathematicians became actively involved in the investigation of numerous problems presented by partial differential equations. The primary reason far this research was that partial differential equations both express many fundamental laws of nature and frequently arise in the mathematical analysis of diverre problems in science and engmeenng. The next phase of the development of linear panial differential equations is characterized by the efforts to develop the general theory and various methods of solutions of these linear equations. In fact, partial differenual equations have bssn found to be essential to develop the hemy of surfaces on the one hand and to the soIulion of physical problems on the other. These two areas of mathematics can be seen as linked by the bridge of the calculus of variations. With the discovery of the basic concepts and properlies of distributions, the modern lheory of linear panial differential equations is now well established. The subject plays a central role in madern mathematics. especially in physics, geomemy. and analysis. Allhough origin of nonlinear panial differential equations is very old. they have undergone remarkable new developments during the last half of the twentieth century. One of the main impulses for developing nonlinear panial differential equations has been the study of nonlinear wave propagation problems. These problems arise in different areas of applied mathematics, physics, and engineering, including fluid dynamics, nonlinear optlcr, solid mechanics, plasma physm. quantum field theory. and condenred-mamr physics. Nonlinear wave equations in parltcular have provided several examples of now soIuli0ns that are remarkably different from those obtained for linear wave problems.The best known examplesof these an Ihe corresponding shock waves, water waves, solitons and solitary waves. One of the remarkable properties of solitons is a localized wave form that IS retained after interaction with other solitons. confirming solttons' 'panicle-like' behavior. Indeed, the theory of nonlinear waves and ~olitonshas experienced a revolution over the past thne decades. During this revolution, many remarkable and unexpected phenomena have d m been observed in physical. chemical. and biological systems. Other major achievements of twentieth-century applied mathematics include the discovery of soliton mteractions, the Inverse Scattering Transform (ISn method far finding the explicit exact soIu1100 for several canonical partial differential equations. and asymptotic perturbabon analysis for the investigation of nonlinear evoluhon q u t i o n s .

XI"

PPCffZe

One of the major goals of the present book is to provide an accessible workmg knowledge of some of the current analytical methods required m modem mathematics, physics, and cngineenng. So the writing of the book was greatly influenced by the emphasis which Lord Rayleigh and Richard Feynman expressed as follows:

"In the mathematical investigation I have usually employed such methods ar, present themselves naturally to B physicist. The pure mathematician will complain, and (it must be confessed) sometimes with justice, of deficient rigor. But to this question there are two Eider. For, however important it may be to maintain a uniformly high Standard in pure mathematics, the physicist may occasionally do well to reit content with arguments which are fairly saurfactory and conclusive from his point of view. To his mind. exercised m a different order of ideas, the more severe procedure of the pure mathematician may appear not more but less demonstrative. And further, in many c a ~ mof difficulty to insist upon highest standard would mean the exclusion of the subject altogether in view of the space that would k required."

LORD RAYLEIGH " ... However, the emphasis should be somewhat more on how to do the mathematics quickly end easily, and what formular, are me. rather than the mathematicians' interest in methods of rigmous proof."

RICHARD P. FBYNMAN Audience and Organization This book provides an introduction to nonlinear partlal differenual equations and to ulc basic methods that have been found useful for finding the solutions of these equations. While teaching B course on p d a l differential equations. the author has had difficulty choosing fextbooLsto accompany the lectures on some modern topics in nonlincarpartlal differential equations. The book war, developed as a result of many ycars of experience teaching partial differential equations at the s e n m undergraduate andlor graduate Ievelr. Pm of this book have also been used to accompany lectures on special topics in nonlinear partial differential equations at Indian universities during my recent visit on a Senior Fulbright Fellowship. Based on my experience. I believe that nonlinear partial differential equations are best approached through a sound knowledge of linear partial differential equauons. In order to make the book self-contained, the first chapter deals with linear p d a l differential equations and their methods of solution with examples of applications. There IS plenty of material in this book for a twosemester graduate level course for mathematics, science, and engineering students. Many new examples of applications to problems in fluid dynamics. plasma physics, nonlinear optics, gas dynamics, analytical dynamics. and acoustics are included. Special emphasis is given to physical. chemical, biological, and engineering problems involving nonlinear wave phenomena. It is not essential for the reader to have a thorough knowledge of the physical aspect of these topics. but limited knowledge of at least some of them would be helpful. Besides. the book is Intended to serve as a reference work for those

seriously interested in advanced study and research in the subject, whether for its own sake or for its applications to other fields of applied mathematics. mathemtical physics, and engineering science. Another reason for adding lhis book to the literatureis the fact that studies an continually being added to the theory. methods of solutianr. and applications of nonlinear panial differential equations. It IS becoming even m m desirable for applted mathematicians, physicists, and engineering scientists to pursue study and research on these and related topics. Yet it ISincreasingly difficult to do so, because major articles appear in journals and research monographs of widely different nature. Some of these occur in papers scattered widely through the vast literatun, and their conneciion~are not readily apparent. Ths diffkulty might be alleviated if a single book on nonlinear partial differential equations contained a coherent account of the recent developmentr, especially if written to be accessible to both graduate and post-graduate students. The field IS growing fast. It is my hope that the book will first interest, then prepare readers to undertake research projects on nonlinear wave phenomena, reaction-diffusion phenomena, soliton dynamres, nonlinear instability and other nonlinear real-world problems, by providing that background of fundamental ideas, results, and methods essentialto understanding the specialized liteatlln of this vast area. The book is aimed 81 the reader interested in a broad perspective on the subject. the wide vanety of phenomena encompassed by it and a working knowledge of the most i m p o m t methods for solving the nonlinear equations. Thore interested in more rigorous ueatment of the mathematical theory of the subjects covered may consult some outstanding advanced books and treatises, listed in the Bibliography. Many ideas, principles, results, methods, examples of applications, and exercises presented in the book are either motivated by, or borrowed from work cited in the Bibliography. The author wishes to express his gratitude to the authors of these works. The first chapter provides an introduction to linear partial differential equations and to the methods of salutionr of these equauonr. and to the basic propertis of these solutions, that gives the reader a clear understanding of the subject and its varied examples of applications. Chapter 2 deals with nonlinear model equations, and variational principles and the Euler-Lagrange equauons. Included me variational principler for the nonlinear Klcin-Gordon equation and for nonlinear water waves. The third and fourth chapters are devoted to the first-order quasi-linear and nonlinear equations and to the methad of characteristics for solving them. Examples of applications of these equatiam to analytical dynamics and nonlinear opuss are included. Chapters 5 and 6 deal with conservation laws and shock waves. and kinematic waves and specific real-world nonlinear problems. The concept of weak or discontinuous solutions is introduced in Sectton 5.4. Several sections of chapter 6 discuss the propeniei of solutions of several nal-world nonlinear models that include uoffic flow, floodweves, chromatographic models, sediment canspon in nvers, glacier flow. and roll waves.

XVI

PdOU

Chapter 7 is devoted to nonlincardispmivc waves. Whitham's equations, and Whitham's averaged variational principle. This i s followed by the Whitham instability analysis and ISapphcatlanr to nonlinear water waves. In Chapter 8, we sNdy the nonlinear diffusion-reaction phenomena. and Burgerr' and Fisher's equations with physical applications. Special attention IS given to traveling wave solutions and their stability analysis. similarity methods and similarity I O I U ~ ~of~ diffusion S equations. Chapter 9 develops the theory of solitons and the Inverse Scattering Transform. Many recent rcsulu on Ihe basic properlies of the Konwq.de Vnes (KdV) and Boussincsq equations are discussed in some detail. Included are Bicklund transformations, nonlinear superposition princrple, the Lax formulation and its KdV hierarchy.

The nonlinear Schrodinger equation and solitary waves are the main focus of Chapter 10. Special attention i s paid to examples of applications to fluid dynamics. plasma phyacr, and nonlinear optics. Chapter I 1 ISconcerned with the theory of nonlinear Klcin-Gordon and meGordon equations with applieauons. "he soliton and anti-sobton solutions of the sine-Gordon equation an described. "he inverse scattering method. the similarity method. and the method of separation of variables For the sine-Gordon equation are developed with examples. The final chapter deals with nonlinear evolution equations and asymptotic methods. Several asymptotic pelrurbauon methods and the method of multiple sealer are developed for the solutions of quasilinear dissipating systems, weakly and strongly dispersive systems. Salient Features The b w k contains 450 worked examples. examples of applications, and exercises which include some selected from many standard treatises as well as from recent research papers. It 1s hoped that they will serve BS helpful self-tests for understanding of the theory and mastery of the nonlinear panial differential equatmnr. There examples and examples of applications were chosen from the areas of panial differential equations. geomeuy, vibration and wave propagation. heat conduction in solids, elecvic circuits. dynamifal systems, fluid mechanics, plasma physics, quantum mechanics. nonlinear optics. physical chemistry, mathematical modeling. population dynamics. and mathematical biology. This varied number of examples and exercises should provide something of interest for everyone. me exercisesm l y complement the text and range From the elementary to the challenging. This book is designed as a new S O U ~ Efor ~ modern topics dealing with nonlinear phenomena and their applications for future development of this important and useful subject. I u main features are listed below: 1. A systematic mathematical treatment of some nonlinear panial differential equations, the methods of the SoIutionS of there equations. and the basic

PreJnce

2.

3.

4. 5.

6.

7.

XYii

pmpenies of these solutions that gives the reader a clear understanding of the subject and its vaned applications. A detailed and clear explanation of every concept and method which IS i n d u c e d . accompanied by carefully selected worked examples, with special emphasis being given to those topics in which students experience difficulty. A wide vanety of modem examples of applications carefully selected from arcas of fluid dynamics. plasma physics. nonlinear optlcr, ~rrlitondynamics. analytical dynamics. gas dynamics, and acoustics to provide motivation. and to illusmte the wide variety of real-world nonlinear problems. Most of the m n t developments in the subject since the early SIXU~S appear here in book form for the first time. Included also is a broad coverage of the essential standard material on nonlinear partial differential equations and their applications that is nor readily found in any texts or reference bwkr. A ruiking balance ktween the mathematical and physical aspects of the s u b p t IS maintained. The book is organized with sufficient flexibility so as to enable insmctors 10 select chapters according to length. emphasis and level of different EOUrses.

8. A wide s p m m of exeiciscs has been carefully chosen and included at the end of each chapter so the reader may funher develop both manipulative skills in the applications of nonlinear equations and a deeper insight into this modem subject. 9. The b wk provides information that puts the reader at the forefront of current research. An updated Bibliography is included to stimulate new interest in future sNdy and research. 10 Answers and hints to selected exercises with original source arc provided at the end of the book for additional help to students.

Acknowledgements I n preparing the bwk, the author has been encouraged by and has knefited

from the helpful comments and criticism of a number of faculty, post-doctoral and doctoral students of several universities in the United States. Canada, and India. The author expresses his grateful thanks to these individuals for their interest m the book. My special thanks to lackk Callahan and Ronee Trantham who typed a manuscript with many diagrams and cheerfully put up with constant changes and revisions. I n spite of the best cffons of everyone mvolved, some typographical erron doubtless remain. I do hope that these are bath few and obvious, and will cause minimum confusion. Finally, the author wishes to express his special thanks to Mr. Wayne Yuhasz, Executi~eEditor, and the staff of Birkhsuser for their help and cooperation. I am deeply indebted to my wife. Sadhana,for her understanding and tolerance while the bw k was k i n g wnlten.

Lokenath Debnath University of Central Florida

Linear Partial Differential Equations 1.1

Introduction

Partid differential equations anse frequently m the formulation of fundamental laws of nature and in the mathematical analysis of a wide variety of problems in applied mathematics, mathematical physics. and engineering science. This subject plays a centel role m modern mathematical sciences. especially tn physics, geometry and analysis. Many problems of physical interest are described by partial differential equations with appropnate initial andlor boundary conditions. 7hese problems are usually formulated as initial-value problems. boundary-value problems, or initial-boundary-valueproblems. In order to prepan the reader for study and reseearchin nonlinear w i d differenual equations. a broad coverage of the essential standard material on linear partial differential equations and thelr applications i s required. 7hir chapter provides a review of basic concepts. principles. model equsuonr and their methods of solutions. 'Ibis is followed by a systematic mathematical teamen1 of the theory and methods of solutions of second-order linear partial differential equations that gives the reader a clear understanding of the subject and its varied applicauons. Linearpartial differential equations ofthe second order can be classified as one ofthree typss, hyperbolic. parabolic, and elliptic and reduced to an appropriate canonical or normal form. The classification and method of reduction is described in Section 1 5 . Special emphasar IS given l o various methods of ~ o h t m nof~ the initial-value andlor boundary-value problems associated with the t h e t y p of linear equations, each of which shows an entirely different behavior in propenies and conswction of solutions. Section 1.6 deals with the solution of linear partial differential equations wing the method of separalron of vanablu combined with the superposition principle. A brief discussion of Fourier. Laplaee. and Hankel uansforms is included in Sections 1.7-1.10. These integral transforms are then applied to solve a large variety of initial and boundary problems described by partial differential equations. The transform solution combined with the convolution theorem provides an elegant represenlation ofthe solution for initial-value and boundaryvalue problems. The final section is devoted to Green's functions for solving a widz variety of inhomogeneous partial differential equations of most common interest. This method can be made considerably easier by using generalmd functions combined with appropriate integral oansforms.

Linear Partial Dz#kentid Equnonr

2

1.2

Basic Concepts and Definitions

A parliol differenrial equation for a function u(x.y ,...) is a relationship ,... and can be written as between u and its partial derivatives u,.u,,u,,u,.u,

F(X.Y.~,~..~,,~,,~,,U,, ,...)= 0.

(1.2.1)

where F is some function, x.y, ... are independent variables and n(x,y, ...) is calleda dependent variable. lk order of B partial differential equation is defined in analogy with an ordinary differential equation as the highest order denvative appearing m (1.2.1). 7he most generalfirst-order partial differential equation can be written as

F ( ~ , Y ~ U , U . ,= U0. ,) (1.2.2) Similarly, the most general second-order panial differential equation in two independent variablesx, y has the form = 0, (1.2.3) F(x,y.u,u.,u,,u,.u,,u,) and so on for higher-order equations. For example, x",+y",=o. (1.2.4) x u , +yu, =x'+y', (1.2.5) U",+U,

=".

+2",

+Urr

(1.2.6) (1.2.7)

":+u; = 1.

are fim-adcr equations,and U"

U"+UD

u.-c1u,

=o, =o, =f(x,y).

(1.2.8) (1.2.9) (1.2.10)

are second-order equations. Rnally,

",+ "U, + "-

= 0,

(1.2.1I)

=o, ( I 2.12) are examples of third-order and founh-orderequations, respectively. A partial differential equation IS called linear if it is linear in the unknown function and all its derivatives with coefficients depend only on the independent vanabler. It is called qunsi-linear. if it is linear in the highest order denvative of the unknown function. For example. (l.Z.4)-(l.2.5),(l.2.8)-(I.2.l0)end (1.2.12)arc linear equations. whereas (1.26)and (1.2.11) an quasi-linear equations. It is possible to write a partial differential equation in the operator form ".+Urn

4 4 x ) =f(x). when L, IS an operator. ?be operator

(1.2.13)

4 is called a lineoroperntor if it sausfies

the Property

&(a"+ hv) = a Lp+ h Lzv for any two functions u and v and for any two eonstantsa and b.

(1 2.14)

1.2 Bosic Concepts and Dcfnioonr

3

Equation (1.2.13)iscalled linearif L, is a linear operator. Equation (1.2.13) is called an inhomogeneous (or nonhomogeneous) linear equation. If f ( r )c 0. (1.2.13) is called a homogeneous equation. Equations (1.2.4),(1.2.8)-(l.2.9)and (12.12) are linear homogeneous equations, whereas (l.2.5), (l.2.lO) are linear inhomogeneous equations. is not An equation which is not linear is called a nonlinror equation. If linear, then (12.13) is called anonlineorequation. Equations (l.2.6)-(l.2.7) and (1.2.1 I) are examples of nonlinear equations. Aclorsicnl solution (or simply B rolution) of (1.2.1)is an ordinary function u = u(x.y ....) defined in some domain D which is continuously diffcrentiable such that i s all panial derivatives involved in the equation exist and satisfy (l.2.l) identically. However, this notion of classical solution can be extended by relaxing the requirement that u is continuously dijjercnriablc over D. The solution u=u(x,y, ...) is called B weak ( ~ ~ . g c n e r o l i ~ c d ) s o l u t iof o n(12.1) if u or its panial derivatives are discontinous in some- or all p i n t s in D. To i n d u c e the idea of B general solution of a panial differential equation, we solve a simple equation for Y = u ( x , y) of the f o m

""=a.

(1.2.15)

Inte&mting this equation with respect tox (keepingy fired), we obtain uy = h b ) . when h(y) is an arbibary function of y. We then integrate it with respect t o y to find u(w)=jh(y)dy+f(x), where /(I) is

an arbiwry function.Or. equivalently, u(x.y)=f(x)+g(y).

(12 . 1 6)

where f ( x ) and g(y) are arbibary functions. The solution (1.2.16) is called lhe general solution of ths second-order equation (1.2.15). In general, the general solution of a partial differential equation is an expression that involves ahitrary functions. This is B striking contrast to the general solution of an ordinary differential equation which involves arbitrary constants. Further, a simple equation (1.2.15) has infinitely many solutions. This can be illustrated by eonridering the problem of constmetion of partial

differential equations from given a r b i b q functions. For example. if u(r.t)=f(x-ct)+g(x+ct), (1.2.17) wherefandgarearbiwryfunctionsof ( x - c t ) and (.r+o),respectively. then,

"" = f"(x-ct)+g"(x+ct).

u,, = c'f"(x -a)+cZg"(x +a) = C'"", where primer denote differentiation with respect to the appropriate argument. Thus, we obtain the second-order linear equation. called the wave equation,

4

Thus. Ihe function u(x.r) defined by (1.2.17) satisfies (1.2.18) ~ m p e e t r v eof the functional forms of f(x-cr) andg(x+cr). provided f and g are at least mice differentiable functions. Thus,the general solution of equation (1.2.18) is given by (l.2.l7) which contain$ arbitrary functions. In the casc of only two independent variables x , y. the solution u(x,y) of the equation (1.2.1) is visualized geomrrricolly a v n surface. called an integral wj'ace in the (x,y,u) space.

1.3

The Linear Superposition Principle

The general solution of a linear homogeneous ordinary differential equation of order n IS a linear combination of n linearly independent solutions with n arbitrary constants. In other words. if .,(.), ",(=) ....,".(.) are n linearly independent solutions of an nth order, linear, homogeneous, ordinary differential equation of the form L u ( x ) = 0, then. for any "bitrary

(1.3.1)

ConStan& c,,c,'.,,. C".

44 =&%(4

(1.3.2)

t=l

represents the most general solution of (1.3.1). This is called the Linear Superposition Principle for ordinary differential equations. W e note that the general solution of (1.3.1) depends on unctly n mldrrmy comrmts. In the ease of linear homogeneous panial differential equations of ulc form L,u(x)=0. (1.3.3) the general solution depends on arbitrary functions rather than arbivary constants. So there are infinitely many solutions of (1.3.3). If we represent this ) u.(x) ...., then, the infinite set of solutions of (1.3.3) by u , ( x ) , u x ( x,_._. infinite linear combinations

+)=

X:."..(X).

(1.3.4)

"4

where en an arbitmy cons1BntS. m general. may nor be again B solution of (1.3.3) because the infinite series may not be convergent. So, for Ihe ease of partial differential equations, the Superposition Principle may not be me- in general. However. if there are only a finire number of solutions ul(x).y(x) ,....u.(x) of the patid differential equation (1.3.3). then,

44=CC"."k).

(1.3.5)

.-I

again is a solution of (1.3.3) as can be verified by direct substitution. As with

linear homogeneous ordinary differential equations. the principle of superposiuon appliesto linear homogeneous partial differential equauonsand u(x) rcpresem B

1.3 T k Linear SupL8pesiiion Principle

5

solution of (1.3.3). provided that the infinite series (1.3.4) is convergent and the operator 4 can be applied to the series term by term. In order to generate such an infinite set of solutions u.(x). the method of separation of variables is usually used. This method, combined with the superposition of solutions. is usually known as Fourier's method, which will be descnbed in a subsequent section. Another type of infinite linear combination is used to find the solution of a given partial differential equation. This is concerned with a family of solutions u ( x . k ) depending on a continuous real parameter t and a function e ( k ) such that

-

(1.3.6)

is convergent. Then,under certain conditions, this intcgral. again. is a solution. This may also be regarded BS the Linear SuperpositionPrinciple. In almost all cases. the general solution of a partial differential qustion is of little use since it has to satisfy other supplementary conditions. usually called initial or boundcry condams. As indicated earlier, the general solution of a linear partial differential equation contains arbitrary functions. This means that then arc infinitely many solutions and only by specifying the initial andlor boundary conditions, we can determine a specific solution of mtcrest. Usually, both initial and boundary conditions arise from the physics of the problem. In the case of panial differential equations in which one of the independent vanables is the time I, an initial condition($) specifiesthe physical state of the dependent variable "(=,I) at a panicular time I= to or I = 0. Often u(r.0) andlor u,(x,O) are specified to determine the function u(x.1) at later times Such conditions are called the Coucky (or initial) onditionr. It can k shown that there conditions are neccsrary and sufficient for thc existence of a unique solution. The problem of finding the solution of the initial value problem with prescribed Cauehy data on the line I= 0 is called the Cauehy problem 01 the initial-value problem. In each physical problem. the governing quation is to be solved within a given domain D of space with prescribed valucs of thc dependent variable u(x.1) given on the boundary JD of D . Often, the boundary need not enclose B finite volume in which case, part of the boundary is at infinity. For problems with a boundary at infinity, boundcdness conditions on the behavior of solution at infinity must be specified. This kind of problem is typically known as a boundq-value problem which is one of the most fundamental problems in applied mathematics and mathematical physics. There an thme imponant types of boundary conditions which arise frequently in formulating physical problems. These are (a) Dirichlet condirionr,when u is prescribed at each point of a boundary JD of a domain D . The problem of finding the solution of B given equation 4.k) = 0 inside D with prescribed values of u on JD is called the Dirichlcr

-

boundary-volus problem;

tiwar Panid Digerenriol Equations

6

(b) Neumonn conditions, where values of normal derivative

*

on the Jn boundary JD are specified. In this case, the problem is called the Neumann boundary-voiuc problem: (c) Robin

conditionr, where

&+ow (x 1

IS specified

on JD. The

enresponding problem is called the Robrn boundn~-volucproblem. A problem described by a partial differential equation in a given daman with a set of initial andlor boundary conditions (or other supplementary condiuons) is said to be well posed (orproperly posed) provided the following criteria are

satisfied: (i) existence: There exists at least one solution of the problem. (18) unipucnesr: There is at most one solution. (iii) stability: The solution must be stable in the sense that it depends continuously on the data. In other words, a small change in the given data must produce a small change m the solution. The stability criterion is essential for physical problems. A mathematical oroblem IS usuallv considered ohvsicallv realistic if a small chance in eiven data pmduccs correspondingly a small change in the solution. According to the Cauchy-Kowalewski theorem, the solution of an analytic Cauchy problem for paRial differential equations'eiists and is unique. However, B Cauchy problem for Laplace's equaiton is not always well posed. A famous example of a non-well-posed (or ill pmed) problem is first given by Hadamard. Hndomord's exnmph deals with Cauchy's initial-value problem for the Laplace equation v*l4I u, +Idn = 0, 0 < y < (1.3.7) with the Cauchv dam .

I

I

"

I

-

u(x.0) = 0

-

and

u) ( x . 0 ) =

(t)

sinnr,

(1.3.8)

where n is an integer representing the wavenumber. These data tend to zeio uniformly as n + . It can easily k verified that the solution of this problem is given by u(x.y)=(~)sinhnysinnr.

(1.3.9)

As n + -, this solution doer not tend to the solution u = 0. In fact, solution (1.3.9) represents oscillations in x with unbounded amplitude n-'dnh ny which tends to infinity as n - + O D . In other words. although thc data change by an arbiwrily small amount, Ihc change in the solution is infinitely large. So the problem is certainly not well posed, that is, the solution does not depend c o n t i n ~ ~ on ~ ~the l y initial data. Even if the wavenumber n is a fixed, finite quantity, the solution is clearly unstable in the sense that u(x.y)+as y + for any fixed X . such that sin ILI # 0 .

-

I 4 Some lmporronf Closslcol Linear Model Equations

7

On the other hand, the Cauchy problem (see Example 1.5.3) for the simplest hyperbolic equation (1.5.29) with the initial data (1.5.35ab) is well posed. As to the domain of dependence for the solution. U(X.I) depends only on those values of f({)andg(c) for which x - n S ~ S x + e t . Similarly, the Cauchy problems for parabolic equations me gennally well posed. We conclude this section with a general remark. The existence. uniqueness, and stlbility of solutions are the basic requirements for B complete description of a physical problem with appropriate initial and boundary conditions However, there are many situations in applied mathematics which deal with ill-posed problems. I n recmt years. considerable progress has been made on the theory of &posed problems, but the discussion of such problems is beyond the scope of this book.

1.4

Some Important Classical Linear Model Equations

We start with a special type of second-order linear panial differential equation for the following reasons.Fmt, second-order equations arise more frequently in a wide variety of applications. Second. their mathematical treatment IS simpler and caSier to understand than that of first-order equations in general. Usually, in almost all physical phenomena, the dependent variable u = Y ( x , ~ , L . I ) is B function of three space variables and time vanable I. Included here are only examples of equations of most common interest. Example 1.4.1 The wave equation is un - 2 V f u = 0 ,

(1.44

Whse

V..dl+d'+d' ax>

mi'

QJ

(1.4.2)

and e is a constant. This equation describes the propagation of a wave (or disturbance). and it arises in a wide variety of physical problems. Some of these problems include B vibrating string. vibrating membrane. longitudinal vibrations of an elastie rod or beam. shallow water waves. acoustic problems for the velocity potential for a fluid flow through which sound can be transmitted, transmission of elecoic signals along a cable. and both elecmc and magnetic fields in the absence of charge and dielecuic.

Exomple 1.4.2 The heat or diffusion equation u. - XVZ" = 0.

IS

(1.4.3)

8

tineoar Poniol Dt@ercnliol Eqmions

the Stokes and Rayleigh problems and also the diffusion of vortieity from B voltex sheet.

Example 1.4.3 The Laplace equation is v% = n_ . ~ ... .., This equation is used to describe elecmstatic potential in the absence of charges, gravitational potential in absence of mass, equilibrium displacement of an elastic membrane. velocity potential for an incompressible fluid flow. temperatun in a steadystate heat conduction problem, and many other physical phenomena.

-

Ewmple 1.4.4 The Poisson equation is v*u= f(X.Y.2).

(1.4.5)

where f(2.y.z) is a given function describing a source or sink. This is an inhomogeneous Laplace equation, and hence, the Poisson equation is used to ENdy all phenomena described by the Laplace equation in 6x presncc of external sources or sinks.

Example 1.4.5 The Helmholtz equation is V’u+h=O, (1.4.6) when I is a constant. This is a time-independent wave equation (1.4.1) with I BS a separation constant. In particular, its rolution in acoustics represents an acoustic radiation potential. Example 1.4.6 The telegraph equation is ug +ow, + bu =u,. (1.4.7) where (1 and bare constants. This equation misees in the study of propagation of electrical signals in a cable of transmission h e . Both current I and voltage V satisfy an equation of the form (1.4.7). This equation also arises in the propagation of pressure waves in the sNdy of pulsatile blood flow in arteries and in one-dimensional random motion of bugs along B hedge.

Ernmap& 1.4.7 The Klcin-Gordon (or KG) equation is ..+($).y=O.

(1.4.8)

whae

2cf Vl rn’

I

(1.4.9)

is the d‘Alembcnianapenilor. A is the Planek con~tant.and rn is a con~tant mass of the particle. Klein (1927) and Gordon (1926) derived a relativistic equation for a charged panicle in an electromagnetic field. It is of conservative dispersive type and played an important role in our understanding of the elementary panicles. This equation is also used to describe dispersive wave phenomena in general.

1.4 Some Impnont Classid Lineor Model 4 u r r o n r

9

Example 1.4.8 The time-independent Schr6dinger equation in quantum mechanics is

(&)V'I+(E-V)~=O, where A is the Planek constant. rn is the function is y(x,y,z,t), E is a constant. and

(1.4.10)

mass of the particle whose wave V is the potential energy. If V = 0 ,

(1.4.10) reduces to the Helmholtz equation.

E x m p k 1.4.9 The linear Korteweg.de Vrics (or KdV) equation is ", + au. +p"- = 0 , (1.4.11) where a and p are constants. This describes the propagation of linear, long. water waves and of plasma waves in a dispersive medium.

Example M I 0 The linear Boussinesq equation is ""-a'v~"-p~v'",,=o.

(1.4.12)

when a a n d p are constants. This equation arises in elasticity for longitudinal waves in bars, long water waves. and plasma waves.

Exnmplc L4.11 The hiharmonic wave equation is uo + c2 =0 , (1.4.13) where c is a. constant. In elasticity, the displacement of a thin elastic plate by small vibrations satisfies this equation. When u IS independent of time I, (I .4.13) reduces to the so called b i h o m n i c cqwrrion viU=0. ( I .4.14) This deqctiber the equilibrium equation for the distribution of ~ l r e s s ein~ an elastic medium satisfied by Airy's s u m s function u ( x . y . r ) . In fluid dynamics,

vU

this quauon is satisfied by the sueam function y(x,y,z) in viscous fluid flow. Exompk 1.4.12 The elecuomagnetic wave equations for the eleenic field E, and the polarization P awe c ~ ( E ~ - ~ : E , ) + =Po. , (1.4.15) (P,+o$P)-E,w~E=O,

(1.4.16)

is the permittivity of free space. wo is the natural frequency of the oscillator, c, is the speed of light in a vacuum, and wpis the plasma fruluency. when

Linear Partial D~FennriolEquations

10

1.5

The Classification of Second-Order Linear Equations and The Method of Characteristics

me generalssond-order linear partlal differential equation in hvo independent variables x, y is given by Au, + Bu, + Cu,+Dux + Euy + Fu = G , (1.5.1) when A, B. C. D, 8.F, and G are given functions of x and y or constants. 'Ike classification of second-order equations is bared upon the possibility of reducing equation (1.5.1) by a cwrdinate wnsformation to cnnonicnl or stondnrd form at a point. We consider the uansfarmation from X. y to 5.9 defined by 5 = 0 ( . , y ) , 9 = V(..Y). (1.5.2ab) where 4 and y are twice continuously differentiable and the Jacobian J ( x . y ) = &y, y,#, is nonzero in a domain of interest so that x, y can be determined uniquely from the system (1.5.2ab). men, by the chain rule,

-

u.==U1C.+u,g.

u"=u15,+u,rl,.

UU=U~~S:+2U1"C~9~++u"~~:+U~Ca+U,B.

lh=u.C:+2u,C,1,+u,,rl:+~~5w+u,9,". un

=" ~~5,5,+~~.(5,s,+C,9.)+~,rl'l,+~15,+~,9,.

Substituting Ihere mulu in quation (1.5.1) gives A'ua+B'u,,+C'u,,+D'u~ +E'u,+F'u=G'. where

A' = A<:

(1.5.3)

+ BCJ, + 4;.

B' =2A<,rl, +B(C,rl, + 5,%)+2cC,rl,. C'=Aq:+Bq,q,+Cq:. D' = AC,

+ 85, + C<, + DC, + 4,.

E' = Arl, + 69, + C?,?+ Drlx + Ev,, F ' = F and C ' = G .

Now.Ule problem is to determine simplest possible form. We choose Or, more explicitly,

< and 9 so that equation (1.5.3) takes the

5: and 9 such Ihat A'

A' = AC: + BC,<, + CC: = 0 , C'=Aq:+Bq,qY+Cq:=O.

= C' = 0 and B'

f

0.

(1.5.4) (1.5.5)

1.5 The Cf~scficarionof Second-OrderLinznor Equoaonr

11

5 =<

These two equations can be combined into a single quadratic equation for or rl

(1.5.6)

We

consider

level

and

<=)(x,y)=constant=C,

curves

q = l(x,y)=con~tant=C,.Ontheseeurves d< = <,&+<.dy =O. dq = q,&+ 7.4,

(1.5.7ab)

(1.5.8ab)

Thus. the slopes of both level curves an the mots of the same quadratic equation which is obtained from (1.5.6) BS

A ( g -B(Z)+C=O.

(1.5.9)

and thc roots of this equation are given by (1.5.10ab)

2 - ; A ( B * m ) .

These equations are known as the chorocIcrisric equafionrfor (1.5.1). and their solutions are called the charocrcrirric curves or simply the characteristics of equation (1.5.1). The solution of the two ordinary differential equations (1.5.10ab) defines t w o distinct families of characteristics ) ( x , y ) = C, and ~ ( xy.) = C, . There are ulrce possible cases to consider.

-

4AC > 0. Equations for which B 2 - 4 A C > 0 arc called hyperbolic. Integrating (1.5.1Oab) gives two real and distinct families of characteristics ) ( x , y ) = C, and ~ ( 1 y ),= C,. where C, and C, arc constsnts of integration. Since A. = C = 0,

Case I. B'

and B'

i0. and dividing

by B'. equation (1.5.3) reduces to the form

ufo = -$(D'ul

+E'u,

+ F'u -G')

(1.5.1 I )

= H, (say).

This is called thepist cononicolform of rhe hyperbolic cquarion. If the new independent variables a=<+% B=<-q m intrdund, rhen u ~ = u p ~ + u u p ~ = ~ . + uUu ~.

(4< =(..). 5 +(.), Co-uently,

B< =(..-.J.

equation (1.5.1 I) becomes

=

U

'

~

~

~

+

U

~

(1.5.12ab) !

~

~

I+(% -u& ' I = urn -us#

=

~

~

-

U

~

Lineor Poniol Diferrntiol Equotionr

12

u.. -uuD =H2(a,S,u.u..ue).

(1.5.13)

This is called the second canonicalform of Be hyperbolic equntion. It is imponant to point out that characteristics play a fundamental role in the theory of hyperbolic equations. Case 11. 6'-4AC=O. There IS only one family of real characteristics whose slope, due to (l.5.lOab)

is given by

4v- B (1.5.14) dr 2A Integration of this equation gives 5 = @ ( r , y ) = c o n s t u t (or q = q/(x.y)=eonslmt). Since B2=4ACandA'=0,weobtain o = A* =A<: + B ~ J , + cg; = (J7i 5, +* 5,)2. It then follows that B' = 2A5.q' + B(5.a + 5,vX)+2cS,q,

=z(J;T<.+*5,)(fi q.+* ' I , ) = o for an arbiwry value of 9 which is independent of 5:. For example. if q = y . the lacobian is nonzeroin the domain of parabolicity. Dividing (1.5.3) by C # 0 yields u,,=H,(C,t).u.u~,u,).

(1.5.15)

This is known BS the comnicalfom ofthc pmbolic rquntion. On the other hand. if we choose 9 = q/(x.y)=constant as the integral of (1.5.14).yuauon (1.5.3)asmmcsthcform Ut< = H $ , W . U < , " " ) . (1.5.16) Equationsfor which B'-4AC=O Case 111. BZ-4AC
arscalledpmbolic.

-

EqusUons for which B' 4AC< 0 are called elliptic. In this cme, equations (1.5.10ab) have no rcal solutions. So there are two families of complex eharaetenstics. Since roofs <,q of (1.2.lOab) are complex conjugates of each other,we i n d u c e the new nal variables BS

a=;(<+11).

B =$< - 7).

sothat < = a + $ and q = a - i S . We use (1.5.17ab) to rransfonn (1.5.3) into the fann A"", +B"u@ +C"u, =H,(a,B,u,u,,u,),

(1.5.17ab)

(1.5.18)

1.5 The Clorsificotion of Second-Onler Imem Equrlionr

13

where the eaeffieicnu of this equation assume the same form as the cocfficienu of (1.5.3). It can easily be verified that A' = 0 and C' = 0 take the form A" 6'fiB" = 0 which M satisfied if and only if A"=C*' and B-=O. Thus. dividing by A", equatlon (I.5.lS) reduces to the form

-

U,+UEE

=+!=Hs(a.B,u,u.."E).

(1.5.19)

This IScalled the cownicalfom ofrhe ellipiic equrion. In summary, we state that quation (1.5.1) is called hyperbolic, parabolic, or

ciiiptic at a point (x,,yo) accordinglyas ~ ( ~ . , ~ . ) - 4 A ( = . . y ~ ) C ( x , , y>, )= c O .

(1.5.20)

If it is m e at all points in a given domain, then, the equation IS said to be hyprrbolie.porobolic or elliptic in that domain. Finally, it has been shown above that. for the ease of two independent variables. a transformation can always be found to mnsform the given quation m the canonical form. However. in the ease of several independent variables, in general, it is nor possible to find such a uansformatian. These three types of partial diffenntial equations arise in many areas of mathematical and physical sciences. Usually, boundary-value problems are associated with elliptic equations, whereas the initial-value problems arise in connection with hyperbolic and parabolic equations.

Excmplr 1.5.1 Show that (a) t k wave equation ua - cauy = 0 is hyperbolic. (b) the diffwmn equation u, - mm= 0 is parabolic. (c) the Laplace equation up + u, = 0 is elliptic. (a) the Tncomi equation um +xu,, = 0 is elliptic for x 1 0 , parabolic for x = O ,and hyprrbolic for x O f o r d l x and I. So. the wave equation is hyperbolic everywhere. Simdarly, the reader can show (b) and ( E ) . Finally, for ease (d), A = l . B=O. C = x . hence, B' -4AC= - 4 x < 0. = 0 , or > 0 according BE x > 0,x = 0,orx < 0, and the result follows. Example 1.5.2 Find the characteristic equations and charactenstics. and h", reduce the equalion xu, +",? =xa (l.5.21)

to canonical form. In this problem. A = x , B = O . C = I , and B2-4AC=-4x. Thus, the equationishyperbolicif x O . The ehsraneristics equationsM

or C = y + 2 6 + constant,

q = y - 2 6 = comlmt. (1.5.23ab)

These represent two branches of the parabolas (y-c)' =- 4x wherc e is B constant. The farmer equation = constant) gives a branch with positive slopes. whereas the latter equation ( q = constant) ~eprcscntsa branch with negative slopes as shown in Figure 1.1. Both branches are tangent to the y-axis which is the single characteristic in the parabolic region. Indeed. the y-axis is the envelope of the characteristics for the hyperbolic region x < 0.

(e

Figure 1.1 Characteristics are parabolas for r < O .

For x < 0 . we use the vansformations C=y+26, q = y - 2 6 to reduce (l.5.2l) to the cmonieal form. We find

(1.5.24ab)

16

Linear PorrrOl Diflenntiol Equolionr

u,+uuu-+=(g

u..

+uuu =

;+(;I. "D

(1.5.27)

This is the desired canonical form of the elliptic equation. Finally. for the parabolic e a ~ e( x = O ) , equation (1.5.21) reduces to the canonicalform ",=0. (1.5.28)

In this case. the characteristicdeterminedfrom = 0 is x = 0. Th&t is, the ydy axis is the characteristiccurve. and it represents a c w e across which a transition from hyperbolicto elliptic form mkes place. Eromph 1.5.3 (The Cauehy Problem for the Wave figunrion). The onedimensionalwave equation

""

-2""= 0 (1.5.29) is a special case of (1.5.1) with A=-c'. B = O. a n d C = l . Hence, B' -4AC=4c2 > O . and therefore. the equation is hyperbolic, as already mentioned before. According to (1.5.10ab). the equations of characteristicsare

&=*I. &

or

e

(1.5.30)

< = x - c r = constant. q = x + c r = constant. (1.5.31ab) This shows that the characteristics are straight liner in the (x,t)-plme. The former represents a family of lines with positive dopes. and the latter a family of lines with negative slopcs in the (x,r)-plane. In terms of new coordinates
("< < - 2 % + 4

so that the wave equation (1.5.29) becomes

-

4c1utq =o. (1.5.32) Since c # 0 , oil = 0 which can be integrated twice 10 obtain the solution

.(C.d=

$(<)+ v(q).

(1.5.33)

where $ and ~yare arbitrary functions. Thus, in terms of Ihe original variables, we obtain

"(1.1) = $(I -cr)+ v(x+cr). (1.5.34) This represents the general solution provided $and v are arbivary but twice differentiable functions. The first term $ ( x - c r ) represents a wave (or disturbance) traveling to the dght with eonrtent speed e. Similarly, Iy(x+cr)

h e o r Pardof Di@renrid Equarionr

18

These waves propagate in the opposite dinction with the same eonstant speed c ~ lshown i in

Figure 1.2.

Spliting of initial data into equal waves.

Figwe 1.2

To investigate ths physical significance of the d'Alcmben solution, it is convenientto rewrite the solution in the form pc, .+" +,f)=i f ( x - c f ) - L I g ( r ) d r + ' f ( x + e r ) + L I g ( r ) d7, (1.5.41) 2c 2 2c 0 =@(x-er)+

whne

@({)

=

f(5)-

2

Y(x+cr),

ig( r)dr, Y(9 )

(1.5.42)

=if( +

Is(r) d7.

9)

zc

( I .5.43ab)

0

Physically, @(x - cr) represents a progressive wave propagating in the positive xdireetion with consLm1 speed c without change of shape as shown in Figure 1.3. Similarly. '?(=+a) also represents a progressive wave uaveling in the ncgativex-direction with the same s p w d Without change of shape.

u = @(I)

u =q x - e t )

x+cr

Figure 1.3

Y

= ct(-2cr)

x+kr

Graphleal representation of solution.

1.5 The Clmsi,icatim of Second-Order tineor Equorionc

19

A more genersl form of ule wave equation is u"-c*(x)u,

=o,

(1.5.44)

where c is a functionof x only. The charactensticcoordinatesM now given by

<=r-j*,

q=r+jL.

44

(1.5.45)

47)

Thus.

"' =--"a'< + U,=ut+""* Ill", n "*=-

i2("

t,-ZuC"+U"")-(u,-u<)-;;i-.

= "<< + ZU<" + "**. Consequently,eq"Bti0" (1.5.44) reduces to

4.4

U"

4ut,, + a ' ( x ) ( u v -ut

In orderto express a' in terms of

) = 0.

(1.5.46)

5 and q , we observe that

11-5=*jdl. 4r) so that x is a function of

(s-5).

(1.5.47)

Thus, a'(*) will be some function of

(7-C). In particular,if .(X)= AX", when A is B constant. so that a'(x)=nAx"'. and when n # I , result (1.5.47) gives

s-<=-l'l A ("-1) z"-'

(1.5.48)

so that

.'(.)=-2".1. s-e

(-1)

Thus. equation ( I ,546) reduces to !he form 4"

Finally. we find that

- " '"- b~- 1 )q (?-
" "t" =--z("-1)

<

)

=

O

1

(s-<)( u m - 4 .

when n = I, a(.) = Ax, and a'(.x) = A and substitution be ued to red-

equation ( I .5.46) D

-.,).

.

(1.5.49)

< =n and 9 =scan A A (1.5.50)

Equation (1.5.49) is called the Euler-Dorboux cpuarion which has the hyperbolic form

Linear Poniol Difenntiol Equntionr

20

(l.5.Sl) where m is a positive integer. we next note that

when m = I, equation ( I S S I ) beeomes (x-y)u,=u,-u, so that (I S.52) reduces to

J’

(1.5.53)

This shows that the solution of (1.5.53) is (x-y)n=$(x)+ ~ ( y )Hence, . the solution of ( I S S I ) with m = I is u ( x . @(.)+V(Y). y ) = 7

(1.5.54)

where 6, and w , are arbiwrv functions

z.so

wemultiply(1.5.51) by (x-y).andapplythedetivativc

that the

J&

result is, due to (1.5.52),

J’

J

J

J

J

(k

S.

+

( x - y) (urn) = ( m + l)(A-*)(uo). (1.5.55) JxaY Jx Hence. if u is B solution of (1.5.51). then. uq is a solutmn of ( I S S I ) with m replaccdby m + l . when m = l . the solution is given by (1.5.54). and hence. the solution of (l.5.51) takes the f m

when $ and

1.6

v are arbih.ary functions.

The Method of Separation of Variables

This mcthad combined with the principle of superposition is widely uscd to solve initial-boundary-value problems involving linear partial differential equations. Usually, the dependent vanable u(x,y) is expressed in the separable fmu(x,y) = X ( x ) Y ( y ) where X and Yare functions of x and y. respectively. In many c w s . !he panid differential equauon rcduccs to two ordinary diffcncntial equations for X and Y. Similar treatment can k applied to equations in three or

1.6 The Method of Sepororron of Variables

21

more independent variables. However, the question of separability of a partial differential equation into two or mom ordinary differential equations is by no means a mivial one. In spite of this question. the method IS widely used in finding SoIutionS of a large class of initial-boundary-value problems. This method of ~olutmnis known as the Fourier method. Thus. the procedure outlined above leads to the important ideas of eigenvalues. eigenfunctmr. and orthogonality. all of which ~n very general and powerful for dealing with linear problems. The following examples illuseate the general nature of this method. Ernrngle 1.6.1 (Tmnsucrsc Vibrnrzon o f 0 Srring). We consider the onedimensional wave equation U" = C%". 0 < x < t, t > 0, (1.6.1) whne c' = T'/p, T' is a eonsmt tension. and p IS the eonitant line density of the shing. The iniual and boundary conditions are "(1.0)= f ( x ) . u,(x,O) = g(x). 0 5 x 5 t , (1.6.2ab) u(O,r)= O = u(t,r), I>O. (1.6.3ab) whmfand g are the initial displacement and initial velocity. respeftlvely. According to the method of separation of variables, we assume B separable solution of the form u(x.f)=X(x)T(r)10. (1.6.4) where X is a function of x only, and Tis a function of r only. Substituting this solution in equation (1.6.1) yields _1 _d2X = _ _ I d'T (1.65) X &' c'T dr' '

Since the Ieh ride of t b s equation is B function ofx only and the right-hand side IE a function of r only. hence, 11 follows that (1.6.5) can be m e only if both sides ~n qua1 to the same consmt value. We then write I d'X 1 d'T (1.6.6) e'T dr' -I' X dr' where I is an arbitrary separation constant. Thus. this leads to the pair of ordinary differentia1equalions -= d'X

&'

U ,!?=,@T.

(1.6.7ab)

dtl

We solve this pair of equations by using the boundary conditions which are obtained from (1.6.3ab) as u(O,r)=X(O)T(t)=O for r > O . (1.6.8) u ( t , r ) = X ( r ) T ( r ) = O for r > O . (1.6.9) Hence.we take T(r)# 0 to obtain X(O)=O=X(l). (1.6.10ab) There an three possible cases: (i) I > 0. ($1) I = 0,and (iii) I < 0. For ease (i), I = a ' > 0.. The solution of (1.6.74 is X(x) =Ad' +Be*'. ( I 6.11)

I?.

Lhcor Partial D8Werentiel Equorions

which together with (1.6.10ab) leads to A = B = O . This leads to B trivial solution u ( x . r ) = O . For case (ii). 1 = 0. In this case, the solution of (1.6.74 is X ( x ) = kr + B . (1.6.12) Then. we use (1.6.10ab) to obtain A = B = O . This also leads to the Vivid solution u ( x , t ) = 0. For case (iii), 1 < 0,and hence. we write 1 =-a' so that the solution of equation (1.6.7a) gives X=Acosm+Bsinar (1.6.13) whence, using (ld.lOab), we derive the nonoivial solution (I .6.14) X(x) = B s i n m , where B is an arbiuary nonzero constant. Clearly. since B # 0 and x(t) = 0, sin x l = 0, (1.6.15) which gives the solution for the parameter

a = q = ( ? ) . n=1.2,3 ._._.

(1.6.16)

Note that n = O ( a = O ) leads to the Vivid solution u ( x , r ) = O . and hence, the case n = 0 is to be excluded. Evidently, we see from (1.6.16) that then m i s s an infinite set of dircnte values of a for which the problem has a nontrivial solution.These values a. arc called the e i g c ~ dand ~ ~the, corresponding solutions are ~ . ( x )= B. sin(?).

(1.6.17)

We next solve (1.6.7b) with 1 =-a: to find the solution for Tm(t)as Tn(t)= Cncm(ancf)+Dn sin(a,cr). (1.6.18) where C. and D. are cmn~tantsof integration. Combining (1.6.17) wth (1.6.18) yields the solution fmm (1.6.4) as u,(x,I)=

["

- '("3 "'("31

a cos "llcf + b sin

(TI

sin - ,

(1.6.19)

when an = C.6.. b. = B.0. are new arbibary constants and n = L2.3 ,.... These

,)=:(

solutions u w ( x , r ) , corresponding to eigenvalucs an

are called the

cigcnfunetions. Finally, since the problem is linear, the most general solution is obmned by the principle of superposition in the fm &t)

= %(an em=+

b. s i n s ) sin(?).

(1.6.20)

"4

provided the series converges and it is twice eontmuously differentiable with respect to x and r. The arbiuary caefflcients a. and b. are determined from the initial conditions (1.6.2ab) which give

1.6 The Method ofSrprorion of Vambles

23

"*,

u(x.O)=f(x)=&a"

,,

u,(x.O)=g(r)=

(I .6.21)

(7)

(92" ) "=, (?!? ).

(1.6.22)

bI a'n

Either by a Fourier series method or by direct multiplication of (1.6.21) and (1.6.22) by sin( am

7)

=$

i

and integrating horn 0 to 1. we can find a" and b.

b.

=&

sin(?)*

=

f(.z)sin(y)dz,

BJ

jg(.).i"(y)di.(l.6.23nb)

in which we have used the result /sin(?)

a_,

(1.6.24)

where S , . are Kronekar delta. Thus, (1.6.20) represents the soluuon where an and b. are given by (l.6.23sb). Hence. the problem is completely solved. We examine the physical significance ofthe solution (1.6.19) in the context of the free vibration of a suing of length 1 . The eigcnfunctions

um(x,t)= (omc o s q r + bn s i n q t ) sin

(y),

(on=?),

(1.6.25)

are called the nth n o m l modes of vibration or the nth honnonie and q reprrsents the dixretespechum ofcimr~r(ormdion)fiercguencyor v. = W- = "c 2x 2 t are called the angular frequencies. The fin1 harmonic (n = I) 1s called the fundnmentol harmonic and all other harmonics (n > I) are called ovcnoncs. The fruluency of the fundamental mode is given by

7 ,-;J-.

&I,=-. Y--

(1.6.26ab)

Result ( 3 . 6 2 8 ) is eonridered as the fundamental law (or Merscnnc h w ) of a smnged musical instrument. The angular frequency of the fundamental mode of uansvem vibration of a suing varies as the square mt of the tension. inversely as the length, and inversely as the square mot of the density. The period of the

=2"=2'which is called the fundamental

fundamental mode is T,

0 1

period.

c

Finally, the solution (1.6.20) describes the motion of a plucked string BJ a superposition of all normal modes of vibration with frequencies which are all integral multiples (on=no, or v. = nv,) of the fundamental frequency. This is the main reason for the fact that svinged instrumens produce mow sweet musical sounds (or toner) than drum instruments.

Limor Poniol Dzfferenaol Eqwrirom

24

In order to describe waves produced in the plucked string with zcm initial velocity ( u , ( x . O ) = O ) , we Write the solution (1.6.25) in the form

n=1,2.3 ,_...

""(x.f)=.si"(~),,,(~),

(1.6.27)

These soluuonr me called sfonding wows with amplitude ansin(?),

which

vanishes 81 x=o,

L, 2,.... 1.

" "

These an called the nodes of the nth h m o n i c . The suing displays n loops separated by the nodes BE shown in Figure 1.4. "=I

n=2

Figure 1.4

Several m d e s of vibration in the string.

It followus frnn elementary trigonometry that (1.6.27) takes thc form

This shows that a standing wave is expressed as a sum of two prognsrive waves of equal amplitudes traveling in opposite directions. This result is in agreement with the d'Alembert solution. Finally, we can rewrite the solution (1.6.19) of the nth normal modes in the f m u,(x.f)=c, sin(?)

cos(~-en),

(1.6.29)

This form of the solution enables us to calculate the kinetic and potential energies of the transverse vibrations. The total kinetic energy is obtained by integrating with respect to x from 0 to I . that IS,

(1.6.30) whcre p is UK line density of the string. Similarly, the total potential energy 1s given by

(1.6.31) Substituting (1.6.29)in (1.6.30) and (I,&l)

when

0"

gives

=E.

Similarly,

= -(n I'T. c")' c o ? ( ~ - Em)= 1 p t m ; c.' c o s ' ( m p &"). (1.6.33) 41 Thus, the total energy of ulc nth normal modes of vibrations is given by

E . = K . ~ V ~ = ~ p l ( m , c , constant. )~=

(1.6.34)

For a given string oscillating m a normal mode. the total energy is pmponional to the squan of the circular frequencyand to the q u a of the amplitude. Finally, the total energy of the system is given by E=

2E. t =

p t 2m:c:,

.=I

(1.6.35)

"=,

which is constant kcause E. = constant.

Example I.6.2 (One-Dimcnsionol Diffusion Equation). The temperature distribution u(x.1) m a homogeneous rod of length t satisfies the diffusion equation U,=KU,,, o < X < f , 1>0, (1.6.36) with the boundary and initial conditions @.t)= o = "(!.t), 12 0 (1.6.37ab) "(X.0) = f ( x ) , 0 s x L 1, (1.6.38) when K is a diffusivily constant. We assume B separable solution of (1.6.36)in the form +,I)= X(x)T(r)# 0. (1.6.39)

Linear Poniol Dz#emntid Equtionr

26

Substituting (1.6.39) in (1.6.36)gives _I _d'X = _ _ 1 aT (1.6.40) x dr' fl d t ' Since the Id-hand side depends only on x and the nght-hand side is a function of time I only. result (1.6.40)can be m e only if both sides are equal to the same constant A . Thus. we obtain two ordinary differential equations

_ d'X _ ,U,=o, dri

dT-Lfl=O. I

(1.6.41ab)

For L 20, the only solution of the form (1.6.39) consistent with the given boundary conditions is "(=,I) I 0. Hence. for negative 1 = -a2, %+lCa~T=o,

$+dX=O,

(1.6.42ab)

which admit solutionsas X(x)=Acosm+Bsinm

(1.6.43)

and (1.6.44) where A. B, and C me constantSof integration. The boundary conditions for X(x) are X(O)=O=X(!) (1.6.45) which are used to find A and B in solution (1.6.43). It turns out that A=OandB#O.Hence. (1.6.46) sin &=O. which gives the eigenvalucs T ( r ) =cexp(-KlI2r).

"II

a = a * = - - , n=1,2,3,,...

(1.6.47)

The value n = 0 is excluded heeaure it leads to a Vivial solution. Thus, the eigenfunctions are given by xn(x) = B. sin(

y),

(1.6.48)

w h a t B. are nonzero comtam.

= E ,we combine (1.6.44) with (1.6.48) to obtain the With a=am solullan for

""(XJ)

as

where am=BnCa is a new constant. Thus, (1.6.47) and (1.6.49) constitute an infinite set of cigenvalues and eigenfunctions. Thus. the moot general solution is obtained by the principle of superposition in the fom

1.6 The Method of Separation of Variables

27

Now,the initial condition implies that f(x)=%a.

(1.6.5 I)

sin(?)

whichdeermines am,in view of (1.6.24). as (1.6.52)

It follows fmm the series solution (1.6.53) that the series satisfies the given boundary and initial conditions. It also satisfies the equation (1.6.36) becausethe series i s convergent for all I (0 S x 6 t ) and f 2 0 and can be differentiated term by term. Physically. tbe temperature dishibution decays exponentially with time f. This shows a sviking conVas1 to the wave equation whose solution oscillates in time 1. The time sale of decay for the nth mode is Td

- i(&y

which is

directly proportional to t' and invcscly proportional to the thcrmal diffusivity. The method of separation of variables is applicable to thc wave equation and the diffusion equation. and also to problems involving Laplace's equation and other equations in two or three dimensions with a wide variety of initial and boundary conditions. We consider the following examples:

Example 1.6.3 (Two-DimcnriomI Difision Equation). We consider that u,=x(u,,+u,). O<xO, (1.6.54) u(x.y,f)=f(x,y)

at

f=O.

on

JD.

(1.6.55)

(1.6.56) when JD is the boundary of the rectangle defined by 0 S x S a, 0 6 y 6 b . The method here IS precisely the same rn in the p n v i w s examplesexcept that we seek a solution of (1.6.54) m the form u ( x , y . r ) = S ( x , y ) T(f)+O. (1.6.57) so that S and 7satisfy the equauons u(.r.y,f)=O

d'S+d'S+&O Jr' J$ c+h-IT=O.

(1.658) (1.6.59)

For I 6 0, separable solution (1.5.57) with the given boundary data leads only to B Vivial solution x(x,y,f)=O. Hence, far positive A , we solve (1.6.58)(1.6.59) subject to the given boundary and initial conditions. Equation (1.6.59)

Liwor Porriol Digrrrnriol Eqvoaonr

28

is an elliptic equauon, end here, we seek a solution S ( x , y) which satisfies the

boundary conditions for 0 I y 5 b. for 0 S x 5 0 . We also reek a separable solution of (1.6.58) in the form S(x,y)=X(x) Y(Y)fO and find that X ( x ) and Y(y) satisfy the equation -X"+ Y + "a = o . X Y that is. S(0.y) = 0 = S(a,y)

S(x,O)

= O =S(x,6)

(1.6.60) (1.6.61) (1.6.62)

or X"+Px=O, Y"+(l-p)Y=O. These equations have to be solved with the boundan, conditions X(O)=O = X(0)

(1.6.64ab) (1.6.65ab)

The eigenvaluc problem (1.6.Mab) with(1.6.65ab)gives the eigenvalues /1.

=o'.

(1.6.66)

(F).

(1.6.67)

and the c o m s p d i n g eigenfunctions x.(~)=A.

sin

where m = 1.2,3,.... Thus,quation (1.6.64b) h o m e r Y + ( L & ) Y =o. (1.6.68) which has to be solved with (1.6.65b). This is mother eigenvalus problem similar La h t almdy considered and leads to the eigenvalucs

a"-&=(g

(1.6.69)

and rhc corresponding eigenfunctiono Y.(~)=B, sin(?),

(1.6.70)

whne n=1,2,3. .... Inoth~rwvords.therolutionofequation(1.6.58) becomes

S,(x.y)= where A,.

X,(x)Y.(y)=A,sin(F)

sin(?).

(1.6.71)

= A-6" at constants together with the eigenvalues

,a

=K,

+(7)1=($+$) XX,

(1.6.72)

1.6 Thr Method ofSepmtion ofvorioblcs

29

where m=1.2.3 ,... andn=1,2,3 ..... With 1 , as eigenvaluer, we solve (1 6.59) to obtain

Tm(r) = s, exp(-i,kr). where B,, are integratingconnants. Finally, the solution (1.6.57) can be expressed as a double series

__

u ( x , y , r ) = x ~ : a , , s i n mm m-l"4

sm '

(3 b

exp(-i,.kr),

(1.6.73)

(1.6.74)

where a,. are constants to be determined from the initial condiuan so that

To find eon it ant^

am, we multiply (1.6.75) by sin

result with respect to x from 0 to (I with fixed y, so that

f %am sin(?)

(5

and integrate the

= j f ( x , y ) sin( ?)h.

(1.6.76)

"-1

The right-hand side is a function ofy and ISset equal to g(y), so that j / ( x , y ) sin(?)

Then. thecoefficients

sin(

T)

.1 ( ,

(1.6.77)

h=g(y).

(m fixed) in (1.6.76) are found by multiplying it by

and integrating with respect t o y from 0 to b, so that

(f)

am = /g(y) sin(?)

d~

I

(1.6.78)

whem a,,

= ( ~ ) ~ ~ / ( ~ , y ) ~ i " ( ~ ) ~ i " ( ? ) ~ d (1.6.79) Y. 0 0

Thus. the solution of the problem is given by (1.6.74) where a, is represented by (1.6.79). The method of consmetion of the solution shows that the initial and boundary conditions an Satisfied by the solution. Moreover. the uniform convergence of the double series justifies differentiation of the series, and this, in turn permits us to verify the solution by direct substitution in the original diffusion equation (1.6.54).

E r m p l e 1.6.4 (Dirichlcr's Problem for a Circle). We consider the Laplace eauatian m oolar coordinates as

tinem Poniol Dr@rmriol Epuorions

30

+.e)=f(e) forail e. (1.6.81) According to the method of separation of variables, we seek a solutron in the fom u(r,B)=R(r)Q(B)#O. (1.6.82) Substihlting this solution in equation (1.6.80)giver E+,E =-B"=1 , R R Q Henee. (1.6.83ab) r'R"+ rR'-U = 0 and W +d Q = 0 . For Q(B). we naturally reqwn periodicboundary conditions e(e+zn)=e(e) for --<e<-. (1.6.84) Due to the penodicity eondltron. for L c 0 , the solution (1.6.82) leads to a mvid solution. So, then are two cases: (i) 1 = 0 and (ii) A > 0. For case (i), we have the solution u(r,e)=(A+Blogr)(CO+D). (1.6.85) Since logr is singular at r = 0, hence, B = 0. For u to be periodic with period 2% C = 0. Hence. the solution u must be con~lantfor . I = 0 . For 1 > 0. the solution of equation (l.6.83b) is @ ( @ ) = AcosJ;ie+BsinJ;ie. (1.6.86) Since Q(B) is periodic with period 2n. hence, must be an integer n so that I = n ' , n = 1 , 2 . 3 ,.... Thu~.solution(l.6.86)kcomes Q(B) = Acosn@+Bsinn@. (1.6.87) The equation (1.6.83a) is the Eulcr equation with rl = n'. It gives solutions of theform R(r)=r'#O sothat(1.6.83a)gives [ a ( a - ~ ) + a - n r* '] = ~ whence. a = f n. Thus, the solution for R(r) is given by R ( r ) = Cr" +Dr*. (1.6.88) Since R(r) + -as I + 0 kcausc of the term 7.". hence, D = 0. Thhus, the solution (1.6.82) reduces to ~ ( re), = cr"(Acosne+ B sinno). (1.6.89) By !he superposition principle, the solution of the Laplace equation within a circular ngion including the origin I = 0 is

1

2,"

u(r.0) = o, + (amcosnO+ b. sinne), (1.6.90) 2 "-1 where o0. an. and bm are constants to be determined from the boundary conditions. and the first term

Lo,

represents the eonstant solution for

a = o (n =O). Finally, using the boundary condition (l.6.81), we derive

31

1.6 The Methai of Scpomlion of Voriobicr

f ( e )= u ( o , e ) = L o+ z a * ( a ncosntJ+b. sinne). 2 "4

(1.6.91)

This is exactly the Fourier series representation for f ( 0 ) . and hence, the coefficients at given by 0

"

=- I

j'f($)cosn$d$,

n = 0 , 1 , 2 , 3,_...

M.0

bn=&Tf($)sinn$d$,

n=1,2.3._...

Substituting the values for a. and b. in (1.6.91) yields the solution

.

,

This formula is known BS Poisson's inregmi formulo representingthe solution of the Laplace equation within fhc circle ofradius a in terms of values prescribed on the circle. It has several imponant eonsequeneco. First, we put r = 0 and 8 = 0 in (1.6.93) to obtain

u(O,O)=&

yf(#)d$.

(1.6.94)

This states that the value of u at the center is equal to the mean value of u on the boundary of the circle. l l i s is called the m a n vaiucpmpcny. We rewrite (1.6.93) in the form u(r,O)= J P ( r . e - $ ) f ( $ ) d $ .

(1.6.95)

Linear Pornol Digcnnriol Eguorionr

32

where P(r, 8 - 0)is called the Poisson kernel given by

p(,,%_+I. (.'-r') 2n n'-2ar eor(%-@)+r' which is zero for r = a but 0 # 0 . Punher.

T[ki P(r.@-0)] f ( W

f(%)=!Eu(r.@)= which implies that

lim P ( r , % - @ ) = 8 ( % - # ) ,

(1.6.97)

,+--

where S ( x )

1.7

IS

!he Dim delta function.

Fourier Transforms and Initial-BoundaryValue Problems

The Fourier rronsfom of u(x.1) with respect to x s R is denoted by B{u(x,r)] = U(k.t) and ir defined by uls integral

-

9[u(x,r)}=U(k.f)=+ jr-'bu(x,t)dr. (1.7.1) 2= when k is red and is called the rrnnrform variable. The inverse Fowter fronsform,denoted by 9.' [U(k,f)} = u(x.r). is definedby

9.' [u(k,f)}= u ( x , r ) =

7 z r- d h U(k.f)dk.

(1.7.2)

Erarnplr 1.7.1 (a)

9{exp(-Qr~)}=& ..p(-$),

@)

d{exp(-ol x I)} =

If

U(X,f)

+ 0 , as 1x1-

E. *-

o>o,

(1.7.3)

>0 .

(1.7.4)

0

-. !hen

which is. by intcgmting by p m . =+[e-%(x,f)]:+& 2n = i k O { u ( x , r ) ] = i k U(k,r).

je- u(x.t)dr (1.7.5)

1.7 Founrr T r m s f o m ond Iniriol.Boundory-Volue Problem

Similarly. if IxI+-

U(X,I)

is continuously n times differentiable and

..__. ("-1).

for k = 1.2.3

* JX'

33

+ 0,as

then,

s{$}=(*T

s{u(x,t)}=(ikT u(t.1).

(1.7.6)

It also follows h m the definition (1.7.1) that

d"

d" 0 =$! ..... B{fi}=". (1.7.7) 5 { % } = 1 * {a,.} at" dt" The definition of the Fourier transform (1.7.1) shows that a sufficient condition for "(%I) to have a Fourier vansfom is that u(x,t) is absolutely integrable in --oo<x<-oo. This existence condition is tw strong for many practical applications. Many simple functions, such BJ a constant function, sinaa, andx"H(r). do not have Fourier uansforms, even though they occur frequently in applications. The above definition of the Fourier transform has been extended for B more general class of fundons to includelk above and other functions. We simply state the fact that there is a sense, useful in practical applications. in which the above stated functions and many others do have Fourier transforms. The following are examples of such functions and their Fourier transforms. B{H(a

~

Ix I))=

(F),

(1.7.8)

where H(x) is the Hceviside unit step function.

S{s(x-a)) =

&

(1.7.9)

eip(-~?k),

where 6 ( x - 0 ) is the D i m delta function. *{"(I

-4 = .])k(6+'[ 2 ink

Theorem 1.7.1 (Convolurion

(1.7.10)

Theorem). If % ( f ( x ) } = F ( k ) and

S { g ( x ) } = G(k). then,

s{f(x)

or, quivalently.

* &)I= F(k) W.

(1.7.11)

* - ' { ~ ( k ) ~ ( k ) } = f ( x )* dx). (1.7.12) where f(x) * g ( x ) is called the convolurion of two integrable functions f(x) and g(x) and is defined by

f(x) * s(4 =

& -i f (1-5) d 5 )

Henee, the result (1.7.12) can also be wriacn as

d5.

(1.7.13)

-

It can easily be verified that Ihe convolution has the following simple pmpcrtier: (commutative) (1.7.15) f ( 4 'd.) = d.) * f(4 (1.7.16) f(x) t & 6 ( . x ) = f ( x ) = & 6(x) * f(x) (identity). The Fourier transforms are very useful in solving a wide variely of initialboundary value problems governed by linear panid differential equations. The following examples of applications illustrate the methad of Fouria transforms.

Exanrplr 1.7.2 (The Couchy Problem for the Wove Equnrion). Obtain the d' Alembert solution of the initial-valuc pmblem for the wave equation u.=e'u,,

--<x<-,

(1.7.17)

1>0.

with the initial data

-

-

< x < -. (1.7.18ab) Application of IheFouriertranrform % [ u ( x , t ) ] = U ( k , t ) to this system gives u ( x , O ) = f(x).

u , ( x , O ) = g(x).

U(k,O)=F(k).

(%)&

=G ( k )

The solution of Ihc transformed system is U(k.t)=A dCb+Be*"', when A and B are co~stanllof integration to be determined fmm the transformed data A + B = F ( k ) end A - B = & C ( k ) . Solvmg forA andB, we obtain the r h SOl"h0"

U(k.r) = i F ( k ) ( c z k

+.-'"')+a 2ick

(e"

-e4).

Thus, the inverse Fourier transform of (1.7.19) yields Ihe solution u(x,r)=$&

-

jF(k){.Y'"'U+.'~.-~'l]dk]

f(x)=F'[F(k)}=-& g ( x )=

{C(k)]=

to obtain the solution in the form

Id* F(k)&,

&-

jerk G(k)dk,

(1.7.19)

-

(1.7.21)

= $f(x -el)+ f ( x + e r ) ] + l Ig(5)dC. z c *-

This is the well known d'Alembcn solurion of the wave equation. Inpanicular, if f(x)=exp(-x') andg(r)=O, then,thed'Alemkrtsolution (1.7.21) rcduces u)

+.r)

=

+

[erp{-(x -er)'J +eip{-(x +ci)2}].

(1.7.22)

This shows that the initial wave profile breaks up into two identical waveling waves of half the amplitude moving in opposite dinetion with speed C. On thC other hand. if f ( x ) = O andg(x)=a(x), the d'Alcmbert solution (1.7.21) beeomcs

=&[H(x

+a)-H ( x - cr)]

= 1H(e'9 -2).

(1.7.24)

ZC

Exampie 1.7.3 (The Linearized Korrewegde Vrieics Equorion). The linearired Kmweg.de Vries (KdV)equation for the free surface elemtion n(x.r) in an inviseid water of commt depth h is q,+eq,+&qm=o.

--<x<-.

r>o.

(1.7.25)

where e = @ is the shallow water speed. We solve equation (1.7.25) with the initial condxtion q(x,0)= f ( x ) . - - c x < - . (1.7.26) Application of the Fouriertransfm Pi{n(x,r)}= E(k,r) to the KdV system gives the solution for E(k.r) in the form E(k.r) = F(k)exp[&t($-l)].

The invene t r a n s f m gives

In panicular, if f(x) = S(x), then (1.7.27) nduces to the A j . integral q(r,r)=~jmi[x(r-cl)+(~)k’]d*

(1.7.28)

which is, in terms of the Airy function,

when the Airy function A i ( r ) is defined by A i ( z ) = ~ j e o s ( k r + ~ k f )dk.

no

(1.7.30)

3

Exmplr I.7.4 (Dirichler’s Problem in the Half Plonc). We eonrider the solution of the Laplace equation in the half plane u,+u,=o. --
with the boundary conditions

-.

u(x.0) = f(x),

--
_ d’u _ k‘U=O dy’

(1.7.34)

U ( k , O ) = F ( k ) , endU(k,y)+O my+-. mus, the solution of this transformed system IS U(k,y)=F(k) exp(-lkly). Applimtim of the Convolution ’Illheorem 1.7.1 giver the solution

=& -

4%~) Whae

]/(5)g(x-5)dS.

(1.7.35ab) (1.7.36) (1.7.37)

(1.7.38) Consequently,the solution (1.7.37) becomes

(1.7.39)

This is the well known Poisson inregrolformulo in the half plane. It is noted that

.

-

,

The curyes in the upper half plane, for which the steady smte temperatun is constant. me known BS isofhcmurl CUWCI. In this case, these mwei represent a family of circula m s x'+y'-uy=n' (1.7.44) with centers on !he y-axis and fired end points on the x-axis at x =fa, as shown in Figure 1.5.

t'

-a

Figore 1.5

0

Isothermal curves representing B family of circular arcs

38

Linear Pomirl Diffrrcntiol Equmionr

Another special case deals with

f(4 = w.

(1.7.45)

The solution for this case follows fmm (1.7.39) and is given by (1.7.46)

Further, we can readily deduce the solution of the Neumnn problem in the half plane from the solution of the Dirichlet problem. fiomple 1.7.5 (Neumonn's Problem in rhc Half Plnne). Find a sdut~ onof lhe Laplace equation u ~ + " , y = o , --oo<x<-oo. y > o . (1.7.47) with the Newmann boundary condition u,(x,O)=f(x), --<x<-. (1.7.48) m i s condition specifies the normal derivative on the boundary. and, physicdly. it describes the fluid flow or heat flux at the boundary. Wedefi"eanewf""ctio" u(.,y)=u,(.,y) sothat (1.7.49)

wherc ao arbimry constant can k added to the right-hand side. Clearly. the function v satisfiesthe Laplace equation

with the boundary condition U(X,O) ="JX,o) =f(x) for --< x < -oo. mus, v(x,y ) satisfies the Laplace equation with the Dirichlet condition on the

boundary. Obviously, the solution is given by (1.7.39), that is, (1.7.50)

men, the solution u ( x . y ) can be obtained fmm (1.7.49) in the form

1.7 Fourier Tmmnsfonnr ond Init~i-Bonn&ryl'aIu Pmblem

39

where an mbiuary eonstant C M be added to this solution. In other words. the solution of any Neumann problem is uniquely determined up to an arbitrary constant.

Example 1.7.6 (The Cauchy Problem /or rhc Diffusion Equorion). We consider the initial value problem for a one-dimensional diffusion equation with no sources or sinks u,=Ku,, --<x<-, r>O, (1.7.52) where IC is a diffusivity constant with the initial condition .(x,O)=/(x), --< x < -. (1.7.53) We solve this pmbkm using the huricr uansform in the space variable x defined by (1.7.1).Application of this transform to (1.7,5Z).(l,7.53) gives r>O. (1.7.54) u,=-rk'U, U(k,O)= F(k). (1.7.55) The solution of the tmnsformed system IS U(k.1)= F(k)exp(-ICk*r). (1.7.56) The inverse Fourier transform gives the solution j ~ ( k e) x p [ ( i ~ - x ~ r ) ] ~

u(x,r)=+

2n .-

which is. by the Convolution Theorem 1.7.1. = = I1f ( 5-) 8 ( " - 5 ) d 5 .

(1.7.57)

wtae

Thus. solution (1.7.57)becomes

% inlegrand involved in the solution consists of the initial value /(x) and the Crcen'sfuncrion (orfundamenrol solution) G(x - <.r) of the diffusion equation

for the infinite interval:

r/41rxr

G(x-t,r)=Leip So.

I"

[

(x,)*]

u(X,r)=lf(5)G(X-5.r)d5.

-

so

(1.7.59)

terms of C(x-5.r). solution (1.7.58)can k written as

that. in the limit as r +0 +. this formally becomer

(1.7.m)

Limeor Pnr&d Differential Equionr

40

-

4 X 3 O ) = f ( x ) = IJ(e)!%G(x-5,f)

Thelimitof G(x-s,f),

4

f-rO+,represcntstheDiracdeltafunction

BE

[

6(x-5)=lim+exp-(-. ,- nlCf

xi:'2] (1.7.61)

It IS l m p 0 ~ to 1 point out that the integrand in (1.7.60) consists of the initial temperature distribution f (x) and the Gnen's function C(x 5 , f) which r ep re~en the t~ temperature response along lhe rod at time f due to an initial unit impulse of heat at x = 5 . The physical meaning of the solution (1.7.60)is that the initial temperature distribution f ( x ) is decomposed into B specmm of impulses of magnitude f ( S ) at each point x = ( to form the resulting

-

tempcrature f ( C ) G(x-5.f). According to the prrnciple of suprrposifion (1.3.6).lhe resulting temperature is integrated to find solution (1.7.60). We make the change of variable

:z-<.

--

d < = d s

2J;i

to express solution (1.7.58) in the fwm

u(x,f)=+

j/(x+zJ;i<)exp(-<')d<.

-

(1.7.62)

The integral solution (1.7.62) or (1.7.58) is called the Poisson integral repre~enrofionof the temperature distribution. This integral is convergent for all time I > 0 . and the inteerals obtained from 11.7.62) . ,bv , differentiation under the integral sign with respect to x and I are uniformly convergent in the neighborhood of the point ( q f ) . Hence. the solution u(x.f) and its derivatives of all orden exist f a f > 0. Rnally. we consider two special cases: (a) f ( x ) = 6(x), and (b) /(I)= ToH ( x ) , where To is a constant. Forcare(a), thesolution(1.7.58)reducesto

=&

w(-&).

(1.7.63)

This is usually called the Green's funcrion or lhefundnmenfol solution of the diffusion equauon and is shown in Figure 1.6.

A*

1.7 Fourier Tmnrfomr and Inrfiof-Boundory-Volu.Pmblemr

41

4x.4

r=z&=l

-2

Figure 1.6

At any time

0

2

The temperstore distribution u(1.f) due to a point source for dinerent values of v=Z&.

f,

the solution n(x,r) IS Gaussian. The peak height of u ( x . f )

6, whereas the width of the ~ ~ l ~ t(xm- n6) 6. In fast. the initially sharply waked profile is gradually

decreases inversely with

increases with

-

Smoothed out as f + under the action of diffusion. There arc remarkable feaNns for diffusion phenomena. For case (b). the initial data is diieontmuour. In this case, the solution is u ( x , r ) =+2 77x7

jup[-q]

d<.

(I .7.64)

lntmducingthe change of variable q=y. we can express solution (1.7.64) 2x7 in the form "(x.+%

=

-',j;"q=+.(-;) 21 .($41.

2 n (1.7.65)

+

Thls shows that, at r = 0, the solution coincides with the initial data

"(1.0) = Tm.The graph uf

1+,r) To

against x is shown in Figure 1.7. As r

Linear Ponid Dt~erenliolEquolionr

42

inerearcs, the discontinuity is gradually smoothed out. whereas the width of the vansition lone increases as &.

-5

0

-2.5

2.5

5

Figure 1.7 The temperature distribution due to discontinuous initid data for different vslucs of 1.=2&=0,1,2,5. Emmplc 1.7.7 (The Schradingrr Equalion in Quonlum Mechanics). The time-dependentSchrCdingcr quation for a panicle of mass rn is

[

iAv,= V ( x )-L] R‘V v=nv,

(1.7.66)

where h =Zzh is the Planek constant, v ( x . 1 ) is the wave function. V ( x ) is the potential, V’ =

z+ E+Lis the threcdimensional Lapladan. end ax2 ayi mf

is the Hamilmian. If V ( x ) = consmnt = V. we seek a plane wave solution of the form v(x.r)= A exp[i(r . x - a ) ] .

H

(1.7.67) where A is a constant amplitude, I( = (k,l.rn) is the wavcnumbcr vector, and 0 IS the frequency. Substituting this solution in (1.7.66). we conclude that this solullon is possible provided the following relation is satisfied: ifi(-iri?)=V-E(iK)’. 2m 01

K*

=k’+l’+d,

1.7 Fourier T m f o m and Inirlnl-Bowdo~-VolueProblem

43

*m=V+K, (1.7.68) 2m This is called the dispersion rclntion for the de Broglie wave and shows that the sum of the potential energy V and the kinetic energy (Ak)Z equals the total 2m energy 1 m. Further. the kinetic energy

K.E.=L(Rr)'=h (1.7.69) 2m 2m' when p = R K is the momentum of the panicle. In the one-dimensional case. the group velocity is

c

'

=*=!!LL=mv=v. a k m

m

(1.7.70)

v

' h s shows that the p u p velocity is equal to the classical particle velocity v .

We now use the Founcr transform method to solve the one-dimensional Sc~ngeraluationforaheepanicle (V-0),that is, d

w,=-&

--.p<x<-.p,

w(x,0)= w&),

--<x<-.p,

w(x,r)+O

as

I>O,

(1.7.71) (1.7.72)

IX]+DD.

(1.7.73)

Application of the Fourier nanrform to (1.7.71>(1.7.73) gives y

=-e

(1.7.74)

p

' 2m ' *(LO)= *&).

(1.7.75)

The solution of this transformedsystem is *(t,r)=*'.(t)cip(-iak*r), m = L . 2m The inverse Fourier transform gives the formal solution

(1.7.76)

w ( x . r ) = L -Y0(t) eip{ik(x-akr)}dk

GL

=- 1 -

r(y,O)dyIcrp{ik(x-~)]dk

1 = - V(V(y,O)dyJexp{i+ 2* 2n

y - a*,)]&.

We rewrite the tntegrand of the second integral in (1.7.77)

(1.7.77)

1.8 Applications of Multiple Fovrier Tronrlomu

45

--

=LI

-LI

S { f ( x , y ) ) = F(k,P) le.p(-i(K.r)}f(x,y)drdy, 2 t -where I =(x,y) and K =(k.P). The inverse Fourier transform is given by

(1.8.3)

8.' [F(k,P)}= f(x, y) = l e i p ( i ( i r))F(k,P)dkde. (1.8.4) 2?Similarly, the three-dimensional Founer transform and 11s inverse are defined by the integrals ?+{f(x,y,r)}= F(k.t.m)

--=m ---_ ___ I

~~~..P{-i(...)]r(..Y~z)

drdy~,

(1.8.5)

W'{F(k.t.rn)} = f ( x , y , z ) =+

(W

I l I e x p { i ( ~ - r ) ]F(k,P.rn)dkdedm.

(1.86)

The operational properties of these multiple Fourier transforms me similar to those of the one-dimensional case. In particular, results (1.7.5) and (1.7.6) relating the Fourier vansformsof derivatives to the Founer transforms of given functions are also valid for the higher dimensional case. In higher dimensions, they are applied to the transforms of partial derivatives of f ( x ) under the assumptions that f ( x , , r ,,....x.)and its panial derivative vanish atinfinity. We illwtrste the multiple Fourier transform method by the following examples of applications:

Emmple 1.8.1 (The Ditichler Problem for the Three-Dimennsionnl Loplncc Equation in the Half-Space). The boundary-value problem for u(x,y,z) satisfies the following quation and boundary conditions: V 2 u ~ u , + u , + u , = 0 , -=-<x.y<-, z>O, (1.8.7)

--<

"(x,Y.O)=f(x.Y).

1.Y

<-

(1.8.8)

F d W +

).,y.(. + 0. as -. (1.8.9) We use the double Founer transform defined by (1.8.3) to Ihe system (1.8.7)(1.8.9) which reducesto

_ d'u -K'u=o d?

U(k,P,O)

forz>0,

= F(k,P).

Thus, the solution of this transformed problem is

u(k,e.z)=F(k,e)erp(-irtz)=F(t.e)c(k.e), when ~ = ( k , e ) a n dc ( t , e ) = e x p ( - i ~ i ~ ) .sothat

(1.8.10)

Example 1.8.2 (The Two-Dimensional Difuusion Equorion). We solve the twedimenaonal diffusion equation u,=KV'u, - - < x , y < - , r>O, (13.13) with the initial conditions u(x,y.O)=f(x,y) --<x,y<-, (1.8.14) u(x.y.r)+o. a s r = m + - , (1.8.15) where K is the diffusivity conslant. The double Fourier transform of u(x,y,l), defined by (1.8.3). is used to reduce the system (l.8.I3).(1.8.l4) into the form

-du =_

r>o,

K~KU,

U(k,e,O)= F(k,t). Thle solution of this system IS

o(k.e.r) = F(k,t)exp(-rKK*) = F(k.e)c(k,e),

(1.8.16)

Whae

C(k.e)= crp(-Kx'r). so that g(x.y) =5 . ' (exp(-tKr')]

&1- If(*,

=& e ; . . p [ - g ] .

-,).I

(13.17)

Finally, using the Convolution Theorem 10 (1.8.16)gives the formal solution u(x.y*t) =

+p[

- (I- 5);;

--

or, eq"l"alC"tlY. u(x,y,r)=' 4*Kr

-J .J -/ ( r , ) e x p [ - G }

65d?.

dr'.

(1.8.18)

(1.8.19)

where r'=(5.1). Wemakethechangeof vmiabb ( r ' - r ) = G the form

R

10

reduce (1.8.19) into

1.8 Applicarionr of Multiple Founcr Tronrfomu

47

_J _

j f ( r + ~ ~ ) e r p ( - ~ * ) f i . (1.8.20)

u(x,y,r)=-+

4Kr .-.-

fl

Similarly. the formal solution of the initial-value problem for the threedimensional diffusion equation u, = ~ ( n , +u,, , +u=), --sc x,y.r < -, r > o (i.n.n) u(x,y.r.0)= f ( x , y . z ) ,

-- < X . Y . 2 <-

(1.8.22)

is given by

d w d = &pJl]f(s,tl,C) ew(-&)&dvdC.

(1.8.23)

whne 72

=(.

- S)*

+(y

- ‘1)l + ( z - C)’.

or,qutvalently, u(x,y,z,r) =

-j

jjf(r’)erp

-

(47W where r=(x,y,z)and r’=(<.q.C).

i-IT-r.liI

d< d q d r ,

(1.8.24)

4Kr

Makmgthechangeofvariable r ’ - r = G R.Solution (1.8.24)reducesto u(x.y,r,r)=

&j-[jf(r++~

R)+R’)~R.

(1.8.25)

This is known as the Fourier solarion. Example 1.8.3 (The Couehy Problem f o r the Two-Dimcnrzonnl Wave Equorion). The initial-vabe problem for the Wave equation in two dimensions is

governed by u.=cl

(u,+u,).

r>o,

--<x,y<-,

with !he initial data u(x,y,O) =0, u,(x,y,O) = f ( x ,y ) ,

(1.8.26)

-- < x , y < -.

(1.8.27ab) where c is a C O I I E ! ~ We ~ . assume that ~ ( x , y , r and ) its first partial derivatives vanish at infinity. We apply the two-dimensional Founer transform defined by (1.8.3) to the system (1.8.26)-(1.8.27ab). which becomes

d’u+ c2k2U= 0, dr’ u(x,e,o)= 0,

k2 = k’+e‘,

(%),+= q k . e ) .

The solution of this transformed system is U ( k , e . t ) =F ( k . 0

a. CX

The inverse Fourier transform gives the formal solution

(1.8.28)

=

&1

i F [ e x p { ix(

?+

ci)}

- exp{ ik(

?-<,)}I

dK.

(1.8.30)

The form of this s a l u t m reveals some interesting features of the wave

t. [ -dl

equation. The exponential terms erp I K ctf

involved in the integral

solution (1.8.30)represent plane wave solutions of the wave equation (1.8.26). Thus, the solutions remain constant OD lhe planes K . I = eonstant that move parallel to themselves with velacity e. Evidently, solution (1.8.30) represents a superposiuon of the plane wave solutions traveling in all possible directions. Similarly. the solution of the Cauchy problem for the three-dimensional wave

equation

u . = c ' ( u , + ~ , , + ~ , ) , --<x.~.z<-. u(x,y,z,O) = 0. ~,(~.y,r.O)=f(x,y.z). --<

r>n, x.y.2 C-

(18.31) (1.8.32ab)

is riven bv

Intermsofthesphericalpolareaordinates ( t , b ' , # ) , where thepolaraxis (the z-axis) is taken along the r direction with =.I= mcosb', we write (1.8.34) ~n the form

(k

1.9 Lopface Tmnrjormr and Inrnol-Bomdnry-Value Problem

49

u(r.r) = L [ 6 ( c r - r ) - 6 ( c r + r ) ] .

4mr For r > 0. er + r > 0,so that 6 ( a+ r ) = 0 and, hence, the solution IS u(r.r)

1.9

=L 6 ( a- r ) = 4mr

4nd

( -9

6 r

(1.8.35)

(1.8.36)

Laplace Transforms and Initial-BoundaryValue Problems

If !&,I)is any function defined in a < x 5 b and r > O . then, its Lnploee rranSformwithrcspecttorisdenotedby % [ u ( x , r ) } = i i ( , r , s ) andisdefinedby x { ~ ( ~ , r ) } = U ( ~ . ~ ) = ~ ~ - " " ( ~Res>O, ,r)df,

(1.9.1)

where s is called the rronsform variable which is a complex number. Under c e m n hdconditions on u ( x , r ) . its mnsform U(x.s) is an analytic function of s in the half plane Res > a. The inverse Lnploce rronsform is denoted by T'[;(x,s)}=u(x,r) and dcfimd by the complex integral 0-

~{+.s)}=

N(x,r)=L

_-

Je' ~ ( x , s ) h . c > O.

(1.9.2)

This integral is evaluated by using the Cauchy Residue Theorem for analytic functions. If Z [ u ( x , r ) } = i i ( w ) , then,

x{%}

(1.9.3)

=~ U ( x , s ) - n ( ~ , O ) ,

x{$}= *'U(x.s)-

( I .9.4)

su(x.o)-",(x,o),

and so on. Similarly, i t is easy lo show that

x{$}=$.

x{$}=s.

(1.9.5ab)

The following examples are useful for appl~eations. x [ f ( r - a ) H(r-o)}=erp(-ra) ,+), 2 [ 6 ( r - a ) ] = erp(-as).

a>o,

a 20,

(1.9.6) (1.9.7)

.(etfc(

}

L, 2Ji = f exp(-a&),

x{eXP(oI)ed(JL?)}=&$,

n 2 0. 4>O.

(1.9.8) (1.9.9)

The Laplace transforms are also very useful in finding solutions of initialvalue problems described by linear panisl differmud equations. The following examples of applications illustrate the m&cd of Laplace transforms.

Example 1.9.1 (The Inhomogeneous Couchy Problem for the Wave Equation). We use the joint Laplace and Fourier transform method to solve the inhomogeneous Cauchy problem u , , - c ' u ~ = ~ ( x , ~ ) .X G R , 1 > 0 ,

(1.9.10)

u ( x , O ) = f ( x ) . u , ( x , O ) = g ( x ) for all X E R , when q(x, I) is a given function representing a source term. We definethe joint Laplace and Fourier wnsform of ~ ( 1 . 1 )by U(k,s) = % [ s [ u ( x .I)}] =

$.-

jc-"dr

I-

je"'u(x,r)dt.

(1.9.11ab)

(1.9.12)

0

Application of the joint tansform leads to the solution of the transformed problem in the form

u ( ~=, ~F ()k )+W)+H k s ) ,

(1.9.13)

(s' +elk')

IheinvcrscLaplaeetranfom of(l.9.13)gives U(k.1) = F ( k ) c01j(&I)

{

+ 1G ( k ) sin (&I) + I 2.' +. ck

ck

H ( k ,s)}

s +c'k'

= F ( k ) c o s ( c ! e ) + ~ s i n ( c k t ) +jsin ~ ck(I-T) Q(k.?)dr. ck Ck 0 (1.9.14) lbe invers Fourier transform leads to !he solulion u(x.1) =

24%

-

+e-""')

e'*F(k)&

+i ](d'* - P" d* .a& 2

G

-

ick

In the ease nfthe homogeneous Cauchy problem, q ( x , r ) s O . the solutm of (1.9.15) reduces to the famous d'Alemberlrolution (1.7.21).

Example 1.9.2 (The Heof Conduction Equorion inn Semi-Infinire Medium ond FracliomI Detiuatives).Solve the equauon ",=xu", x > o , f,O (1.9.16) with the initial and boundary conditions "(x.o)=o. x>o. (I .9.17) "(o.r)=f(t). f>O. (1.9.18) u(x,r)+0, BJX+OD. r>o. (1.9.19) Application of thc Laplace mansform with mpsct to Ito (1.9.16)giver -d'J --u s = (1.9.20)

d2

k

O.

The general solutionof this equation is P ( x , s ) = Acrp(

-xg)+

B ex.(

xg),

where A and B are integrating con~tants.For bounded solutions. B I 0, and wing S(O,s)= f(s), we obtain the solution

F(x,+f(s)exp(-xg].

(1.9.21)

The inversion theorem gives the solution u(x.1)

= +bf(r 2

ZK

whichis, byputting A=-$2 x 1

- r)r->nexp(-')

4 x 1 dr,

or d A = - ; r - v 2 d r , 4 K

This ISthe formal solution of the heat conduction problem. In pmicnlar. if f(:) = To = constant, lhc solution (1.9.23) becomes

(1.9.22)

Clearly, the temperature distribution tends asymptotically to the constant value

To,= I + - . W e consider another physical problem that is concerned with determinating the temperature distribution in B semi-infinite solid when the rate of flow of heat is prescribed at the end I = 0. Thus, the problem is to solve diffusion equation (1.9.16) subject to conditions (1.9.17). (1.9.19). and - k ( zJu )=g(f) alx=O, f > O ,

(1.9.25)

where k is a consmt called t h e m 1 conducfiviry. Application of the Laplace transform gives the solution of the uansformed problem

.(...)=;g..p(-.g). I(.)

(1.9.26)

The inverse Laplace transform yields the solution u(x,f)

= ki zp /p(f-

r)r$ exp)(-'

4K7

dr.

(1.9.27)

which is. by the change of variable 1 =

In pmieular, if p(r) = TQ= constant, this solution becomes

Alternatively, the heat conduction problem (1.9.15)-(1.9.18) can be solved by using fiaetionalderivatives (see Debnath, 1995). We recall (1.9.21) and remte st

"+

(1.9.30)

JX

This can be expressed in terms of a fracuonal derivative of order

L BS

1.9 Inploce Tmnsfomuand Inilio6EowdaryVolur Problem

53

Thus. the heat flux 1s expressed in terms of the fractional derivative. In particular, when "(0.1) = constant =To. then. the heat flux at the surface is given by

Example 1.9.3 (Diffusion Equation in

Finite Medium). Solve the

(I

diffusionequation U,=kUU.

(1.9.33)

I>o,

o<X
with the initial and boundary conditions

=o, o< x < a ,

"(X,O)

(1.9.34) (1.9.35) (1.9.36)

u(O,t)=U, I > O , u.(a,r)=O, r > O ,

where U is a constant. We introduce the Laplace wnsfom of tz(x,t) with respect tot to obtain -d% - - u s = 0, (1.9.37) o<x
-

**

S(O,+U S

'

(")dr

=o.

(1.9.38ab)

r_.

The general solution of (1.9.37) is Z ( x , s ) = A cosh(xE)+BYnh(x@

(1.9.39)

where A and B are constants of mtegration. Using (1.9.38ab). we obtain the value^ of A and B, so that the solution (1.9.39) becomes msh[ (a- x $ ]

qx,.)=U.

*

(1.9.40) cosh(nE)

'

The inverse Laolace uansfom eives the solution (1941) seosh

a-

The inversion can te carried out by the Cauehy Residue Theorem to obtain the solution

XWp{ - ( 2 n - l ) 2 ( E r

m]].

(1.9.42)

n}].

(1.9.43)

By expanding the cosine term. this becomes,

XeXP{-(2"-I)f$r

This result can be obtained by solving the problem by the method of reparation of variables. Example 1.9.4 (Difision in (I Finite Medium). Solve the one-dimensional diffusion equation in a finite medium 0 < L < a. where the concentration function C(z.1) satisfies the quation

c,=xc=,

o0,

(1.9.44)

forO O ,

(1.9.45) (1.9.46)

forz=0,1,0.

(1.9.47)

and the initial and brmndary data C(z,O)=O

c(r,l)=c. -=

0

w h m C, is a constant. Application of the Laplace wnrform of C(z.1) with respect to f gives

iz-( +) F dz'

= 0, 0 < z < 0 ,

The solution of this differential system is C, corh( F(z,s)=-, s

zE)

+)

(1.9.48)

+%(-I)"

exp[-a((Zn+l)o+z)]

"-0

}

(1.9.49)

Using the result (1.93). we obtain the final solution

This solution represents an infinite sdsr of complementary c m r functions. The successive terms of this Series are, in fact, the concentrations at depths 0 - 2 , n+z, 3 a - r , 3o+& in the medium. The series converges rapidly for

...

all except large values of

(3

Erornplr 1.9.2 (The Wave Equation /or the Transverse Vibration of a Scmr-Infinite String). Find the displacement of a semi-infinite suing, which is initially at rest in its equilibrium position. At time I= 0. the end I = 0 is consbained to move so that the displacement is u ( 0 , r ) = A f(t) for I > 0. where A is a constant. The problem is to solve the one-dimensional wave equation u,=c%_,

O<X
I>O,

(1.9.51)

with the boundary and initial conditions u(x.t)=A/(r)

atx=O, 1 2 0 .

u(x.t)+O

as x + -, r 2 0 .

u(x,t)=O=&

att=O

3t

forO<x<-.

(1.9.52)

(1.9.53)

(1.9.54ab)

Application of the Laplace wnsform of u(x.1) with mpect to Igives d'ii s2 ----u *2 c1 = 0, rorosx<-,

-

ii(x,s)=Ai(s)

at x = O ,

B(X.S)+O

asx+-.

The S O I U U O ~of this differential system is i i ( x . s ) = A f ( s ) erp

Inversmn gives the solution

(1.9.55)

lrncor Parid Digrnnriol Equations

56

(1.9.56)

~(~,r)=Af(t-~)"(I-~).

,1.

In other words, the ~olutionis

r>X

Af(r-z),

o,

+,r)=[

(1.9.57)

t<-

This solution represents a wave propagating at a velocity c with the characteristic X = C I .

Exmmple 1.9.6 (The Couchy-Poisson Wove Problem in Fluid Dynamics). We consider lhe two-dimensional Cauehy-Poisson problem (Debnath, 1994) for an invirid liquid of infinite deplh with a honzanlal f n e surface. We assume that the liquid has constant density p and negligible surface tension. Waves a n generated on the free surface of water initially at rest for time t < O by the prescribed free surface displacement at f = 0. In terms of the velocity potential $ ( x , z , t ) and the free surface elevation

q(x,t), he linearized surface wave motion in Canerim coordinates ( x , y , z ) is governed by h e following equation and fne surface and boundmy conditions: V'$=$u+&=o. -- O , (1.9.58)

$+,c- +R g, =qO= o }

onz=0,

$,+O

r>o,

(1.9.59ab) (1.9.60)

as z + - - .

The initial conditions are and ?(.LO)= 'lo(x), (1.9.61) w h q&) IS a given initial elevation with compact support. We innodwe the Laplace transform with respect to r and the Fourier transform with respect ~ O defined X by $(z,O,O)=O

-

[;(k,i,s),

-

t ( k . s ) ] = L je='"dzje"[$,q]

Jz;;.-

dr.

(1.9.62)

0

Application of he joint transform mehod to the above system gives . . &-k'a=O,

K+O

--
aSz+--,

where

q%(41.

Ij&)= The bounded solution of equatlon (1.9.63) is

(1.9.63)

(1.9.65)

57

1.9 h p h m T r m s f o m a d InitioCBoundory-Value Pmblemr

%(k,s) = X . exp(lklr), where

(1.9.66)

X = X ( s ) isanarbiwryfunctionofs.

Substituting (1.9.66) into (1.9.Mab) and eliminating equations gives

4 from the resulting

X.~ e n c c the . solutions for % and 5 are

[$,?f]=[-w (1.9.67)

where the disprsion relation is

mi =glkl.

(1.9.68)

The inverse Laplace and Fouriertransforms give the solutions 0 ( ~ , ~ , f ) = - ~ ~ ~ ~ ~ ~ p ( i ~ + ~(1.9.69) k ~ ~ ) ~ ~ ( k

Il(X.f)

=

6 1-Ti&)

EOS (uf C b d k

1 =-JZ;;ITi,(t)[."""+."*"']dk.

(1.9.70)

in which fio(-k)= rjo(k) isassumed. Physically. the first and second integrals of (1.9.70) represent waves traveling in the positive and negative directions of I. respectively. with phase velocity

z. These

integrals dcscribc superposition of all such waves over the

wavenumber specrmm 0 < k < -. For the classical Csuchy-Poisson wave problem, tl,(x) = L? S ( x ) where S ( x ) is the Dirac delta function. so that f i o ( k ) = a / 6 . Thus, solution (1.9.70) becomes q ( x . 1 ) = "I[e'"-"'

+ e"*"']dk.

( I .9.71)

22 0

The wave integrals (1.9.69) and (1.9.70) represent the exact solution for the velocity potential 0 and the free surface elevation 7 for all x and t > O . However, they do not lend any physical interpretations. I n general, the exact evaluation of these integrals is almost B formidable task. So it is necessary to reson to asymptotic methods. It would be sufficient for the determination of the principal features of the wave motions to investigate (1.9.70) 01 (1.9.71) asymptotically for large time t and large distance x with (x/f) held fired. The asymptotic solution for this kind of problem is available in many standard books (for example. see Debnath, 1994, p. 85). We state the stationary phase approximation of a typical wave integral. for I --f

-.

-f(k,)

[

I exp i [rli:k,)f

~ W ( k , ) + z s g nW"(\)}],

(1.9.73)

4

where W ( k ) = z - m ( k ) , r > O , and k = k , is a stationary point that satisfies rhe equation

W'(k,) = 1-m'(k,) = 0.

a < k, < b .

(1.9.74)

Application of (1.9.73) to (1.9.70) shows that only the first integral in (1.9.70) has a stationary point for x > O . Hence, the stationary phase approximation gives the asymptotic solution, as I + -, x > 0,

whne k, = ( g 1 1 / 4 x ' ) istherootofthequation m'(k)=l.

On the other hand. when x
[&]

-

fio(k,)exp[ r(to(k,) - k, lxl] + %spa m"(k, )] . (1.9.76) Im ( k J In particuler. fw the classical Cauchy-Poisson solution (1.9.71), the mymptmie representationfor q(x.1) followr from (1.9.76) in the form f

q ( x , r )2 - z 6A X'P cos($),

gr* >>4x

andgivesasimilarreEultfor q ( x , t ) , when x < O and

(1.9.77)

I+-.

1.10 Hankel Transforms and Initial-BoundaryValue Prohlems The Honkcl Imnsfom,of a function f(r) is defined formally by

~"[f(.)}=i.(K)=jrI.(n)f(r)dr.

(1.10.1)

1.10 Hankel Trom/ommsand l n i t i ~ f - B o ~ ~ - V aProblems fue

59

where J.(m) is the Bessel function of order n and the integral on 6x right-hand

side is convergent. The invrnc Hnnkel rmnrfom is defined by

X ; ' [ j n (r)]= f(r) =

K

Jn( kT) jm( K) d k

(l.lO.2)

provided that the integral exists. Integrals (1.10.1) and (l.lO.2) exist for certainlarge classes of functions that usually oecur in physical applications. In particular, the Hankel Vansfoms of orderzsro ( n = O ) and of order one ("=I) are useful for solving initial- and boundary-value problems involving Laplace's or Helmholtz's equations in an arisymmede cylindrical geomeby.

In particular,when n = 0 and n = I , (1.10.3) reduces to s p i e l results which M very useful for appheatlanr.

Example 1.10.2 Obtain the solution of the boundary-value problem

u,+~u,+u,=o, r

oO,

u(r,O)=u, for O s r s n , u,isacansmt. u ( r , r ) + 0 as z - i - .

(1.10.5) (1.10.6)

(1.10.7)

Linear Poniol Dtb4errnaol Epuollons

60

Application of the zero-ordcr Hankel uansfarm, wlth respect to r, l o the problem giver

iZ+&=O dz'

i ( r . 0 )= uo

j r Jo( m ) d r = 'uo J , ( a r )

Thus, the solution of !he transformed problem is i(K,Z)=

'".J , ( a K ) eXp(-KZ). K

Using the inverse Hankel uanrform gives the formal soluuon U ( r , Z ) = a U , ~ J , ( r K ) J , ( G K ) exp(-Kz) d K .

(l.lo.8)

Example 1.10.3 (Arisymmrtrie Wave Equntion). Find the solution of the free vibration of a large circular membrane governed by the inilid-value problem C+"+;",)="..

o
(1 J0.9)

f>O,

-.

(1.10.10) where c'=(T/p)=constant, T is the tension in the membrane, p is the surface density of the m c m h e . f ( r )and g ( r ) are arbluary functions of r. Application of the mo-order Hankel transform i ( x . r ) of the displacement function u(r,r) to (1.10.9)-(1.10.10) gives the solution "(SO)

=f(,), u,(r,O) = g ( r ) . 0s r <

-

i( K.1) i ( r )cos(cn)+ &Q 8i"(cm).

(1.10.1 I)

CK

The inverse Hankel transform leads to the solution u ( r . r ) = i K j ( K ) c o s ( c m )J , ( n r ) d K + L j g ( ~ ) s i n ( c m ) Jg(kl)dC.(l.IO.lZ) c o

In panicular, we consider the initial data u(r.0)=f(r)=Ao(r'+02~i, u , ( r . O ) = g ( r ) = O , so that g( K) 0 and

i(r)= Ao[r(n'

+ r 2 ) - iJ,(n)dr=%--r,

(1.10.13ab)

by Example l.lO.l(a)

Thus. the formal solution (l.lO.12) becomes n(r,I) = Aoje"

J,( m)cos(cm)dK = Ao Rejexp[-r(o +icI)]J@(m)dK

=An Re(r2+(o+icI)2}i,

by Example l.lO.l(a).

(1.10.14)

1.10 Honkel Tmnsfomr and In~tiol.Boundary-Valu~ Pmblemms

61

Eronplc 1.10.4 (Sleody Temperorwe Disfribufionin (I Semi-lnfinile Solid with (I Siendy Heal Source). Find the solution of Ihe Laplacc equation for the steady temperature distribution u(r,z) with a steady, symmetric heat SWTCO

%dr): u,,+Lu,+u-=-&q(r),

O
O
(1.10.13

u(r.0) =o. o< r<-. (1.10.161 where is a con~mnt.This boundary eoodition represents zero temperature at theboundary r = O . Application of the zero-order Hankel transform to (1.10.15) and (1.10.16) gwes

d'i

_ _K 2 i

= -& G(Z).

dr' The bounded general solution of this system is

i ( K , O ) = 0.

i(K.z)=~erp(-Kz)+~;(K), where A is a constant to bc determined fmm the transformed boundary condition. I" this present case,

A=-%-(,). rl

Thus. the formal solution is given by (1.10.17)

i ( K + M p ( I - P ) .

The inverseHankel transform yields the exact integral solution u(r,z)=Q.~'") O

(l-e-=)JO(k?)dK.

(1.10.18)

K

Erornplr 1.10.5 (Axisymmelric Diffusron Equalion). Find the solution of the arisymmevle diffusion equation

u , ( " + 5)

u,=Ku

O
f>O,

(l.lO.19)

where x (> 0) is B diffirivity constant, with the initial condttion

for O < r < - .

.(r,O)=f(.),

(1.10.20)

We apply the rem-order Hankel transform to obtlin %+k'Ki

=0,

i(k,O)=j(k),

where k is the Hankel transform vanable. The solution of this transformed

system is i ( k , f ) = j ( k ) erp(-Kk'f).

Application of Ihe inverse Hankel transform gives

(1.10.21)

62

- _

Lineor Pom’d Diflmntiaf 4 u t m r

. ( ~ . I ) = j k j ( k ) J ~ ( b ) ~ ~ ~ ” d k = ! k [J! ~l ~ k l ) j ( f ) d l ] ~ - ~ ” J ~ ( k ~ ) d k

-

-

which is, interchanging the order of integrauon, =Ilf(f)dlJkJ,(kf)J,(k~)~~p(-lrX‘t)dk. 0

(1.10.22)

0

Using a standard table of integrals involving Bessel functions. we state that

when lo(.) is the modified Bcssel function and lo(0)= I . In pmicular, when

1 = 0 , J,(O)=I.andinlegral(1.10.23)bccomcs j k J , ( k r ) e x p ( - k ’ K f ) d k 2C1 = l c x p ( -4K1 z).

We next use (1.10.23) to rewrite the solution (1.10.22) as u(r,z)=&jfj(f) I(&;) e x p (?+f’) [-~]df.

(1.10.24) (1.10.25)

We now assume that f(r) represents a heat source conscnwtcd in a circle of radius n and allow (I+0,so that the heat source IE conccnhavd at r = 0 and

~ Or, equivalently,

~

Zf ( r s) d r~= l . r

r(,)=l w, Zx

whne

r

6(r) IS the D i m delta function.

Thus, the final solution due to the coneenvatedheat soulcc at I = 0 is

=-4:k-l

’”(-&)’

(1.10.26)

E x m p f e 1.10.6 (ArisyrnrnerrfcACOUSI~C Radiation Problem). Obtain the solution of the wave equation e’(u,,+tu,+u,)=u.,

O
u,=F(r,r)

z>O,

onr=O,

r>O,

(1.10.27)

( I . 10.28)

1.10 H d d Tmnsfomr and 1nitiol-Boundory.Volu. Problem

63

when F(r.1) is a given function and c is B constant. We also assume that the solution is bounded and behaves as outgoing sphnical waves. We seek a steady-state sohtion for the acoustic radiation potential u=c'"@(r.z) with F(r.r)=e"f(r), so that @ satisfies the Helmholtz equation @,+~@,+@=+*@=0. O < r < - ,

2

z>O,

(1.10.29)

with the baundary condition $,=f(r) w h m f ( r ) is a given function of r.

onr=0,

(1.10.30)

Application of the Hankel transform % e , { @ ( r . z ) } = ~ ( k . z )lo (1.10.29)(1.10.30)gives Z>o. 6"=K26, &=j(k),

onz=O,

Whae

+$'. The solution of this differential system is

6(k,r)= -Lj(k)exp(-KZ),

(I. 10.31)

is real and positive for k > mle and purely imaginary for k < mlc. The inverse Hankel w n s f m yields UK formal solution

whm

K

@(r,z)=-j

j ( k ) Jo(kr) exp(-Kz)dk.

(1.10.32)

O K

Since the exact evaluation of this integral is difficult for M arbitrary j ( k ) , WE choose a simple form of f(r) as f ( r )= A H(o - r ) , (1.10.33) whercAisaconstant.and hence, j ( k ) = E J,(ok).

Thus, the solution (1.10.32) takes the form @(r,z)=-AajL J,(ak)JJkr)exp(-n)&.

(1.10.34)

OK

For an asymptotic evaluation of this integral, it is convenient to express (1.10.34)intermsofR,thedistaneefromther-aris.roUlat. R 2 = ( r 2 + z 2 ) a n d L = Rcor.9. Using the asymptotic result for the Bessel function in the form J&r)-($)'eoo(k-~)

as r + - .

(1.10.35)

64

Linear Partial Difermtid

Equtions

where I = Rsin.9. Consequently, (1.10.34)combined with u==erp(iax)@ becomes u A-a & P I~ J,(ok) m ( k R sine-:) e r p( - a ) dk.

ix

This integral can be evaluated asymptotically for R + - , by using the stationary phase approximation formula to obtain the final result

”---

izi

J,(&,)exp[i( ax

-

y)],

(1.10.36)

where k, =m/(csine) is the stationary point. Physically, this solution represents outgoing spherical waves with constant velocity c and decaying amplitude, BS R + -.

Excmple 1.10.7 (Axirymmcrric Biharmonic Equation). We solve the aisymmetrie boundary-valueproblem

V%(~,Z)=O.O < I < - . with the boundary data u(r.O)= f(r).

au 0 -=

(1.10.37)

Z>O

05 r < - ,

(1.10.38)

O
onr=o,

(1.10.39) 11.10.40)

..

The use of the Hankel transfom %e,(u(r,z)} = i ( k . z )

(2-

k 2 1 : i(k,z)=O,

i(k,O)=j(k),

fi=O dr

to this

z>O, on r = O .

problem gives

(1.10.42) (1.10.43)

The bounded solution of equation (1.10.42) is i(k,z) = ( A + zB)cip(-b), (1.10.44) where A and B are integrating ~ o n ~ t a n tlos bc determined by (1.10.43)as A = j ( k ) a n d W=kj(k).Thus, rolution(1.10.44)becomes i (k ,r) = (I + b )i (k )e x p (-b ). The inverse Hankel Wanrfm gives the formal solution u ( r , z ) = l k ( l + b ) j ( k ) J,(kr) exp(-b)&.

(1.10.45) (1.10.46)

Example 1.10.8 (The Axisymmelrie Cnuehy-Poisson Water Wave Problem). We consider the initial-value problem (Debnath, 1994)for an invivid

1.10 Honkel Trmsfomts and Initial-Boundary-ValueProblems

65

water of finite depth h with a free horizontal surface at i = 0 and the z-axis positive upward. We m u m e that the liquid has constant density p with no surface tension. The surface waves are generated in water which is initially at rest for f < 0 by the prescribed free surface elevation. In cylindrical polar coordinates (r,b k ) , the arirymmetric water wave equations for the velocity potential @ ( r , z , andthe ~) freesurfaceelevation q ( r , ~ are )

V’$=@,+~@,+@==O. O
-h
f>O,

(1.10.47) ( I .10.48ab)

@z=O onr=-h, t>O. (I. 10.49) where g i s the constant gravitational acceleration. ”k initial conditions are $(r.O.O)=O and q(r,O)=qo(r), for OLr<=-, (1.10.5Oab) where qo(r)is the prescribed free surface elsvatlon. We apply the joint laplace and the zero order Hankel transform defined by $ ( k . L , S ) = ~ e - ” d I ~ r , ~ ( k ~ ) @ ( ~ , ~ . f ) d r . (1.10.51) ”

”

to (1.10.47)-(1.10.49) sothat these equations reduce to

!!-re=-eo(k) dz s;+g?j

on z = o ,

=0

gZ=O

anz=-h.

where a ( k ) IS the zem-order Hankel transformof q.(r). The solutions of this system are $(k,z,s)-

g i l . ( k ) c o sh kk+ h)

($’+m*) coshkh e(k,s)=silo(k)

(s~+m~)3

’

(1.10.52) (1.10.53)

When

d = g k tanh(kh), (I. 10.54) is the famous dispersion relorion between frequency 0) and wavenumber k for water waves in a liquid of depth h. Physically, this dispersion relation describes the interaction between the inertial and gravitational forces. Application of the inverse transforms gives the integral solutions

q(,,r)=~tl,(t,.)ii,(k)eorwa.

(1.10.56)

These wave integrals represent exact solutions for 0 and q for all I and f, but the physical features of the wave motions cannot be described by them. In general. the CXBCI evaluation of the integrals is almost a formidable task. In order to resolve this difficulty, it is necessary and useful to resort to asymptotic methods. It will be sufficient for the determimuon of the baste features of the wave motions 10 evaluate (1.10 55) or (1.10.56) asymptotically for B large time and distance with (111) held fixed. We now replace J,(kr) by its asymptotic fomula(1.10.35)fa kr-+-, sothat (I.lO.56) gives

where the stationary point k, = (gf1/4?) is the mot of the equation d ( k ) = L.

For suffrcicntly deep water, kh -+

(1.10.59)

-. the dispersion relation becomps

(1.10.60)

d=gk.

The solution of the arisymmeuie Cauehy-Poisson problem is obtained by using a prescribed initial displacement of unit volume eoncenwted at lhe origin, which means that qo(r)=(o/Zm)S(r),so that f i , ( k ) = a , Thus. the asymptotic 2x

solution is obmned fmm (1.10.58) in the fom,

-

q(r,,) ngfl 4x.\i?r, cos($)

gr2>>4r.

(1.10.61)

It is noted that solution (1.10.58) is no longer valid when w"(k,)=O. This case can be handled by B modification of the asymptotic evaluation (see Dcbnath, 1994, p. 91).

1.11 Greea's Functions md Boundary-Voluc Problam

67

1.11 Green's Functions and Boundary-Value Problems Many physical problems are described by second-order nonhomogeneous differential equations with homogeneous boundary conditions or by second-oder homogeneous equations with nonhomogeneous boundary conditions. Such problems can be solved by a powerful method based on B device known as Green's functions. We consider nonhomogeneouspartial differential equation of lhe form

44.) = fk).

(1.11.1)

when r = ( x , y , z ) is a vector in three (or higher) dimensions, L, is a linear partial differential operator in h e or more independent variables with eonsmt cocfficisnu. u(x) and f (x) are functions of three or more independent variables. 'The Gncn's function C(x, S) of this problem satisfies the equation

-

4 G ( x . E ) = S(X S) (1.11.2) and rcpresenls the effect at the point i of the Dirac delta function source a1 the point 6 = ( 5 , ~ , 5 ) . Multiplying (1.11.2) by f(6) and integrating over the volume V of the 6 space, so that dV = dCdTd5. we obtain jL.G(x.E)f(E)dE= JS(1 -S)/(S)dS=

fk).

(1.1 1.3)

lntmhangingtheardcrof the oprmtor L. and integral sign in (1.11.3) gives (1.11.4)

Asimpleeomparison of (1.11.4) with (l.lI.l)le&to the solution of (1.11.1) in the f o m 44= JG(r.C)f(E)dS. (1.11.5) Clearly (1.11.5) is valid for any finite number of cornponenu ofr. Accordingly. the Green's function method can be applied, in general. to any linear, constant coefficient, inhomogeneous panial differential equations in any number of independent vanahles. Another way to approach the pmhlem is by looking for the inverse operator G'.If 11 is possible 10 find G',then. the solaion of (1.1 1.1) can be obtained as u ( x ) = K'(f(x)). It turns out that, in many important eases, it is possible. and the inVerne opcramr can be expnsred BS an integral oprrslor of the form

+) = G'(f(E)) = fG(%E)f(E)dC.

(1.1 1.6)

68

The kernel G(r,E)

Lineor

Parlid D!ffkmdal Equalions

called the Green’s functron which is, in fact, the charactenstic of the operator L, for any finite number of independent variables. The main goal of this section is to develop a general method of Green’s function from several examples of applieauons. IS

Example 1.11.1 (Green’s Function /or the One-Dimensionof Diffusion Equnlion).We consider the inhomogeneous. one-dimensional, diffusion equation u,-xu,=/(x)6(1). xcR, r>O. (1.11.7) with the initial and boundary conditions u(x,O)=O for x s R and u(x.t)+O as [XI+=-. (1.11.8ab) We take the Laplace transform with respect to f and the Fourin transform with respecitox to(1.11.7)-(1.11.8ab), so that

(#+&) b(k,I)+O

’

as I k l - + = - .

(1.11.9) (1.11.10)

The inverse Laplace vansfom gives i ( k , r ) = f(k)exp(-&t) when g.(k)=exp(-n*’t),

so

= f(k)i(k).

(1.11.11)

that

Application of the inverse Fourier uanrform combined with Convolution Theorem 1.7.1 gives

whcreule Oreen’s function G(x,r,S) is given by

Evidently. G(x.r) = G(x,f;O) is an even function of x, and at any time 1, the spatial distribution of G(x.t) is Gaussian. The amplitude (or peak height) of G(x,r) decreases inversely with &, whereas the width of the peak increases with &.?be evolution of G(x,t)= u(x,t) has already been plotted againnx

for different values of r = 2&

in Ftgure 1.6.

1.11 Green's Fumtiom and B o W d o y V o l u Problems

69

Example 1.11.2 (Green's FU~CI~O" for rhc Two-Dimcnsio~IDifjiiusion

Equnrion).We consider the two-dimensional diffusion equation u, - KV'u = f(x.y)6(r), --< x , y < -, r > 0 with the initial and boundary conditions u(r.y,O)=O. for all ( ~ . y ) e R ' ,

vm+-,

(1.11.15)

(1.11.16)

u(x.y.1)+0 as (1.11.17) where K is the diffusivity constant. Application of the Laplace transform and the double Fourier vansfom to the above system giver (1.11.18)

where K = ( k . l ) . lbe inverse Laplace vansform gives ~(K,I)= ~ ( ~ , + x ~ ( - K K * I =j(k,i)i(k,i), )

(1.11.19)

whue i ( k , l ) = c i p ( - K s * r ) , so that

Finally, the convolution theorem of the Fourier transform gives the formal solution

[

u(x,y.r) = 4nKr-: J ~ f ( c , q ) e x p -(.-')*

2

4KI +(Y- 7)' ] d < d %

--

= ]]f(C)G(r.E)dE

(1.11.21) (1.1 1.22)

where r = ( x , y ) a n d g = ( < , l ) ) andthcGrecn'sfunction G(r,g) isgiven by

Similarly. we can construct the Green's function for the three-dimensional diffusion equation u , - K V * u = j(r)s(I) for --<x.y,z<-, I>O (1.11.24) with the initial and boundary data "(r.o)=o for <x.y,z<-. (1.1 1.25)

--

u(r.I)+O

as r = ( x * + y 2 + z * ) ~ + - ,

(1.11.26)

where r = ( x , y , r ) . Application of the Laplace vansfom of u(r.t) wrth respect to Iand the threedimensional Founermnsform with respect to x,y,z giver the solution

w h a t E=(5,1,5),andlheGreen'sfunction isgiven by

(4nKt)i In fact, the same method of eonsvuetion-can be uied to find the Green's function for the n-dimensional diffusion equation u , - K V i = f ( r ) 6 ( t ) , reR", f > O , (1.11.30)

"(r,o)=o u(r,t)*O

for all BS

rep,

,=(=:+I:+

(1.11.31)

...+x:); +-,

(1.11.32)

where r = ( x , , x , ,....1.) and V' i~thendimensionalLspla~ian given by

L,

V ' = L + L + ...+ " ax: Jx: &: The solution of this problem is given by +J) =

4 jf(E)G(r&)dB. (4nKf)i -

.<2,...,cn) and the n-dimensional Green's

when 6=(<,

(1.11.33)

(1.1 1.34) function G(r.6) is

ExcMpk 1.11.3 (The Three-Dimmsioml Poisson Eguotion). The solution of the Poisson equation -vzu = f(r), (I. 11.36) w h m r=(.t.y.z)isgivenby

+) =

J -j J c(~.E) f ( ~4, )

(1.11.37)

when lhe Creedsfunction C(r.6) of the operator. -Va, is given by

G(r,E)=L-L

4nlr-El To obtain the fundamental solution, we need to solve the equation

(1.11.38)

71

1.11 Gncn’s Fwtctionr and Boundoly-Volw Problems

-V’G(r,5)=S(x-5)S(y-11)6(2-~). r+<. (1.11.39) Application of the hree-dimensional Fourier Wnsform. defined by ( 1 . 8 3 , to (1.11.39) gives r ’ C ? ( K , E ) = A CXP(-iK’6). (2@ what 6(~,6)=S{G(r&)} and ~ = ( k , e . m ) . The inverse Fourier transform gives the formal solution G(r.S)=+ jJe%P+(-E)] dK

(2K)

(1.11.40)

; ;

-

where R = r - &

We evaluate this mtegal using the rphcneal polar ewrdinates in the r-spare with the axis along the R-axis. In terms of spherical polar coordinates ( K, 0.0). so that K - R = KR cos B where R = - f 1. Thus. (1.1 1.41) becomes

11

pmviddulat R > 0 . In electrodynamics, the fundamental rnlution (1.11.42) has a well-known mtcrprctation. Physically, it represents the potential at point r generated by the unit point charge dirfribution 81 point 6. Th~hais what can be expected because S(r 6) is the charge density conerponding to a unit point oharge at 6. The solution of (1.11.36) is then given by

-

u(r) = jJG(r&)/(E)dE =

jjE.

(1.1 1.43)

The intcgrand in (1.1 1.43) mnsim of the given charge dimibutionflr) at I = 6 and Green’s function G(r,E). Physically, G(r,E) f(6) represents the resulung potentials due to elementary p i n t charges. and the total potential due to a given charge distribution flr) IS then obtained by the integral superposition of the resulting potentials. This IS called the principle ofsuperposition. Example I.11.4 (The Two-Dimcnaionol Helmholtz Equation). Find the fundamental solution of the two-dimennonal Helmholtz equation (1.1 1.44) -ViG+ I’G = S(x - <)a().- q). < x,y c -.

--

72

tinrarPorriarDiffrren,~14~”~~

It is convenient to change variables x - 5 =x‘. y-q=y’. droppingthe asterisks. (1.11.44) reduces to the form C, +G, - a*G = -6(x)6(y).

Consequently, (1.11.45)

Application of thedoubleFouriervan~farm 8(r)= S(C(x,y)) to (1.11.45) gives the solution as

8 ( . ) = L L

(1.11 46)

277(r2+a’)’

where ~ = ( k , t ) ~ nK’=k’+e’. d The inverse Fourier uanrfom yields the solution

__

--

G ( x , y ) = l j jexp(ir.x)(K‘+a’)-’d~d(.

4n

(1.11.47)

of polar coordinates (x,y)=r(co~.9,sinB). (k,t)=p(eosp,sin$), !he integral solution (1.11.47) becomes I n terms

cos(p-b’))dQ,

G(x.y)=Aj&Texp[irp 477 o ( P + a

1

D

which is, replacing the second integral by Zn JQ(rp),

=..I1 -P

JdrP)dP,

(1.11.48)

In t e m s of the original coordinates. the fundamental solution of equation

.-

(1.11.44)isgiven by

Accordingly, the soluuon of !he inhomogeneous equation (V - a * ) u = -f(x, y) is given by

.(.a) =fJ‘3r.E)f(E)dE3

-

(1.1 1.50)

(1.11.51)

where C(r.E) is given by (1.11.49). Since the integral solution (1.11.48) does not exist for a = O , the Green’s function for the two-dimensional Poisson equation ( I 11.44) ann not be derived from (1.11.48). 1nstead.wedifferentiate (1.11.48)withrespccttortoobtlin

whlchxs, for a=O,

1.11 Green's Funci~onrand Boundary-Value Problems

73

1 _dc dr= _ 2nfJ;(r~)d~=-L.

2m Integrating this result gives the Green's function

G(r,O)= - I l o g r .

an

In terms of the onginal coordinates, the Oleen's function bcomcs

G(r.6) = -Llog[(x - C)' +(y - g)']. 4n

(1.1 1.52)

This is the Green's function for the two-dimensional Poisson equation V' = -f(x,y). Thus, the solution of the Poisson equation is

.(x.Y)=

I ~G(r&)f(E)&

(1.1 1.53)

where G(r.6) isgiven by(1.11.52).

Example 1.11.5 (Grrm's function for the Three-DimensionalHelmholtz Equation). We consider the rhne-dimensional wave equation -["" -e'V'u]=q(r,t), (1.11.54)

where q(r.t) is a source. If q(r.t)=q(r)exp(-iwr) represents a source oscillating with a single frequency m , then, as expected, at least after an iniual Uanatent penod, the entire motion reducts to a wave motion wnh the same freouency m so that we can write u(r..t)=u(r)eip(-iw). Consequently, the wave equation (1.1 1.54) reducer u) the three-dimensional Helmholtz equauon - ( V ' + k * ) u ( r ) =f ( r ) . (1.1 1.55)

where k =? and f ( r )= c-'q(r). The function u ( r ) satisfies this equation in e some domain D c R with boundary dD, and it also satisfies some prescribed boundary conditions. We also assume that " ( r ) satisfies the Sommerfeld radiation condition which simply states that the solution behaves like outgomg waves generated by the source.I n the limit BS m +0.so that k + 0 and f ( r ) can be interpreted as a heat sauce, the equation (1.11.15) results into a threedimensional Poisson equation. The solution " ( 7 ) would r e p e n t the steady temperature disbibution in D due to the heat source f ( r ) . However, in general. "(r)can bc interpreted as a function of physical interest. We constmct a Green's function G(r.6) for equation (1.11.55) so that G(r.6) satisfies the equation -(V* +k')G = &(x)&(y)&(r). (1.11.56) Using the spherical polar coordinates, the three-dimensional Laplaccan em k expressed m terns of radial coordinate roely so that (1.1 1.56) assumesthe form

q,0

-[

= 4m $$(r2 gar) + k ’ G ] with the radiationcondition r(G, +rlrG)=O.

c r < -,

pi

(1.1 1.57) (1.1 1.58)

For r > 0 , the function G satisfies the homogeneous equation

_rl_ir( r ’ $ ) + k ’ G = O .

(1.11.59)

dari t(fl)+k’(&)=O.

(1.11.60)

or. equlvalenuy, This equation admits a solution of the form rC(r)=Ac” +Be-’*. a B,’

G(,) = A-+

I

(1.11.61)

**

(1.11.62) I

where A and Bare arbivary constam. In order to satisfy the radiation condition, weneedtoset A=O,andhense, thcsolution(1.11.62)becomes

+,

C(r)=Bc.

(1.11.63)

To determine B, we use the spherical surface Ssof radius

E,

so that

limI$dS=-limIBc’”b(~+ik)dS=l, c 4

I.

from which we find B=-’

(1.11.~)

=*QS.r

4x

BS E +

I A C S l h e form

0 . Consequently. the Green’s function

p

C(r)=--. (1.11.65) 4m Physically, this represents outgoing spherical waves radiating away from the SOURIC at the origin. With B point soume at a point E, the Green’sfunction is re~-nmd by (1.11.66)

whererand Sanposrtion vectors in R’. Finally, when k = 0 , this result reduces exactly to the Green’s function for the three-dimensional Poisson equation. Exnrnplc 1.11.6 (One-DimensionalWove Equorion). We first consider the one-dimensional inhomogeneouswave equation -[u,-$u.]=q(x.r).

xsR,

r>o,

(1.11.67)

1.11 Green's Functionrad Boundo~VolucProblems

75

with the initial and baundary conditions u(x,O)=O.

for x s R .

u,(x.O)=O

u(x,r)+O

(1.11.68ab) (1.11.69)

IxI+-.

as

The Omn's function G ( x , r ) for this problem satisfies the equation

-[.--

S G , , ] = 6(x)6(r)

(1.11.70)

and the samc initial and boundary conditions (1.11.68ab)-(1.11.69) satisfied by G(x.0. We take the joint Laplace transform with respect to r and the Fourier vanrfurm with respect tu I tu cquatiun (1.11.70). so chat (1.11.71) whcn k and s rcpnrent the Fourier and Laplace oansform variables nspeetively. The inverse Fourier uansform rives

Finally, the inverse Laplacs Wansfom yields the Green's function with a s o w e at the origin

G ( . . r ) = ~ ~ ' { ~ c r p ( - j l r l ) } = ~ H ( r - ~ ) . (1.11.73) whne H(1) is the Heaviside unit step function.

With a point souce at

(5. z), the Onen's functionhas the form

G(r,r:{.r)=fH(r-r-

v].

(1.1 1.74)

This function is also called the Ricmann function for the wave equation. The result (1.11.74) shows that G = O unless the point (x,r) lies within the chnmrmstic cone defined by the inequality c(r r ) > x 5 Thc~olutionoftheequation(1.11.67)i~

-

]G(.r.r;<.c)4(5.7)dr

u(x.r)= -

=f which is, sin-

I - I.

0

-b. .

jH(r-7

H = I for x - c ( r -

-

v)

4 5 . 7 ) d7

(1.11.75)

r ) c 5 < x + e ( r - r) and ram outside.

tineor Pornhl Diffemntial Equoiions

76 x*l,-d

I

=:

Jq(<,r)d5=:

Idr

Jjq(<,r)drd5. (1.11.76)

.
where D IS the triangular domain made up of two points X T cf on the x-axis and another point ( q f )off the x-axis in the (x,t)-plane. Thus, the solution of the general Cauchy problem described in Example 1.9.1 can bc obtained by adding (1.11.75) to the d'Alcmberl solution (l.7.21),and hence. it reducer to (1.9.15).

Emmple 1.11.7 (Green's Function for on Arisymmetrie Wave Equation). We consider the two-dimenaonal wave equation in polar ewrdinstes -pa-$un]=

f(r.1). O < r < - . I > O .

(1.11.77)

where the Laplacian 0' in cylindrical polar coordinates without B-dependenceis given by

v

'-t dJ$)

(1.11.78)

=--

and u =u(r,r) andf(r,r) are the given function of I and appropriate initial conditions u(r,0)= 0. u,(r.O) =o. nK Oreen's function G(r,r) must satisfy the equation

-pG-+Gn]=t

1.

We also prescribe

6(r)S(f)

(1.11.79) (1.11.80)

with the s m e initial conditions (1.11.79).

We apply the joint zero-order Hankel transform with reSFct to LaplaeeuanEformwithres~cltorto(1.11.80)toohta~n E(K,S)

=

e ' 2Z(?

I

and the

(1.11.81)

+C2K')'

where K and s are the Hankel and Laplace uanform variables, respectively. The inverse Laplace transform gives

e ( x . r ) = l . sin(cn). ZZK

Then,the inverse Hankel transform yields the solution for G ( r , f )BS G ( r - . t ) = e~ ' { ; s i n ( c m ) }

(1.11.82)

78

Linear Parlral Digcrenrial Equn'om

In terms of the spherical polar coordinateswith the polar axis along the vector z , so that ~ . r = k ? c o s 8 ,r=lxlandd.=x2drsin8d8d~, integral (1.11.93)assumesIhe form

-

xr

I d $ IKsin(ern)drjexp(inFosB) sin8d8.

G(x,r)=+

(24

0

(1.11.94)

0

For I > 0, cl + I > 0. and hence, S(cr + r ) = 0 . Thus, (1.1 1.96)

in which the formula 6(m)= IS(,)

ISused.

Ifthesourceisloeatedat ({,q.<,?)=(&r),thcdcsiredcreen'sfunctionis given by G(x,r;& %) = 4 + - s 1

~ [ 1 - % - ~ ) - 6 ( 1 - r + ~ (1.11.97) ) ] .

It should be noted that Green's function (1.1 1.96) for the hyperbolic equation is a generalized function. whereas in the other examples of Green's functions, it was always B piecewise analytic function. In general. the Green's function for an elliptic function is always analytic, whereas the Green's function for a hyperbolic equation is a generalized function.

1.12 Exercises 1. Classify each of the panial differential equations below BS either hyperbolic. parabolic, or elliptic, determine the characteristics, and transform the equations to canonical farm: (a) 4u, +5u, +u, +u, +up = 2. (C)

u,+yun=0.

(e) x*u,

+ yzu,

=0 ,

(0U"

(g) x * u , + 4 y u , + u n + Z u , = 0 .

(i)

, U

+ ZXU,

(b) Zu, -3u,

+u,, = y ,

(d) yu,+(x+y)u,+xu,,=O,

+ a 2 u,>

+u = 5 ,

+XU,"

= 0,

(h) 3yu,-xu,=O,

0) y u ,

-ZU,

+c' u,,

+ x2 u,

-u

= 0.

6L

Lineor Porrkl Differential E p d a n r

80

10. Solve the Cauchy problem for the linear Klcin-Gordon equation -2"= +a2 u = 0. < x < -, > 0,

,

--

""

u ( x . 0 ) =f(x),

($),*

=&-(I)

for

--

<x<

-.

I I. Solve the telegraph equation ",-c~u"+u,-ou~=o,

(2)

.(x.O)=f(x).

--
I>O,

for --<*<-.

=&)

N

Show that the solution is unstable when c' d,show that the bounded integral solution is given by u ( x , I) =

1- ~ ( k erp[-kx(c' ) JZ;;!

-01)

r+ik(x+ot)]l,

where A ( k ) is given in terms of the mnsformed functions of the initial data Hence, deduce the asymptotic solution. BS I + -, in the form

12. The transverse vibration of an infinite elastic beam of mass m p ~ unit r Ien& and the bending stiffness EI is governed by

d=N.- - < x < - ,

u,+n'u,=O,

r>o.

Solve this equation subject to the boundary and initial data

"(0,r)=0 fnr all I > O . u ( x , O ) = @ ( r ) , and u,(z,O)=yr"(x) for O < x < Show that the Fourier transform solution is ~ ( tI) .= @ ( k ) cos(ork')- ~ ( ksin(&). ) Find the integral solution for u(x, I). 13. Solve the Lamb (1904) problem in geophysics that satisfies the Helmholtl equation in an infinite elastic half-space I(

=

-

+" + o4 i2u = o ,

--<x<-,

z>o,

where o is the fnquency and c, is the shear wave speed. At the surface of the half-space ( 1 = 0),the boundary condition relating the surface stress to the impulsive point load distribution is given by

p du=-P+)

ar

a1

r=O,

when p is one of the Lam& constants. P is B constant and

81

1.12 Exercises

as z+for -=+<x<-. Show that the solution in tsrms of polar cwrdmates is U(X.Z)+O

-P Zip

(q$ $) rwr

.p(

-

for

0 1 >> c2

14. Find Ihe solution of the Cauchy-Poisson problem (Debnath, 1994, p. 83) in an inviseid water of infinite depth which is governed by

$=+$n=o. --
1> 0.

$ * + O as z+--. and q ( x . O ) = P S ( x ) . whcra 4 = $(x,z,l) IS the velocity potential, q ( 1 , r ) IS the free surface elevation, and P is a consmt. Denvc the asymptotic solution for the free surface elevation in the limit as $(x.O,O)=O,

t 4 s .

15. Obtain the solutions for the velocity potential $(x, L. I) and the free surface elevation q ( x , I) involved in t k two-dimensional surface waves in water of finite (or infinite) depth h. The governing cquatmn. boundary. and free surface conditions and initial conditions (see Debnath 1994. p. 92) are @"+$==O, - h S r < O , - - < x c - , t > O

0, + g q =-P p (x )e x p (i o t )

ez- l l , = O

I

z=o, 1 > 0

@(x,z,O)=O=T)(~.O)f~sll~~nd~. 16. Solve the steady-state surface wave problem (Debnath. 1994. p. 47) on a running stram of infinite depth due to an external steady pressure applied to the h.ee surface. Tlx governingquation and the free surface conditions are &+@.=O. --DD<x<-. - - < z c o , l> O ,

82

Liwar Poniol Dtflercnriol Equations

when U IS the stream velocity, 9(x, z, r) is the velocity potential, end q(x, r) is the free surface elevation. 17. Apply the Fourier transform to solve the initial-value problem for the diss,pative wme eq"BU0" u,, = 2"" + au,, < x < -, t > 0.

--

u(x.o)=f(x), + . ~ ) = a f " ( x ) for - - ' c x < - . when a is B positive constant. 18. Ux the Fourier transform to solve the boundary-value problem " +", , =--x exp(-x'), < x c - , a < y <-. u ( x , 0 ) = 0,

for -

Show that

-

+ . Y ) = x1

--

< x < -, u and i o derivative vanish as y + ![1-ev(-~)] -

-_

s i n k exp(-+)dk. 1

19. Solve the mirial-value problem (Debnath, 1994. p. 115) for the twodimensional surface waves at the free surface of a running stream of velocity U.The problem satisfiesthe equation, boundary, and initial conditions 9=+9.=0, - - < x < - . -hszsO. r>O,

7, + u%- h = O 9(x.z.O)=q(x.O)=O. forallrandz. 20. Apply the Founer transform to solve the equation u-+un=o, --<x<-, yto, satisfying the conditions u(x.O)=f(x). ",(x.o)=o for - - < x < - , ind u(x,y) and i o p m a l derivatives vanish as 1x1 + -. 21. The vansvcrsc vibration of a thin membrane of great extent satisfies the

waveequation

--

c'(u= +u,) =un. < x, y <-. r >o, with the initial and boundary conditions u(x,y.r)+O as IxI--t-. IyI+forall r20.

.(x.y.O)=f(x,y). u,(x.y,O)=O forall X.Y. Apply the double Fourier vansfom, method to solve this problem. 22. Solve the diffusion problem with a source q ( x , 1)

1.12 Err,c;s
83 U, = X U ,

+ q ( x . I), -- c x c-,

"(x,o)=o for

r >O,

--
Show that the solution is

u(x. 1 ) =

FJ(I 4nm,

7)-+ d7 j q ( k , 7)er p

23. Apply the hiple Fourier transform to solve the initial-value problem u,=r(u,+u,+u,).--<x. y. r < - , r > o . u ( x , O ) = f ( x ) forall x,y,z, where x = (1.y. 2 ) .

24. Use the double Fourier transform to solve the telegraph equation un+ou,+bu=eiu,.

--<x.r<-,

u(0,r) = f(r). u,(O,r) =&), for - - < I <-, where o , b , c areeonitantsand f(r)m d g (r) arearbitrslyfunctionsof I. 25. Use the Fourier transform to solve the Rorsby wave problem in an inviscid 8-plane ocean bounded by walls a1 y = 0 and y = I , where y and x represent vertical and horizontal directions. The fluid is initially at mt and then, at I = 0+, an arbitrary dishlrbanee localized to the vicinity of x = 0 is applied to generate Rossby waves. This problem satisfies the Rosrby wave equation

$(v

-r ~ ) i ] + & =o, ~ --<

with the boundary and initial conditions y.(x,y)=O for O < x < - ,

x

<

-.

o s y s I, r > 0.

y=Oandy=l,

y ( x , y . t ) = v,(x,y) at I = O for all xand y.

Examinethecase for y,(x)=&

exp{ikoz-(:y}.

26. The equations for the cum01 I ( x , r ) and potential V ( x , r ) at B point x and time r of B timsmisSion line containing resistance R, inductance L, capacitance C, and leakage inductance G me L I, + RI = -V=, and CV, + GV = -1,. Show that both I and V satisfy the telegraph equation

f u. -u,

+au, +bu = 0,

where c'=(LC)-'. o = L G + R C , a n d b = R G . Solve the telegraph equation for the following cases with R = 0 and G = 0:

84

linear Porriol Difl&ential Equations

(a) V ( x , r ) = V o H ( r ) at x = O , r > O .

v ( x . r ) + O as x + - , r>O, where V . is constant. (b) V ( x ; , t ) = V . e o s a r at x = O . l > O , v(x,r)+O

as I+=-, r > O .

27. Solve the telegraph equation m Exercise 26 with V ( x , O )= 0 for (a) the Kelvin ideal cable line (L= 0 = C) with the boundary data V(O.t)=Vo=constant, V ( x , r ) + O BS x + for r > O . (b) the "on-inductive leaky cable (L = 0) with the boundary conditions V ( O , r ) = H ( r ) and V ( x , r ) + O BS x + for r S 0 .

28. Solve the telegraph equation in Exercise 26 with V(x.O)=O = V,(x,O) for the Heaviiide distonionless cable

conitant it ant= L

C

k) with the

boundary data V ( O , r ) = V , f ( r ) and V ( x . r ) + O BS x + for f > O , where V, is constant and f ( 1 ) is an arbitrary funclion of 1. Explain the physical significance of the solution. 29. Solve the inhomogeneous equauon u,,=-rusinor, r>O u(x.o)=x. . .

u (.O , r ) = O

30. Find the solution of the inhomogeneous equation

1 .-u,=ksin("). 2 "

O<xO

u(r.O)= 0 =",(&a) for o < x < 0 , u(o,r)=o=u(a,r) for r > O . 31. Solve the Stokes problem which is concemcd with the unsteady boundary layer flows induced in a scmt-infinite viscous fluid bounded by an infinite horizontal dirk at z = O due to nontor~maloscillations of the disk I* 11s own plane with a given frequency a.The equation of motion. the boundary and iniual eandiuons an U , = Y U ~ .

z>O, r > O .

u(r,r)=Ue'Y' an r = O , t > O , u(r,r)+o when u(2.r) constant.

BS

z+-

for r>O.

u(r.o)=O for I s 0 and z>O, is the velocity of the fluid of kinematic viscosity Y and U is

85

1.12 E~erciser

Solve the Raylcigh problem ( m = O ) . Explain the physical significance of both Stokes and Rayleigh solutions. 32. Solve the Blasius problem of an unsteady boundary layer flow in B semiinfinite body of viscous fluid enclosed by an infinite horizontal dirk at z = 0. The governing equation and the boundary and initial conditions are

"+,

JU -= at

f>O,

L>O.

JZ

"(z.f)=uf

on

"(Z.f)+O

Bs

z=o, f > O , *+-, f > O .

u(r.r)=0 at f S O , r > O . Explrn the significance of the solution. 33 Obtain the ~ o l ~ l i oofnthe Stokes-Ekman problem of an unsteady boundary layer flow in a semi-infinite body of viscous fluid bounded by an infinite horizontal disk at L =0, when both the fluid and the disk rotstc with B uniform angular velocity I2 about the z-axis. The governing boundary layer equation,the boundary and the initial condiuons aye *+zaiq=vfi,

mi

q(i,f)=oc'"'+be"' q(r,1)+0 as z+-.

Z>O,

on

r=O.

f>O.

f>O.

q(r.r) = 0 at t S 0 for d l L > 0, w h e ~q = u + i v, a, is !he frequency of oseillauons of the disk and a. b are complex constants. Hence, deduce the steady-state solution and determine the strnc~reof the associated boundary layen. 34 Show that. when a, = 0 in Exercise 33. the steady-flow field is given by

-

q(2.f) ( a + b ) c r p { ( - ~ ) p z } .

35

Hence, determine the thickness of !he E h a n layer. Solve the telegraph equation

U.-c'",+20",=0. --<x<-, f>O, "(x,o)=o, ", (x.0) = p ( r ) . 36 Use the Laplace vanrform to solve the initial-boundary-value problem u,=c'u,.

O<x
u ( x . 0 ) = x +sin(%)

for

f>O.

o <x
"(O,f)=o="(a,f) for f > O .

37 Solve the diffusion equation u,=tu,.

u(x,0)=1

-a<x
for

f>O.

-LI<.r<(I.

86

h e o r Partial Di&wnrioi Eqwrms

for t > O . 38. Use the joint Laplace and Fourier transform to solve the mtial-value problem for transient water waves which sattsfies (see Debnath. 1994. p. 92) u(-n,r)=o=u(o.r)

V'$=@m+$z=o,

--<x<-.

--czco.

r>o,

m, = 11, $, +gll = -PpG).".

P

On2

=o, r>o,

t

O(x.r.0) = 0 = q(x.0) for all x and I . where P and p are constants. 39. Show that the solution of the boundary-value problem

u,,+L,+".=o,

o
o
41. If f ( r ) = A ( a ' + r ' Y i . where A is a constant. show that the IolUtiOn of the bih-onic equation described m Example 1.10.6 is

42. Solve the arisymmetrie biharmonie equation for the free vibration of an elastic disk

87

1.11 Exrcirrr

b.($+;$"+""=o, "(T,

w h a t b'

0 ) =f(.),

=x is the 20h

o
q(r.0)= 0

I>O.

fm 0 < I < -,

ratio of the flexural rigidity of the disk and its mass

2 h a pcr unit area. 43. Show that the zero-orderHankel transform solution of the arisymmetic

problem

",+lu.+",=o,

0<1<-, --
Iim ( r ' u ) = ~ , lim ( ~ l r r ) u , = - f ( z ) , - - < z < - , ,+D

,4

i(k,z)= I jcip{-kIz - O]f(C)dC. 4 ~ . k-

Henee. show that u(r,z)=

1jp+(z-&f(C) d

~ .

4r-

44. Solve the nonhomogeneous diffusion pmblcm

(

A

u,=Ku,,+'u,

+Q(r,t), O c r < - .

u(r,O)= f(r)

f>O,

forO
w h m K is a constant. 45. Solve the pmbkm of the elecmfied unit disk in the (x,y)-plane with center at the origin. me electric potennal "(1.2) is arisymmcvic and satifies the

boundary value problem ",+~u,+",=o,

o c r < - , o
u(r,O) =%. 0 s r < a ,

L o , onz=O J2

for n < r < - ,

u ( r . z ) + 0 as z+for all I , w h a t ug is constant. Show that the solution is " ( ~ , z ) = ~ j I . ( b ) ~ r - ~ h d k . = o k 46. Solve the aiisymmcvic surface wave problem in deep water due to an oscillatory surface pressure. The governing equationsu e

timar Poniaf Differenad E p d o m

88

"'0

= 0"

+1, + & = 0.

4, + g q =-'p(r)

erp(iwr)

0.-R,=O

0 < I < -,

-- < z s 0,

1

onr=o.

1>0. 2 so

$(r.r,O) = 0 = q(r,0), for 0 s I < - , --< 47. Solve thc Neumann problem for the Laplsce equation

o < r < - , o
",,+l",+uu=o. uz(r.O)=-$H(a-r),

u(r,z)+O

O < r < -,

fcrO
Bsz-+-

Show that

limu(r.2) = -!-(r' 0-0

Zn

+zz)-+.

48. Solve the Cauchy problem for the wave equation in a dissipatingmedium

(

A

u.+znU,=c~ u W + ' u , , o < r < - ,

r>o.

u(r.0)= f(+ u,(r.O) = p(.) for 0 < r < -* where K ISa constant. 49. Use the joint Laplace and Hankel vansform to solve the initial-boundaryvalue problem

o
c~(u,+~u,+u,)=u..

o<,-<-.

u,(r.O.r) = H(n-r)H(r),

0 <,O, and u(r,z,r)+O

u(r,z,r)+O

%r+-

r>o. sz+-,

*(r,z,O)= 0 = ",(r,z,o), and show that u , ( r , z , r ) = - m H ( i - r ) !Il(&).[c!+$)

Io(kr)d.

50. Findthesteadytemperature u ( r . 2 ) inabeam 0 6 r < - , O s z s a , when the face r = O is kept at temperature u(r.O)=O and Ule face z = n is insulated except that heat is supplied through a circular hole, such that u ,(r,a )= H(b-r). The t e m p e ~ ~ ru(r, e z ) satisfies the axisymmehic equation

1.12 Excrcms

89

o s r < - , oizsa.

u,+~q+u,=O,

51. Find the integral solution of lhe initial-boundary-value problem

o s r < - , osz<-.

u,,+iu.+u==u,,

r>o,

for all rand z.

u(r.z.O)=O

-

U ( T . L . I ) + 0 as r t - or z + 52. Solve the Cauchy-Poisson wave problem (Debnath, 1989) for a viscous liquid of finite or infinite depth governed by the equations. free surface.

boundary. and initial conditions

0" +'a + h = 0.

v,

=.(

V"+!

us

-5 r +

K).

whas $ ( r , z , f ) and r(r,z,r) represent the potential and stream functions,

rerpeetively, O s r < - , - h s z s O ( o r - - < z i O ) m d The hec surface conditions are q,-w=o p(u,+w,)=o

$,+gq+2"

wz = o

I

r>O.

on z=o, r > O .

where q = q ( r , r ) is the free surface elevation, u = $ , + y, and w = 0,

-w-yr, are the radial and vcnical velocity components of liquid

panieler, p = pv is the dynamic viscosity, p is the density, and kinematic viscosity of the liquid. The boundary conditions at the rigid boltom are u = $, + VI, = 0 on z = - h . w=

#r

The initial conditions are

-+

(v), =o

I

q=aw, 4=w=o

atr=O,

when (1is a constant and S(r) is the D i m delta funetlon. If the liquid IS of infinite depth, the bottom boundary conditions M

Y

is the

90

Lineor Parlid Di&mnrial €+uiions

(@,yr)+(O.O)

as L+--.

53. Use the joint Hankcl and Laplace uansform method to solve the initialboundary-valueprobkm a,,

+i ",-a"

- 2 E u, = a

?!dS(l),

o
120,

as r + - , u(0,r)isfinivfor r > O . u(r,r)+0

u(r.O)=O=u,(r.O)

far O < r < - .

54. Surface waves are generated in an inviscid liquid of infinite depth due to an explosion (Sen. 1963) above it which generater the p r e ~ ~ ufield re p(r,f). The velocity potential $(r,z,r) satisfiesthe Laplace equation

""+lu,+uz=o.

O
1>0

and the free surface condition u.+gu,=i

P

" (2

[H(r)-H{r,ro(I)}]

on z = O ,

when p is the constant density of the liquid. rg(i) is the extent of the blast. and the liquid is initially at nst. Solve this problem. 55. Use the joint Laplace and Fourier transform to show that the solution of the inhomogenwus diffusion problem u,-r?~"=q(x,l), XER, f>O, u ( x . O ) = f ( x ) for all x e R , canbceipnssedin termsoftheOnen'sfttnction C(q1;c.r) as

~ ( ~ . ' ) = j d ~ j q ( S , r ) G ( x , r ; 5 , r ) dj f5( + 5)G(x,f:5,0)d5, 0

-

-

Whae

56. Find the Green's function G(x,r) of the Bernoulli-Euler equation on an elastic foundationwhich satisfiesthc equation

J'G

J'G

E I y + K G + r n y = W S ( x ) S ( t ) , XER, r > O 3"

with initial conditions C ( s 0 )= O = G,(x.O). 57. Solve the initial-boundary-valuepmhlem u, Ku, = q ( l ) s ( x - Vr), x s R , 1 > 0 ,

-

91

1.11 &wc!srs

u ( x , O ) = O and u(x,r)-+O

as l x l + - ,

whereq(r)=O for r
C , +G, = S ( x - c)S(y- q), Oc x , < < a , 0 < y. I)< b. C ( x . y ) = O on x = O . and x=o;G(x.y)=O on y = O a n d y = b 59. Show that the solution of the two-dimensional Illffusion equation u, - K ( u , + U n ) = q ( l , y , r ) ;

--CX.y<-.

1>0.

with u(x.y;O)= 0

60. Find the Green's function for the one-dimensional Klein-Gordon equation

XER,r>O

u,-c'u,+d'u=p(x,r). with the initial and boundary conditions u(x.O)=O=y(x,O) u(x.r)+o

as

forall x s R . IxI+-.

t>o.

where c a n d d a n M ~ S ~ U . Show that the Green's function for this problem reducer to that of the wave equation in the limit as d 0 &eve the Green's functions for both two- and thm-dimensional KleinGordon equations.

-,

Chapter 2

Nonlinear Model Equations and Variational Principles 2.1

Introduction

Tbis chapter deals with the basic ideas and some imponant nonlinear model equations which arise in B wide variety of physical problems. This is followed by variational principles and the Euler-Lagrange equations. Also included are Plateau's problem, Hamilton's principle, Lagrange's equations, Hamilton's equations, the variational pnnciple for nonlinear Klein-Cordon equations, and the variational principle for nonlinear water waves.

2.2

Basic Concepts and Definitions

The most general first-order nonlinear panid differential equation in two independent variables x and y has the form

".

(2.2.1) F(x, Y . w,)= 0. The most general second-order nonlinear partial differential equation in two independentvariables x and y has the form (2.2.2) F ( r . Y,U, U,. UI Y". Un. u , ) = o . Simrlarly, the most general ftrst-order and the second-order nonlinear equauons in m m independent variables can be mtmduced. More formally, it is possible to write these equations in the operator form L. 44=f(4. (2.2.3) whem L, is a panial diffenntial operatorand f(x) is a given function of two a m m independent variables x = (x. y.. ..). It has already been indicated in Section 1.2 that if 4 is nor a linear operator, (2.2.3) is called a nonlinear p a n i d diflercnriol equorion. Equation (2.2.3) IS called an inhomogeneous nonlinear equation if f(x) t 0. On the other hand, (2.2.3) is called a homogeneous nonlinear equationrf f(x)= 0. In general, the linear superposition principle can be applied to linear partial differential equations if certain convergence requirements are satisfied. This principle is usually used to find a new solution as a linear combination of a given set of solutions. F a nonlinear partial differential equations, however, the linear superposition principle cannor be applied to geneate a new solution. So,

Nonlinear Model Equnrzou end V o n o i m d Princrplcr

94

because most solution methods far linear equations cannot be applied to nonlinear equations, there is no general method of finding analytical ~olutionsof nonlinear partlal differential equation. and numerical techniques are usually required for their solution. A uansformation of variables can sometimes be found that fransforms B nonlinear equation into a linear equation, or some other ad h w method can be used 10 find a solution of a particular nonlinear equation. In fact, new methads are u~uallyrequired for finding solutions of nonlinear equations. Methods of solutions far nonlinear equation represent only one aspect of the theory of nonlinear partial differential equations. Like limar equations, questions of eriitence. uniqueness. and stability of solutions of nonlinear partial differential equations are of fundamental importance. These and other aspectsof nonlinear equations have I d the subject into one of the most diverse and active areas of modem mathematics.

2.3

Some Nonlinear Model Equations

Nonlinear panial differential equations arise frequently in formulating fundamental laws of nature and in the mathematical analysis of B wide variety of physical problems. Listed below are some important model equations of most common interest Example 2.3.1 The simplest first-order nonlinear wave (or kinematic wove) equation is

u,+e(u)ux=O,

xeR.

t>O.

(2.3.1)

when c(u) is a given funcum of I I . This equation describes the propgation of a nonlinear wave (or disturbance). A large number of nonlinear problems governed by equation (2.3.1) include waves in vaffiic flow on highways, (Lighthill and Whitham, 1955: Richards, 1956), shock waves, flood w a ~ e s , waves in glaciers (Nyc 1960, 1963). chemical exchange processes in chromatography. sediment transport in rivers (Kynch, 1952). and waves in plasmas.

Erornple 2.3.2 The nonlinear Klein-Gordon cquolion is "" -c'V'u+ V'(U) = 0. (2.3.2) when e is B constant, V'(u) is a nonlinear function of u usually chosen as the denvative of potential energy V(u). It arises sn many physical problems including nonlinear dispcrsion (Scott. 1969: Whitham. 1974). and nonlinear meson theory (Schiff, 1951). Exomplc 2.3.3 "hc Sine-Gordon equation

an --c2u_

+C

S ~=0, U

x E R, t > 0.

(2.3.3)

2.3 Some N o n l m m Model Equorionr

95

where cand K are constBnts. has arisen Classically in the study of differential geometry. and m the propagation of a 'slip' dislocation in crystals (Frenkel and K O I I ~ O ~ O Y1939). B. More recently. it arises in a wide variety of physical problems including the propagation of magnetic flux in Josephson type super conducting tunnel junctions, the phase jump of the wave function of superconducting electrons along long Josephson junctions (Josephson, 1965; Scott. 1969). chain of rigid pendula connected by springs (Scott, 1969). propagation of short optical pulses in resonant laser media (Arecchi et sl. 1969; Lamb, 1971), stability of fluid motions (Scott et al. 1973; Gibbon. 1985). m fmomagnetism. ferroeleetric materials. in the dynamics of certain molecular chains (such as DNA) (Bamne et al. 1971). in elementary particle physics (Skyme. 1958, 1961; Enz. 1963), and in weakly unstable banclinic wave packet m B two-layer fluid (Gibbon el el. 1979). ErmpIe 2.3.4 The Burgers equation is u,+uu~=vu_. xsR. r>O, (2.3 4) where v is the kinematic wscos!ty. This is the simplest nonlinear model equation for diffusive waves in fluid dynamics. It was first introduced by Burgers (1948) to describe one-dimensional turbulence, and it also arises in many physical problems including sound waves m a viscous medium (Lighthill, 1956). WBYCS in fluid-filled viscous elastic tubes. and magnetohydrodynamic waves in a medium with finite electrical conductivity.

&& 2.3.5 The Fisher equation u,-vum=k(u-:),

whcre v, k and

xeR. r>O.

(2.3.5)

.%reconstants, is used BS a nonlinear model equation to study wave propagation in a large number of biological and chemical systems. Fisher (1936) first inwodueed &is equation to investigate wave propagation of a gene in C

It IS also used to study logistic growth-diffusion phenomena. In recent yem, the Fisher equation has been used as a model equation for B large variety of problems which include gene-culture wave$ of advance (Aoki. 1987). chemical wave propagauon (Arnold ct al. 19871, and spread of early f m m g I" Europe (Ammeman and Cavali-Sforva, 1971, 1983). It also arises in the theory of combustion. nonlinear diffunon. and chemical kinetics (Kalmogaov et al. 1937; Aris, 1975: and Fife. 1979). a population.

Example 2.3.6 The Boussrnrsq equation "" -um + ( 3 4 " - = 0 (2.3.6) descnbci one-dimensional weakly nonlinear dispersive water waves pmpagaung in both positive and negative x-directions (Peregrine, 1967; Toda and Wadati,

"-

Nonlinear Model Eqvoooru and Vorialionol Pnncipler

96

1973: Zakharov, 1974; Ablowitz and Haberman. 197% and Prarad and Ravindran. 1977). It also arises in one-dimensional lattice waves (Zabusky, 1967) and ion-acoustic solitons (Kako and Yajima, 1980). In recent years, considerable attention has been given lo new forms of Boussinesq equations (Madsen et al. 1991: Madsen and Sorensen, 1992, 1993)dealing with water wave propagation and to madified Boussinesq equauons (Nwogu, 1993; Chen and Liu, 1995) in terms of a velocity potential on an arbitrary elevation and fnc surface

displacement of wafer.

Exampk 2.3.7 The Koneweg-de Vrics (KdV) cquanon u,+auux+fly=O,

x s R . 1>0,

(2.3.7)

where a and fl are constants, is a simple and useful model for describing the long time evolution of dispersive wave phenomena in which the steepening effect of the nonlinear term is counterbalanced by the dispcrsion. It was originally introduced by Kortcwcg and de Vries (1895) to describe the propagation of unidirectional shallow watpr waves. It admio the exact solution called the soliton. It arises in many physical problems including water waves (Johnson. 1980; Debnath, 1994), intcrnal gravity waves in a svatified fluid (Benney. 1966: Redckopp end Weidman, 1978), ion-acoustic waves in a plasma (Washima and Tamti, 1966). pressure waves in a liquid-gas bubbles (van Wijngarden. 1968). and rotating flow ~n a tube (Lerbovrch. 1970). There are other physical systems to which the KdV equation applies as a long wave approximation include acoustic-gravity wavcs in a compressible heavy liquid, sxisymmetric waves in a "onuniformly rotating fluid, acoustic waves in anharmonic cry~tal~. nonlinear waves in cold plasmas, arisymmeme magnetohydradynamic waves, and longitudinal dispersive waves in elaslle rodr. Erample 2.3.8 The modified KdV (mKdV) equation u,-6u2u,+y=0.

xsR, t>O

(2.3.8)

describes nonlinear acourtie waves in an anharmonic lattice (%bulky, 1967), Alfv6n waves in a collisionless plasma (Kakutaoi and Ono. 1969). It also arises in many other physical situations.

Example 2.3.9 The nonlinear Schrddmgcr ( N U )equation iu,+u,+ylur u=O, X E R , 1>0. (2.3.9) when y is a constant. describes the evolution of water waves (Benney and Roskes. 1969; Hasirnoto and 000, 1972; Davey, 1972; Davcy and Stewartson, 1974; Peregrine, 1983: Zakhmv, 1968; Chu and Mei, 1970; Yuen and Lake. 1975; lnfcld ct al. 1987). It also arises in some other physical system3 which include nonlinear optics, (Kelley, 1965; Talanov, 1965; Bespalov and Talanov,

98

Nonlinear M&l

Equations and Vo&tiomI Primipks

To eliminate this unrealistic feature of the KdV equation, Benjamin et al. (1972) proposed the equation (2.3.12). The dispersion relation of the linearized version of (2.3.12) is UJ-

k

(I+k*)'

(2.3.17)

Thus phase and group velocities of waves associated wtth thts model are given by C

'

= g= (I+k')-', k

C, = ( I - k ' ) ( l + k * ) - * .

(2.3.18ab)

Both C, andC, tend to zero, as k - r - - . showing that short waves do Dot propagate. In other words, the BBM model has the approximate features of responding only insignificantly to shoe wave components introduced in the initial wave form. Thus, the BBM equation seems to be a preferable model. However, the fact the BBM model is B better model than the KdV model has not yet teen fully confirmed.

Exompk 2.3.12 The Gmzburg-Landau (GL) equorion is A,+aA,

= b A + e A IAr,

(2.3.19)

where L? and b are complex eonitanls determined by the dispersion relation of linear waves, and c IS determined by the weakly nonlinear interaction (Stewanson and Stuart, 1971). This equation describes slightly unstable nonlinearwaves and has arisen onginally in b e theories of superconductivity and phase transitions. The complex Gineburg-Landau equation simplifies significantly if all of the cocfficienti are real. The real Ginzburg-Landau equation has k e n extensively investigated 10 problems dealing with phase separation in condensed matter physics (Ben Jacob et al. 1985; van Saarloos. 1989; Balmfad?, 1995). Example 2.3.13 The Burgers-Hurley equnlion u,

+ auu, -YU=

= B(I - u ) ( u - y)u. X E R , 1 > 0.

(2.3.20)

where n. p 2 0. y(0 < y < I). and Y are parameters, describer the interaction bctween coovection. diffusion, and reaction. When a=O. equation (2.3 20) reduces to the Hodgkin and Huxley (1952) equation which dercnbes nerve pulse propagation in nerve fibers and wall motion I" liquid crystals (Scon. 1977; Satsuma, 1987; Wang, 1985, 1986, 1990). Because of the complexity of the Hurley equation, the Fitzhugh-Nagumo (Fitzhugh. 1961. Sleeman. 1982; Nagumo CI al. 1962) proposed simple. analytically tractable, and particularly useful model equations which contain the key features of the Huxlcy model On the other hand, when p = 0 , equation (2.3.20) reduces to the Burgers equation (2.3.4) describing diffusive waves in nonlinear dissipating systems. Satsuma

2.4 Yononom1Principles and rk Euler-Logmnge Equorions

99

obtained solitary wave solutions of (2.3.20) by using Hirota's method in soliton theory.

ExampIe 1.3.14 The Kodomrscv-Pc~iashvili (KPJ equorion (u,-6uu,+u,).+3a2u,=0

(2.3.21)

is a two-dimensional generalization of the KdV equation. Kadomtsev and Pelviashvili (1970) first introduced this equation to describe slowly varying nonlinear waves in a dispersive medium (Johnson, 1980). Equation (2.3 21) with u' = + I ansees in the study of weakly nonlineardispersive waves in plasmas and also in the modulation of weakly nonlinear long water waves (Ablowitz and Ssgur, 1979) which travel nearly in one-dimension (that is, nearly in a venieal plane). Satsuma (1976) showed that the KP equation has N line-soliton solutions which describe the oblique interaction of Solitons. The equation with 6' =-I arises in acoustics and admits unstable soliton solutions, whereas for 6' = + I the solitons are stable. Freeman (1980) presented an interesting review of soliton interactions in two dimensions. Recently. Chen and Liu (1995) have derived the unified KP (uKP) equation for surface and inlerfacml waves propagating in a rotating channel with varying topography and sidewalls. This new equation includes most of the existing KP-type equations in the literature as spccial cases. We close this section by mentioning the Yang-Millsfield equon'ons whish Seem to be a useful m d e l unifying clecvomagnetic and weak forces. ?hey have solutions, called instmfons. localized in space and time, whleh M mterpreted as quantum-mechanical VMsitions between different states of a panicle. Recently, it has been shown that the self-dual Yong-Mills equations are multidimensmnal integrable systems, and these equations admit nductions lo well-known solston equations in (1+1) dimensions, that IS, the Sine-Gordon, NLS. KdV and T d a Lattice equations (Ward, 1984, 1985, 1986).

2.4

Variational Principles and the EulerLagrange Equations

The most remarkable classical Buler-Lagrange variational problem is to determine a function u ( x , y ) in a domain D c R' satisfying the boundary condrtions preraibed on the boundary JD of D and exmemizing the functional I[u(x.~)]=~~F(x,~,u,u,,u,)dr4v,

(2.4.1)

where the function F is defined over the domain D and assumed to have eonunuou~second-order pmial denvatwer.

Nonlmm Model E ~ U I L ~ O and N Voriatioml Principles

100

In accordance with the case of functionals depending on a function of one independent variable (for details. see Debnath and Mibsinski, 1990), the fin1 variation 61 of 1 is defined by 6l(v,h)=l(u+h)-l(u).

(2.4.2)

In view of Taylor's eipanim Iheorem, this reduces to

61 =

jj(u Fw+ v, F, + uy F,)dx dy,

(2.4.3)

to vanish on dD and p = u, and 9 = uy. A necessary condition far the functional I to have an exmmum is that the lint variation of I vanishes. that is. 0 = 61 = u Fu+ ur Fp+ u, $) dx dy where u = u(x, y) which

IS assumed

Ij(

+ dj[$(u F,)+$(u

Fq)]dx dy.

(2.4.4)

We assume that boundary curve dD has a pieccwirc. continuously moving tangent so that Green's theorem can be applied to the second double integral in (2.4.4). Consequently. (2.4.4) reduces to

Since v = 0 on aD,the second integral in (2.4.5) vanishes. Moreover, since u is an arbitrary function. 11 follows that the inlegrand of the first intcgral in (2.4.5) must vanish. Thus, the function " ( 2 . y ) eruemizing the functional defined by (2.4. I) satisfiesule p d a l differential equation

2

:F,-~F*=o.

dy

(2.4.6)

This is called the Eubr-Lagrange cquntion for the variational problem involving two indcpcndcnt variables. A necessary condition for an extremum of I[u(x. y ) ] is that u(x, y ) satisfies the equatlon (2.4.6). If u ( x , y ) IS presrribed on dD. the solution of the Eulcr-Lilgrangcequation with Ule given boundary condition can be found.

2.4 Vanoliono1 Principles ond the Euler-LOgron~eEqunoru

101

The above variational fomulauon can readily be generalized for functionals depending on functions of three or more independent variables. Many physical problems require dctemining a function of several independent variables which will lead to an exmemum of such functionals. In parlicular, if u = u ( x ) dcpcnds on only one indepcndent vanable x , then. the functional (2.4.1)becomes I[u(x)]=IF(x,u,~')dr,

(2.4.7)

where d = $ , and the Euler-Lagrange equation (2.4.6) reduces to the standard f m (2.4.8)

Example 2.4.1 Find u(x. y ) which eramizes the functional dr dy.

I [ u ( x , y ) ] = l l ( u , ' +u:)

(2.4.9)

D c R'.

The Euler-Lagrange equation with F = p1 + q2 is

"(") ap +% J~

J

(JF),o.

h u,, +un = 0. (2.4.10) This IS the two-dimensional Laplace equalion. Similarly, the functional I[U(X,

'.(I

Y.z)]=JJ

+u; + ~ ; ) d x d y r*, D C R'

(z.4.11)

will lead to the lhne dimensional Laplace equation u,, +an + = 0.

"-

(2.4.12)

In this way. we can derive the n-dimensionalLaplace equation v'U=u,,+",~,+

...+" =o. ,

(2.4.13)

Example 2.4.2 (Ploreou's Problem). Find the surface S in ( x . y. z) space of minimum ivea passing through a given plane curve C. The direction cosine of the angle between the z-axis and the normal to the s l n f a a ~ = u ( x , y ) i . ( l + u . ' + u , ' ) - ~ .Thepmj~tionoftheelemenlds of the

area

of

the

surface

onto

the

(x,y)-plane

is

given

(leu,' + u , * r i &=dr dy. The a n a Aof the surface S is given by

by

102

Noniineor Model Eguorionr oed Voriotionoi Princrpkr A =JJ(I+.;

+";$

drdy,

(2.4.14)

where D IS the area of the ( x . y)-plane bounded by the c w e C. Thc Euler-Lagrangeequation with F = (1 + p'

+ q')?

is given by

"[+]+"[A] axJl+p+q dy~l+p+q = 0.

(2.4.15)

This is the equotion of minimal m t f a c ~which reduces to the nonlinear elliptic panial differenual equation (l+u,')u,-2u,u,u,+(l+u,')u,=O.

(2.4.16)

Therefore, the desired function u(q y) should be determined as the solution of the nonlinear Diriehhtprobhm for (2.4.16). This is difficult to solve. However, if the equation (2.4.15) is linearized around the zero solution, the square root term is replaced by one. and then, the Laplace equation IS oblained Example 2.4.3 (Lagmange's Equotion in Mechanics). According to the Hamilton principle of mechanics, the first variation of the time integral of the Lagrangian L = L(9,.4,. t) of any dynamical system must be stationary. that is, 0 = SI=

S r L(9,,4,,

t)

(2.4.17)

dt.

whoe L = T - V is the difference between the kinetic energy T and the potential energy V. In coordinate space. there arc infinitely many possible paths joining any two positions. From all these paths, which EM at a point A at time 1, and

end at another point B at time I,, nature selects the path 9, = 9,(t) for which SI = 0. Consequently. in this case, the Euler-Lagrangeequation (2.4.8) reduces to

.-.(%)=a, r = 1 , 2 ,....n. (2.4.18) a9. dt Jq. In classical mechanics. thcse equations are universally known as the Lagrange equotionr of motion The Hamiiron function (or simply Hamiltonion) H is defined in terns of !he gcncraliredcoordinates 4,. generaliredmomentum p, =

z, and L by 34,

2.4 Vorioriomi Pnncqia and thr Euier-lagmge Equmnr

103

] =kd

C=d p q -L) ddL-4 =O. (2.4.20) dr d t [ g ( ' " ,=, '( dt Jq, J q , ) Thus, H IS a constant, and hence. the Hamiltonian is the eonsmt of motion. Ernrnplr 2.4.4 (Hamilton's Equations in Mechanics). To derive Hamilton's equations of motion, we use the concepts of generalized momentum p, and generalized face F, defined hy

aL p, =and F =&

(2.4.21ab)

aq,

24,

Consequently,UK Lagrange equauons of mmon (2.4.18) reduce to -=-_ JL dp, 2% dr

#,,

(2.4.22)

In general. the Lagrangian L = L ( q , . G , , r ) IS a function of q,. 4,. and I when 4, enters thmugh the kinetic energy as a quadratic term. It then follows from the definition (2.4.19) of the Ha m,ltonm that H = H ( p , . q , , t ) ,andkncc,iudiffcrcntial d H = c E d p p+ c E d q , +* dr. a , as, dt Differentiating (2.4.19) with respect to I giver dH

or dH =

(2.4.23)

zp.4,+ z g , d p ,- 1a,,s -cs aql dq,

dq, -%-dt.

which, due to equatlon (2.4.21a). =cq.dp,-cdLdq,-sdr. aq. a:

(2.4.24)

We next equate the coefficients of twa identical expressions (2.4.23) and

Using the Lagrange equations (2.4.22). the first two equations in (2.4.25) @"eS

. dH , JH (2.4.268h) q =p, =--. I ap.' Jq, Tlese we universally h o w " BS the Hamilton canonical equottom of motion.

104

Nonlinmr Model Equntions ond Voknnbnrl Prinopks

E x m p k 2.4.5 (me Wave Equation of LI Vibrating String). We assume that, initially, the string of length t and line density p is stretched along the x-axis from x = 0 to x = t. The string will be given a small lateral displacement, which is denoted by "(x, I) at each p i n t along the x-axis at time I . The !&etic energy T of the string is given by

(2.4.27) and the potential energy is given by

dr,

(2.4.28)

when T' IE the constant tension of the string. Aecordinr to the Hamilton ninciole

I n this case,

L = i(pu:

- T*u;) which does Dot depend explicitly on x, 1, or U.

and hence. the Euler-Lagrange is given by

(2.4.30) w h a ~c* = (T'/p). This is the wave equation of thc vibrating suing. We close this section by adding an important comment.Many equauons in applied mathematics and mathematical physics can k derived from the EulerLagrange variational principles. the Hamilton principle. or from some appropriatc variational principle.

2.5

The Variational Principle for Nonlinear Klein-Gordon Equations

me nonlimar Klcin-Gordon equation is U" -us + V'(u) = 0, (2.5.1) when V'(u) is some nonlinear function of u chosen as the derivative of potential energy V ( u ) . The variational principle for equation (2.5.1) is given by cSjJL(", U,.",)dt dr. (2.5.2) whne L

ISthe associated Lagrangian density

2.6 The Voriorional Pnnciplefor Nonlinear Wolrr Wmes

105

L ( ~ , ~ , , " , ) = ~ ( " ~ - ~ ~ ) - v (2.5.3) (~). b lme Euler-Lagrange equation associated with (2.5.2)is

dL au - "( ax

")-a", "(a

2%) JU, = 0

which C M be simplified to obtain the Kleln-Gordon quation (2.5.1). We consider the variational principle SJJL dr dt = 0 ,

(2.5.4)

(2.5.5)

with the Lagrangian L given by Lei(";-"

2

.

'-u')-yu'.

(2.5.6)

where y is a constant. The Euler-Lagrange equation associated wlth (2.5 5 ) gives thc special casc of the Klein-Gordon quation Ua -u,+u+4yu'

2.6

=a.

(2.5.7)

The Variational Principle for Nonlinear Water Waves

In his pioneering work Whitham (1965a.b)fvst developed a general approach to linear and nonlinear dispersive waves using a Lagrangla". It is well known that most of the general idear a b u t dispersive waves have originated from the classical problems of water waves. So it is important to have a variational principlc for water waves. Luke (1967) first explicitly formulated a variational

principle for two-dimensional water waves and showed that the basic equations and boundary and free surface conditions can be derived from the Hamilton principle. We now f m u l a l e the uarhtionol principle for threedimensional water waves in the f m SI = SJJLdr dt = 0, (2.6.1)

wkn the olgmngion L is assumed to be equal to the pressure. so that L=-p~~,~,[P,+f(VP)'+gz]dz,

(2.6.2)

when D is an arbiuary region in the ( x . 1 ) space. p IE the density of water, g

is the gravitatronal acceleration, and @ ( x , z , t ) is the velocity potential of an unbounded fluid lying between the rigid bottom a1 z = - h ( x , y ) and the free surface L = q(x. y. I ) as shown in Pigwe 2.1. The functions P(x. z, I ) and

Nonlmor Model Equations and Vormsonol Principles

106

q ( r , t ) are allowed to vary subject to the restrictions S$=O and

67 = O at I,.x2. y , . y2. t, , and r2.

Figure 2.1

A general surfpee gravity wave problem.

2.7 Exercises

107

The second term within the square bracket integrates out to the boundaries JDof Dandvanishesd S9 3s chosen 10 be zero on JD. If we take Sq = 0, IS@],-, = [S$],-., = 0.since S9 is otherwise arbitrary; it Nrn~out that V'$=O,

--<x,y<-,

-h
(2.6.5)

Since Sq,[S9]z-q, [S@],-, may be given arbitrary independent valuer, it follows that

$,+;(V+)'+gq=O

on z = q .

' 1 , + r l , h + ? , 9 , - 9 ~ = 0 on z = n 9zh. + 9"h, + = 0 on L = -h.

(2.6.6) (2.6.7)

(2.6.8) li?lhus, (2.6.5)-(2.6.8) represent the well-known nonlinear system of equations for classical water waves. Finally. this analysis IS rn perfect agreement with that of Luke (1967) for two-dimensional waves on water of arbitraly but uniform deph h.

2.7

Exercises

1.

Use the Hamilton principle to derive (I) the Newton second law of motion, and (ii) the equation for a simple harmonic oscillator.

2.

Dcnve the equation of motion of an elastic beam of length t , line density p , cross-sectional moment of inenia I, and modulus of elasticity E which is fired at each end and performs small uansverse oscillations in the horizontal

(x,y)-pIane. 3. Apply the variational principle sJJLadr=O, wheretheLagrmgian L = i ( u i -u,u.)-u:.

toderive theequation

uX,+6u,u,+u,=O.

4.

Show that this equation leads to the KdV equation when 7 = u x . Use the variational principle

SJJ(I -

":+ u:): dr d, = 0

to denve the Born-lnfeld (1934) equation

(I-u:)u,+2u,u,",,-(l+u:)u,,=o. 5 . Show that the variational principle (Whitham, 1967)

1.7 Exercise*

109

When p=u.,

I=",,,

q=u,.

m=u,,.endn=u,7.

10. I n each of the following cases apply the variational principle or it-r simple extension with appropriateboundary conditions to derive the cornspndmg

equations: (a) F=u: +u: + Z u L .

(5)

(b) F = ; [ # : - a ( u ~ + ~ ~ ) - ~ ~ u ~ ] ,

(d) F = i ( u : -a'&),

+nu: - P u L ) ,

(c) F = +(up.

F = p(x)u" + $ ( 9 ( x ) u 2 ) - [ r ( x ) + k ( x ) ] u 2 ,

wherep. q. r. and s me given functions of x and a and P an constants. 11. Derive the SehrCdinger equation from the variationalprinciple

"!I[f(v:

+

v: + w : ) + ( V - E ) w '

I = d.r&&

0.

whue h = ZnR is the Planck constant, m is the mass of a panicle moving under the action of a force field described by the potentla1 V(x,y,z),and E is the total energy of the panicle.

12. Derive the Poisson equation V'u = F(x, y ) from the variational principle with the functional

I@) =

II[d +u:

+ ZuF(x,y)] &&,

where u = " ( x . y ) is given on the boundary JD of D. 13. Rove that the Eukr-Lagrange equation for the functional

I = I!JF(x,y,z,u, p.9.r.l.m.n.o. b,c)drdy& IS

F - d F - d F -dF,+d'F

"

% X P

a y q

a2

ax''

+dip

ayz-

+dfF

&"

+ a' F. + LF* + L F . = 0, ax+ ayaz dZm when (p.4.r) =(u..u7,u.),

(l.m.n)=(u_.un,un). (a,b.c)=(uq.u,,uy).

14. Derive the equation of motion of a vibrating smng of length I under the acuon of an external force F(x.t) from the vanattonal principle 8

,

6 ] j [ ( + p u : -T'u:)+puF(x,t)]drdr= 0 . 8, 0

where p

IS the line density and

T' 1s the consmnt tension of the string.

Nodincar Modrl Eqmtwu Md Voriorionnl Principles

I10

15. The kinetic and potential energies associatedwith the transversevibration of a thin elastic plate of constant thickness h an

T = 'PJJU'dr

dy,

2 ,

v = 'p,

JJ[(V")' -2(1- U)(Un", -u;)]drdy.

2 " whne p is the surface density and p, = 2h'E/3(1-

u').

Use the variational principle

sjJJ[(T-")+fu]*dydr=O I

D

to derive the equation of motion of the plate

pi + P,V% = f (1.Y.l)

1

w h m f i s the uans~crscforce per unit area acting on the plate. 16. The kinetic and potential energies mroeiated with wave motion in ela~tic sol,& are T=$JlJp(u:+u:+w:)drdyt.

v

=iJJJ[I(.. + y +?a(.: + ")

+wz

u;

+):.

+p((v,+u,)l+(w3+uz)f +(u, + w , ) ' ] ] drdydz. Use the variational principle

6jjlj(T-v)drdydzdt=o 10 derive the equation of wave motion in an elastic medium

(I+p)grad divu+pV*u=pun. when u = (u, v. w ) is the displacementvector.

Chapter 3

First-Order, Quasi-Linear Equations and The Method of Characteristics 3.1

Introduction

Many problems in mathematical. physical, and engineering sciences deal with the formulation and the solution of first-order partial differential equations. Fmm a mathematical point of view, first-order equations have the advantage of providing a c ~ n ~ e p tbasis ~ a l that can k utilized for second-, third-. and higher brder eq"iti0nS. This chapter is concerned with tint-order. quasi-linear and linear partial differential equations and their solutions by using the Lagrange method of characteristics and its generalizations.

3.2

The Classification Equations

of

First-Order

The most general, first-order, partial differential equation in two indepcndcnt variables x and y is of the form F ( X ,Y,U, u,, ur) =o. (x. Y) E

D C R',

(3.2.1)

where F is a given function of its arguments. u = u ( x , y) is an unknown function of the independent variables x and y which lie in some given domain Din R',

u,

du

=-

ax

and ur

=$.

Equation (3.2.1) is often written in terms of

standard nohtionr p = u, and q = u7 so that (3.2.1) takes lhe form F(x,Y.",P,q)=o. (3.2.2) Similarly, the most general. first-order, partial differential equation in three independent variables x , y , r can be written as

F(x,y.r.u.u,.u,.u,) = 0.

(3.2.3)

Eqation (3.2.1) 01 (3.2.2) is called qwi-lmcorpnnioldiferenfiol quation if 11 is linear in first-partial derivatives of the unknown function u(x.y). So, the most general quasi-linear equation must k of the form

a ( x , y , u ) u , +b(x,y,u)u, = c(x,y,u),

(3.2.4)

11 2

FirsfOr&r. Qwi-tineor Eguotionr o d T h r Uerhodof Chorocrcnslicr

where its coeflicients a, b, and c are funeuons of x. y. and u . The following are examples of quasi-linear equations: .( y' + ") ",- y(r' + a ) u7 = (2- yf ) u ,

(3.2.5)

uu,+u,+nu'=O,

(3.2.6) (3.2.7) (y' -uZ)*. -xyuy = X U . Equation (3.2.4) is called semilinear PDE if its coefficients n and b are independent of u, and hence, the sem-linear equation can be expressed in the form n(x,y)u,+b(x.y)u,=c(r.y.u). (3 2.8) Examples of semilinear equations are (3.2.9) xu, + yu, = "1 + 2. (X+l)'U,

+(y-l)'u, ",+nu, + u *

= ( x + y)u'.

=o,

(3.2.10) (3.2.11)

where LI IS a constant. Equation (3.2.1) is said to be lineorif F is linear in each of the variables a. u. and u,, and the coefficients of these variables me functions only of the independent variables x a d y. The most general, first-order, linear partial differential equation has ule form a(x.y)u, +b(x.y)u, ++, Y)" = d ( x , Y). (3.2.12) where the coefficients a, b, and e , m general. are functions of x and y and d(x, y ) IS a given funcuon. Unless stated otherwise. thcse functionsare assumed to be continuously differentiable. Equations of the form (3.2.12) are called homogeneousif d ( x , y) I 0 or inhomogeneous if d(x, y) f 0 Obviously. linear equations are a special kind of the quasi-linear equation (3.2.4) rf a,b are independent of u and e is a linear function I" U. Similarly, semilinearequation (3.2.8) reduces to B linear equation if e is linear in u. Examples of linearequations am (3.2.13) XU' + yu, -nu = 0,

"", +(x+y)u,

(3.2.14)

-u=e',

Y%+*"y=lY,

(3.2.15) (3.2.16) (y -r)., + ( z - X)", + ( x -y)"< = 0. An equation which is not linear IS often called B nonlinear equation. So, the tkt-order cquationsam often classified as linear and nonlinear.

3.3

The Construction Equation

of

a

First-Order

We consider a system of geometrical surfawsdescribed by the equation f(x.y,z,a,b)=O,

(3.3.1)

3.3 The Conrtmfion of n Firs!-OrderEquolion

113

where nand b are arbitrary parameters. We differentiate (3.3.1) with respect to x and y to obtain f .+ P I , = 0. f, + 4 f, = o . (3.3.2ab) where p=*

+'

and q=* Jx The set of th n c equations (3.3.1) and (3.3.2ab) involves two arbitrary parameters a and b . In general. these ouo parameters can be eliminated from this set to obtain B first-order equation of the form

(3.3.3)

Thus the system of surfaces (3.3.1) gives rise to B first-order partial differential equation (3.3.3). In other words. an equation of the farm (3.3.1) containing fwo arbitrary parameters is called a mmplcrc solurion or B complete i n f e g d of equation (3.3.3). Its mle is somewhat similar to that of a general solution for the case of an ordinary differential equation. On the other hand, any relationship of the form f(0. = 0. (3.3.4)

w)

which involves an nrbirrnryfunerionfof two known functions

0 = @(x.y,r)and

y = ~ ( xy,,L) and provides a solution of a firsborder panid differential equation is called B general solurion or general in1egr01 of this equation. Clearly. the general solution of a first-order PDE depends on an orbitmryfunction.TIUSis m striking contrast to the situation for ordinary differential equations w he n the general solution of a first-order ODE depends on one arbitrary constant. The general solution of a PDE can be obtained from its complete integral. We obtain the general solution o f (3.3.3) from its complete integral (3.3.1) as follows. First, we p n -k the second parameter b as an arbmary function of the fin1 parameter a in the complete solution (3.3.1) of (3.3.3). that is, b = b ( o ) . We then consider the envelope of the one-parameter family of solutionsso defined. l h i s envelope is represented by the two simullanmus equauons f(x,y,r,o,b(o))=O. (3.3.5) where the second equation (3.3.6) is obtained fmm the first equation (3.3.5) by partial differentiation with respect to a, In principle, equation (3.3.5) can be solved for o = o ( x . y , r ) as a function of x , y . a n dr . We substitute this result back in (3.3.5) to obtain

(3.3.7) f { x ,y.2 n(x, y.2,). b(o(x.y , r ) ) } = 0 , where b IS an arbitrary function. Indeed, the fwo equations (3.3.5) and (3.3.6) together define the general solution of (3.3.3). When a definite b(o) is prescribed. we obtain a prticulor solution fmm Ule general solution. Since the general solution depends on an arbilrary function, there are infinitely many SoIulions. In practice. only one s d u t i m satisfying prescribed conditions IS

Frmt-Order, Quai-Linear Eqwrionsand The Method of Chorociensiics

1 14

required for a physical problem. Such a solution may be called a particular SOl"ti0".

In addition to the general and particular solutions of (3.3.3), if the envelope of the two-parameter system (3.3.1) of surfacer exists. it also represents a solution of the given equation (3.3.3); the envelope is called the singular solulion of equation (3.3.3). The singular solution can easily be constructed from the complete solution (3.3.1) representing a two-parameter family of surfaces. I h e envelope of this family is given by the system of three equations f(x.y,z,n,b) =O. f.(x.y,z,o,b) = 0, f,(x,y,r,o.b)= 0. (3 3.8) In general. it IS possible to eliminate a and b from (3.3 8) to obtain the equation of the envelope which gives the singular solution. It may be pointed out that the singular solution connot be obtained from the general solution. Its nature is similar to that of the singular solution of a first-order ODE. Finally, it is imponant to note that solutions of B PDE are expected to be represented by smooth functions. A function is called smooth if all of its denvativcs exist and are continuous. However, in general, solutions are nor always smooth. A solution which IS nor everywhere differentiable is called a w o k solution. The most common weak solution is the one that has discontinuities in i u first panial derivatives ~ C ~ aE curve, E so that the solution can bc represented by shock waves as surfaces ofdiscontinuity. In case of a firstorder PDE, th en are discontinuous solutions where z itself and not merely p=-

m ax

an d q =$

arc discontinuous. In fact. this kind of discontinuity is

usually known BE a shock wove. An imponant feature of quasi-linear and nonlinear partial differential equations is that their solutions may develop discontinuities as they move away from the initial state. We close this Seetion by considering some examples.

Exampk 3.3.1 Show that a family of spheres x* + yz +(L - c)' = r* (3.3.9) satisfiesthe first-order linear p m a l differential equation yp-xq = 0. (3.3.10) Differentiating the equation (3.3 9) with respect to x and y gives x + p(r - e ) = 0 and y+q(z - e ) = 0. Eliminating the arbitrary conimt e from these equations. we obtain the firstorder, p m a l differentialequdon

yp-xq=o. Example 3.3.2 Show that the family of spheres ( x - 0 ) ' +(y - b)' + z* = 1' satsfieo the fint-order, nonlmear.partial differential equation z*(p* + q * + I ) = 2.

(3.3.11) (3.3.12)

115

3.3 The Connnretia, of ~1 FirscOrdrr Equnlion

We differentiatethe equation ofthe family of spheres with respect to x and y to

obtain ( x - o ) + r p = O . ( y- b) + r q = O. Eliminatingthe two d i t r a r y constants a and b , we find the nonlinear PDE z’(p’+q’+l)=r’. All surfacer of revolution with the z-axis as the axis of symmetry satisfy the

eq”ali0“

z = f(r’+y’).

(3.3.13)

where f is an arbitrary function. Writing u = x1 +y’ end differentiating (3.3.13) with respect to x and y , respectively.we obtain p = 2 x f’(”), q = z y f’(”). Eliminating the arbimry function f(u) from these redts, we find the equation yp-xq = 0. More generally. if 0 = $ ( x , y, z ) and v = ~ ( xy,, z) me two given functions of X. y. and z and if f (0, v) = 0. where f is an arbitrary function of then, T = r(x, y) satisfies a first-order. paRlal differentialequation

p-+qJ(0, -- v) J(y,z)

J(0. v) J(2.x)

a(#. 4 , J(x,y)

0 and v. (3.3.14)

when

.

.

.

.

(3.3.15)

This result can he pmrcd as follows: We differentiale f (0, v) = 0 with respect to x and y respectively to obtain the followingequations $($+pz)+g(z+p$)=O.

(3.3.16)

$(~+q$)+g(~+q$)=o. (3.3.17)

Nont”vial ~dutionsfor

3 and 3 can be found if the determinant of the a+

JV

eocffiicientsof Ihese equations vanisher. that is,

;;:;1

(3.3.18) v, V ”++ W p , y 0. Expanding this determinantgives the first-order.qu a s i h e a r equation (3.3.14).

First-Order, Qmi-timar Equorions Onz Tlu Methai o/Chorocr~rirncr

1 16

The Geometrical Interpretation of a FirstOrder Equation To investigate the geometrical content of B first-order, partial differential equation, we begin with a general. quasi-linear equation (3.4.1) 4.x. Y , U) uX+b(x, Y. x ) U ) -c(x. Y. U) = 0. We assume that the possible solution of (3.4.1) in the form u = "(ay ) or in an implicit form / ( A Y. u ) = u(z. Y)-" = 0 (3.4.2) represents a possible solution surface in (I,y. u ) space. m i s is Often called an infcgrol smrface of the equation (3.4.1). At my point (I,y, u ) on the solution surface, thegradient vector V / = ( / ' , / , , / ~ ) = ( ~ ~ . " " , - l ) is normal to the solution surface. Clearly. equation (3.4.1) can be wvritten ar the dot prduct of two YeCtDrs

ou,+bu,-c=(n,b,e).(u,.u,.-I)=O.

This clearly shows that the vector

( 0 , b, c)

(3.4.3)

must be a tangent vector of the

integral surface (3.4.2) at the point (x. y. u ) , and hence. it determines a direction field called the ehnmcreristic direction or Mongc axis. m i s direction is of fundamental importance in determining a solution of equation (3.4.1). To summariie,wehaveshownthat f ( x . y . u ) = u ( x , y ) - u = O , as asurface in the ( x , y . u)-spsce, is a solution of (3.4.1) if and only if the direction vector field (a, b. c ) lies in the lamgent planes of the integral surface / ( Ay , u ) = 0 at each point ( x , y , u ) , w h e n V/#O. as showninFigure3.1.

Figure 3.1

Tangent and nannal vecIor fields of (he solotion surface at P point (I&").

3.4 The GeonwtricolInierprnoiion of (I First-Order Equorion

117

A C U N ~ in (x, y.u)-space, whose tangent 81 every point coincides with the characteristic direction field (a,b,e), is called a ehnraercristie C U N C . If the parameme equauons of h s characteristicCUNC are x = x ( t ) , y=y(r), u = u ( f ) * (3.4.4)

then. the tangent wcu)rto this curve is

(2.2,$)

which must be equal to

(n.b,e). Therefore. the system of ordinary differential equations of the

characteristic cuwcs is given hy g=a(x,y.u),

*=b(x,y,u),

*=e(x,y,u).

(3.4.5)

These are called the charocrc&ic equnrionr of the qumi-linear equation (3.4.1). In fact, there are only two independent ordinary differential equations in the system (3.4.5): therefore, its Solutions consist of a two-parameter family of CUNS in (x,y,u)-space.

The projection on u = 0 of a charsetstirtie EWVC on the (z.y)-plane is called B chamcterirtic h e c u m or simply chamcre&ic.

Equivalently, the characteristic equations (3.4.5) in the nonparameuic form m

dy_ . du -dr= _ = o b c

(3.4.6)

The typical prohlem of solving equation (3.4.1) with a prescribed u on a given plane curye C is equivalent to finding an integral surface in (x.Y.N) space, satisfying the equation (3.4.1) and containing the threc-dimensional space E U N ~ r defined by Ihe values of u on C. whish is Ule projection on u = 0 of r. Rerncrrk I. The above geomeuieal interpretation can he generalized for higher order panial differential equations. However, it is not easy to visualize geomevical arguments that have been described for the case of three space dimensions.

R e m r k 2. The geometricalinterpretation is more complicated for the case of a nonlinear pmial differential equations, because the normals to possihle solution surfacesthrough a point do not lie in a plane. The tangent planes no longer intersect along one smight line, hut instead. they envelope along a curved surface known m the Mongr cone. Any funher discussion is beyond the scope of this hook. We conclude this section by adding an important observation regarding the nature of the characteristics in the (x.y)-plane. For a quasi-linear equation. characteristics are determined by the first two equations in (3.4.5) with their SlODeJ

1 18

Firsl.Or&r. Qwi-Linear Eguolionr ond "k Uethal of Chhorrurmsrics

If (3.4.1) is a linear equation, then. a and b are independent of u , the chmctenrtics (3.4.1) a m p h e curve3 with s l o p dy - b ( x d (3.4.8) dr a(x,y)' By integrating this equation,we can determine the characteristics which represent a one-parmeter family of curves in !he (x,y)-plane. However, if n end b are constant, the characteristics of eqution (3.4.1) are straight lines

3.5

The Method of General Solutions

Characteristics

and

We can w e the geometrical interpretation of the first-order, partial differenual equation and the pmpenies of characteristic C U N ~ S to develop a method for finding the general solution of quasi-linear equations. This is usually referred to as the method of characteristics due to Lagrange. Th~smethod of solution of quasi-linear equauons can be dexribed by the following result. Theorem 3.5.1 The general solution of

B

first-order. quasi-linear PDE (3.5.1)

n(x,~,u)u,+b(x,y,u)u,=c(x,~,u)

f(0,v)=0. (3.5.2) where f is an arbitrary function of @(x,y,u)and v(x,y.u) and @ = consmt = c, and v = constant = c, are solution C U N ~ S of the characteristic eq atianr _&= _ dy = _ ,du (3.5.3)

"

o

b

e

The solution curves defined by @ ( x ,y, u ) = c, and ~ ( xy,, u ) = c2 are called the families of characrcrisriccurves of quation (3.5.1)

Proof. Since @(x.y, u ) =el and v(x. y.u) =c2 satisfy equations (3.5.3). these equations must be compatible with the equation @, dy+@* du = 0. (3.5.4)

&+A

This is equivalent to the equation

0. + b 4., + e 4.-= O . Similarly. equation (3.5.3) is also compatible with B v. + b v, + c y. = 0. We now solve (3.5 W3.5.6) for a,b. and c to obtain LI

(3.5.5) (3.5.6)

3.5The Methd of Chorocnristics and General Soluriorrr

119

It has been shown earlier that f(+,v)=O satisfies equation similar to (3.3.14), that is.

a(+,

J(m.

p-++.---. w ) w ) a(+, w ) (3.5.8) J(Y, u ) J(u, 1) J(x, Y) Substituting (3.5.7) in (3.5.8). we find that f(+, y r ) = O is a solution of (3.5.1). This completes the proof. Note that an analytical method has been used to prove Theorem 3.5.1. Alternatively, a geomeuical argument can be used to prove this theorem. The geometricalmethod of pmof is Ieh 10 the reader as an exercise Many problems in applied mathematics, science, and engineering involve panial differential equations. We seldom !qto find or discuss the properties of a solution to there equations in its most general form. I n most cases of interest, we deal with those solutions of pmial differential equations which satisfy cenain supplemenmy conditions. In the case of a first-order partial differential equation. we determine the specific solution by formulating an inirinl-volue problem or a Couchy problem.

Theorem 3.5.2 (The Couehy Problem for (I First-Order PDE). Suppose that C is a given curve in the (x,y)-plane with its parametric equations x=..(t), Y=Yo(t), (3.5.9) where t belongs l o an interval I c R, and the derivatives .:(I)and y;(t) me piecewire continuous functions, such that +y: # 0. Also, suppose that u =%(I) is a given function on the curve C. Then, there exists a solution = u ( x , y) Of the equation

12

"

F ( x , Y,~ . u ~ . u , ) = O

(3.5.10)

in a domain Do f R' containing the curve C for all t e I and the solution " ( A y ) satisfies the given initial data, that is,

u(%(:).Yo(f))=%(f) (3.5.ll) for all values of f E I. In shon, the Cauchy problem is to determine a solution of equstion (3.5.10) in a neighborhood of C . such that the solution u = ~ ( xy ,) t a b s a prescribed value %(t)on C.The curve C is called the initial CUNC of the problem, and %(t) is called the initial &fa. Equation (3.5.1I) is called Ihe inirial condition of the problem. The solution of the Cauchy problem also deals with such questions as the conditions on the functions F, xo(t).y,(r)and u&) under which a solution exists and is unique. We next discuss a method for solving B Cauehy problem for the first-order, quasi-linear equation (3.5.1). We first observe that geometrically

12 0

Fwst.Order, Qmsi-LimmEqmrionr and Thr MerhDd oJChorocferisficr

x=x,(t),y=y,(t). andu=u,(t) represent an initial CUWE r in ( x , y , u ) space. The curve C, on which the Cauchy data is prescribed, is the pqeetion of r on the (x.y)-plane We now present a precise formulation of a Cauchy problem forthe firsborder, quasi-linear equation (3.5.1).

Theorem 3.5.3 (The Couehy Problem for L? Quosr-Linear Equofion). Suppose that xo(f),y,(t), and ".(I) are continuously differentiable functions of f in a closed interval. 0 5 t S 1. and that o, b, and c are functions of x , y. and u with continuous first-orderpanial derivatives with respect to their arguments in wmc domain D of (x, y.u)-space containing thc initial curve r: x = w , y = y , ( r ) , u=%(r), (3.5.12) w h a t 0 5 f 5 I, and satisfying the condition Y X t ) a ( s ( f ) ,y&). .&)) - 4(f) b(xo(f).y&). 4)) 0. (3.5.13) Then, there exists a unique solution u = u(x. y ) of the quai-linear equation (3.5.1) in the neighborhood of C : x = xo(f), y = y.(f), and the solution satisfies the iniual condition (3.5.14) "Jt) = "(+,(f). y&)), for 0 5 f s 1. Note: The condition (3.5.13) excludes the possibility that C could be a characteristic.

*

Erornple 3.5.1 Find the general solulion of lhe first-order linear partial

differentialquation.

X " , + y ", =u_ (3.5.15) The characteristic NWCE of this equation are the solutionsof the characteristic

WpZ3UO"S

dr= _dy _ = _ ,du X

Y

(3.5.16)

U

This system of equations gives the integral surfaces

) = l = C , and $v=!=C,, x when C, and C, are arbiwry constants.Thus. the general solution of (3.5.15) is

f(f.;)=o.

(3.5.17)

where f is an arbiwry function. This general solutmn can also he written as " ( x . Y ) = x .)e;.(

(3.5.18)

where g IS an arbitrary function. Eranrplr 3.5.2 Obtain the general solution of the linear Euler equation x ", + y U" = n u . (3.5.19) The internal surfacer are the solutions of the characteristic equations

3.5 7'hr Method of Chomdcrirficrand Genemi solut!onr

dr= _dy _ = _ ,du nu

y

x

121

(3.5.20)

From these equations,we get

L C , ,I l = c , , x

x"

where C, and C, are a r b i w constam. Hence. the gennal solution of (3.5.19) IS

f(:,g)=o.

(3.5.21)

This can also be uliten as x"

or

= g( f).

u(x,y)=x"B p ) .

(3.5.22)

This shows bat the solution u(x,y) i s B homogeneous function of x and yof degree n. Example 3.5.3 Find the general solution of the linear equation 1 2 " x +y2 = ( x + y)". The charaercristicequations associated wiIh(3.5.23) are

",

(3.5.23)

$=*=A. (3.5.24) Y (.+Y). Fmm the fmt two of the% equations, we find x-'-y-'=c,, wherc C, is an s r b i w constant. It follows h o r n (3.5.24) tha1

(3.5.25)

dr-dy=du xf-ya

(x+y)u

(h

d(x-y) -=-. x-y

du

"

This gives

-y= " c2.

(3.5.26)

whue C, i s a constant. Funhemon, (3.5.25) and (3.5.26) also give *y = C,,

where C, is a constam.

Thus, the general solution of (3.5.23)

i s given by

(3.5.27)

1 7.2

Firsf-Order, Qmi-linear Equrriorua

,:(

d The Method of Choraererisrics

y)

f = 0, (3.5.28) where f is an arbitrary function. This general solution representing the integral surface can also be Wnlte" as

.(.,y)=.y

g(y).

(3.5.29)

when g is an arbitrary function.

Or, equivalently,

.(.,y)=.y h ( 7 ) .

(3.5.30)

when h is an arbitrary function.

Example 3.54 Show that the general solution of the linear equation (3.5.31) (y - z)u. +(z - x ) u r + (1-y)u* = 0

u(x.y, z ) = f ( x + y + r . 2 +y1 + , I ) , when f is an arbivary function. The charactenstie c w e s satisfy theehmaeistic equ~lions dr dy-dr-du -= y - r z--l x-y 0

(3.5.32)

(3.5.33)

or du=O, dx+dy+dz=O. xdr+ydy+zdz=O. lntcgmlionof thue equations giver u = C , . x + y + z = C , . andx'+y'+z' =C,, when C,, C, and C, are arbitrary consmb. Thus, the general solution can be written in terms of an arbitrary function f I" the form

u(x,y,z)=f(x+y+r.x'+y~+z~). We next verify that this is a general solution by inVoducing three independent vanabler 5, q. defined m terns of I, y. and z as 5 = x + y + z , 7 = X I + yl +L', and <= y +z. (3.5.34) where 5 ii an arbittary combination of y and L. Clearly the general Solution becomes

<

.=f(C.d, and hence,

-.,d"+. a+,+ < - J< ' J< a<'

Y

It follows from (3.5.34) that

(3.5.35)

3.5 The Method of Chomeferirricsand Cenrml Solutiom

I t follows fmm the first and the third nsulu that

.=ydy+,dl, JC

JC

JC

123

= -1 and. therefore.

y=y4+ym, ,=,4+,m JC JC JC’ J<

Clearly, it follows by substractionthat z-y=(z-y)d”,

*,*. * JC

Using the values for

JC JC

and

JC

x-r=(y-z)* I”

a<

’

(3.5.35). we obtam

( ~ - y ) &JC = ( y - z ) d . + (Jx z - x ) & + ( x - JY y)$.

If u = u ( e , q) satisfies (3.5.31). then, * = O

JC

(3.5.36)

and hence, (3.5.36) reduces to

(3.5.31). This shows that the general solution (3.5.32) satisfies equation (3.5.31). Exmpie 3.5.5 Find the solution of the equation u(x+y)u, +u(x-y)u, = x * + y * with the Cauchy data u = 0 on y = Zx.

The characteristicequations am dr - dy du .(.+y)

“(x-y)

x’+yl

ydr+xdy-udu

(3.5.37)

xdr-ydy-udu 0

-o=-

Consequently. d [ ( xy -in’)] = 0 and d[$(r’ - y’ - u ’ ) ] = 0.

(3.5.38)

These give two integrals u’-x’+y’=C, and 2 r y - u ’ = C l , where C, and C, a n constants. Hence, the general solution is

(3.5.39ab)

f(x‘ -y2 -u*.2ry-u’)=0, when f is an arbivacy function. Using the Cauehy data m (3.5.39ab). we obtain 4C, = 3C,. k f m 4(u’ - z2 + y ’ ) = 3(2xy- u’). Thus, the S O ~ U Uof O~ equation (3.5.37) is given by 7u’ = 6xy+ 4(x’ - y’). (3.5.40)

Erontplc 3.5.6 Obtain the solution of the linear equation

12 4

Fmf-Oder. Quasi-Linear Eqwfiou and 7'k Ucrhd of Chomererirricr u, - u y

=1

(3.5.41)

with the Cauchy data u(x,O)=x'.

The characteristicequations are dx= _dy _ = _ ,du (3.5.42) I -I I Obviously, _ dy= - I , a n d ~ = l . dx dr Clearly x + y = constant = C, and u - x = constant = C,. Thus,the general solution IS given by u--x= f ( x + y ) , (3.5.43) where f is an arbitmy function. We now use the Cauchy data to find f ( x )= x1 --x, and hence, the solution is .(.,y)=(.+y)'-y. (3.5.44) The characteristics x + y = C, are drawn in Figure 3.2. The value of u must be given at one point on each characteristic which interseelsthe line y = 0 only at one poinl, as shown in Figure 3.2.

't

Exclmpk 3.5.7 Obtain the solution of the equation (y +(" - x ) u y = x - y with the condilion u = 0 on ly = 1.

-+,

(3.5.45)

The characteristicequations for quation (3.5.45) are

*=A=*.

y-" Il--x x-y Thepararne~ieformrofthescequatiansare

I=,-. dy=.-,,

dr ' dr ?bere lead to the following equations

du=x-y, I

(3.5.46)

3.5The Muhod of Choroerrristics md Gcncml Solutions

i+j.+i=O

and x i + y j + u i = O

where the dot denotes the derivative with respect to Intcpra-Blmg(3.5.47ab). we obtain

125

(3.5.47ab)

I.

x+y+u=constant=C, and x2+y2+u1=constant=C2. These equationsrepresent circles. Using the Cauchy data, we find that c: = ( x + y)' = x* + y 2 + 2 q = c, + 2 . Titus, the intepralsurface is describedby (x+y+u)' = * ~ + y ' + u+2. ~ Hence. the S O I U U O ~is given by

"(X,y)=l-xy

(3.5.48ab)

(3.5.49)

X+Y

Example 3.5.8 Solve the linear equation yu, +*u3 = "

(3.5.50)

with the Cauchy data

u(x,O)=z'

and u(O,y)=y'.

(3.5.51ab)

(x

du= _dr-dy _ _ = _ _ , dr+& y-x y+x

"

Solving these equations. we obtain

" = LL= C*(. + y). x-Y

(X

" =C,(.+y),

2 -y1 = 4= E0"Ita"t = c

c,

So thc characteristicsine nctangular hyperbolasfor C > 0 or C < 0. Thus, the general solution is given by X+Y f[-lL,x'-y')=O.

OI equivalently " ( A Y ) = (1+Y) g(x' -y*).

(3.5.52)

Using tho Cauchy data, we find that g ( x ' ) = x ' , thatis, g ( x ) = x . Consequently,the solution beeomes

126

Fwst-Order, Quari-timarEqunrionrand T k Methdof Chnmclerisltcs "(X,

y ) = ( x + y)(x' - y ' )

"(X,

y ) =(.+

0"

2 - y* = c>0.

Similarly, y)(y'

-2)

onyx

-2

= c>o.

It follows fmm these results that u + 0 tn all regions, as x +f y (or y + ix), and hence, u is continuous acrn~iy= f I which represent asymptotes of the rectangular hyperbolas x 2 -y' = C. However, u, and uy are nor continuous. y +f X . For x1 - y l = C > O . u, =3x' + Z x y - y ' = ( x + y ) ( 3 x - y ) + 0, (IS y +-x. uI.= - 3 2 - 2 q + x' = (x+ y ) ( x - 3 y )

Hence.both

u, and u2 are continuous as y

u,+-4x'

u,+4x'.

+0, as y + -x.

+ --x..On the olher hand, y+x.

Bs

This implies !hat u, and rzy an drreontinuousaemss y = X . Combining all these results, we conclude that u ( x , y ) is continuous everywhere in the (x. y)-plane, and u,, ur an continuous cverywhcrc in the ( x , y)-plane except on the line y = x. Hence, the partial derivatives u,. u7 .we discontinuous on y = X. Thus, the development of diaconrinuirics across characteristics is a significant feature of the solutions of panial differential equations.

hkarplr 3.5.9 Determine the integral surfacer of the equation

- y(x* + ")"y = ( 2-y+

x(y2 +a)".

(3.5.53)

with the data x+y=o.

"=I.

l k characterisuc equationsare

dr=dy=du x(y'+u)

-y(.'+u)

or dr x (y'+u)

&

dy Y

(X'.-Y*)"'

-

u

-

-(2+") ( x ' - y l )

Conrequently.

l o g ( q u )= log c,

or qy'=C,.

From (3.5.54), we obtain

&+dy+d" x

y

u 0

(3.5.54)

whence we find that XI

+ y2 - 2 u = c,.

Using the given data. we obtain C, = -2 and C, = 2x' - 2.

so that c, = -2( c, + 1). Thus the integral surface is given by 2 + y= -2u = -2 - 2 q u .

or

zqu+ xl+

y1-

2u+2 = 0.

Example 3.5.10 Obtain the solution of x u , + y u , =xexp(-u)

(3.5.55)

(3.5.56)

with the data u=o ony=x*.

Thc characteristicquatlons BTC

& ' = d y = d u x

xcxp(-u)'

y

(3.5.57)

or L=c, We also obtain from (3.5.57) that dr = e* du which can be nntegmtedto find e" = X+C,. Thus. the general solution is given by

'.(f

--x,

Or. equivalently,

e" = x +

f)=0.

p).

Applying the Cauchy data, we obtain & ) = I - x . (3.5.56) IS given by

(3.5.58) Thus. the solutton of

C'=X+l-Y.

or "=log( x + I-;). Ercnnple 3.5.11 Solve the initial-value problem

(3.5.59)

12 8

FirrcOnLr, Qunri-LineorEquotionr ond T k M & d of Chomt~rirricr

u,+uu,=x, u(x,O)=f(x), when (a) f(x) = 1 and (b) f ( x ) = x. The ehmcteristic equations are dr = dr = du = -d, ( x + u ) -

1

Integrationgives

"

x

(3.5.60)

(3.5.61)

*+?A

f = log(x+u) -loge,

or ("++-

=c,

Similarly, we get ua - 9

=c,

For e a ~ e(a). we obmn 1+x = C, and I --x2 = C,, andhenee C, =ZC, -C:.

Thus. ("1

or

- 2 )= 2("+ X

) P

- ( u +x)'e"'.

"

- x = Ze-' - ( u + x ) P . A simple manipulation gives the solution tanh f + sech f . an exercise.

u(x,f)=x

Case (b) is lee to the reader

(3.5.62)

Ernrnplr 3.5.12 Find the integral surface of the equation (3.5.63) Ui +", = I . representedparametricallyby x=x&), y=y&). u=a(s), (3.5.64) when s is a parameter. The charactensticequations forthe given equation are

"

so that the surface pmses through an rnitial CUNC

dr= _dy _ = _ du " 1 1

which are. in the parametric form,

&=.. & = I . d y = 1 (3.5.65) dT dT dr when r is B parameter. Thus the solutions of this parametric system, in g a d , depend on two parameters s and T. We solve this system (3.5.65) with the initial data r(s.0)= x&). Y(S.0) = Y&), "(S,O) = a(*). The solutions of (3.5.65) with the given initial data are

3.5 T k Method o/Chonutmsl~c~ and Genrml Solutionr

+;

x ( s , T) =

y(..7)=

r u&)+

129

x.(s)

(3.5.66)

7+Y,(4

u(s,7)= T+&)

We choose a particular set of values for the initial data BS x(s.0) = 2s’. y(s.0) = 2s, “(S.0) = 0, s > 0. Therefore. the solutions am given by x=Ll+2s’.

y = r+2s,

u = <.

(3.5.67)

Eliminating T and s horn (3.5.67)gives the integral surface (u-y)’+u*=Zx.

or (3.5.68)

2u = y f ( 4 x - y ’ ) ; .

me solution surface satisfying the data u = 0 on y l = 2 x

is given by

(35.69)

~ ~ = ~ - ( 4 . - ~ * ) i .

This represents the solution surface only when y1 c 4 x . Thus. the solution does not exist for y’ > 4 1 and is not differentiable when y‘ = 4 s . We verify that y1 = 4 x represents the envelope of the family of characlerisues in the ( x , y ) plane given by the r-sliminant of the first two equations in (3.5.67). ulat is. F(x,y,s)=2x-(y-2s)’-4s2 =O. (3.5.70) This representsa family of parabalas fadifferent valves of the parameters. m~hus

the envelope is obtained by eliminating s fmm equations

2= O

and

F=O.

JS

This gives y’ = 4x which is the envelope of the characteristics for different s, as shown in Figure 3.3.

Figure 3.3

I

L

Dotted curve is the envelope of the ehanetcrirties.

(a) Show that the family of right circular cones. whose axis coincides with the z-axis, x z yx =(z-c)’tan’ a satrsfies the finborder,p d a l differential equation y p - xq = 0.

+

(b) Show that all surfaces of revolution, L = f (x‘ + y‘) with the z-axis as the axis of symmeuy. when f is an arbitrary function, satisfy the PDE y p -rq = 0. (c) Show that the two-parameter family of C U ~ Y ~eS - u - b y - o b = O sausfies the nonlinear equation xp+yq+pq=o. Find the panial differential equation arising from each of the following surfaces

(a) r = x + y + f ( r y ) ,

(b) z = f ( x - y ) .

(d) r = q + f ( x * + y ’ ) , (e) Z z = ( a r + y ) ’ + B . Find the general solution of each of the following equations (b) a u , + b u , = O , w h e r e o,bareconsmno. (a) u, = O ,

(4 ” , + Y ” , (e)

=o,

2qu.+(x’+y’)U”=o.

(d) ( I + x * ) u , + u , = O ,

(0 ( y + ” ) ” ’ + y ” , = X - y .

(g) y * u , - x y u , = x ( u - Z y ) . (h) Y U , - X U , = ~ . (i) y*up+ u’xq = -q2, (i) ( y - L U ) P + ( X + y u ) q = x 2 + y * . Show that the general solution of the equation ‘U + Zry’., =0 is given by u = f(2

+

]; .

where f is an arbitrary fundion. Verify by differentiation that u satisfies the original equation. Find the soluuon of the following Cauchy problems: (a) 3u,+2u, =O.with u(x.O)=sinx. (b) y u , + x u , = O . with u(O,y)=erp(-y’).

(c) I U , + Y U , = 2 q , with u = 2 o n y = x ’ . (d) u , + x u , = O , with u ( O , y ) = s i n y . (e) y U ’ + X U ” = q . x 2 0 , y z o

131

3.6 Erercises wsth ~(O.y)=exp(-y') for y>Oand u(x,0)=erp(-x2)

(0

",+xu,

=(y-fx')l,

for x > O .

with u(O,y)=exp(y).

(g) x u , + y u , = u + l , with u(x.y)=x'

on

y=z2.

(h) uu,-uu,=u*+(x+y)'.with u = l on y=O. (i) xux+(x+y)u, = = + I . with u(x.y)=r' on y=O. Solve the mtial-value problem u, + u u,

=o

with the initial curve

,=r. u = c

x='r',

Find the solution of the Cauchy problem Zryu.+(x'+y')u,=O.with

u=erp

Solve the following equations

x",+yu,+zu,=o. (b) x1 us + y2 u7 + z ( x + y) uy = 0, (4x(y -r)u, +y(r-.) U" +z(x-y)u,

(el

on x + y = l

(a)

=o ,

(d) yz u, --u u, +q(2 + y') uc = 0 . Solve the equallo" ".+XU =y with thc Cauchy data: (b) u(l,y)=ly. 10. Show that u, = e' and u2 = e-' are solutions of he nonlinear equation

(4 u(0,y)=y2,

("'+"J

-uZ

=O

but that their sum (2+ e-?) is not a solution of the equation. 11. Solve the Cauehy problem (y+u)u,+yu,=(x-y). with u = I + x on y = l . 12. Find the integral surfaces of the equation uu,+u, = 1 for each of the following initial dam: (a) x(s,O)=i, y(s,O)=Zr. u ( s , o ) = s , (b) x(s.0) = s*, y(s,O) = 2s. u ( s . 0 ) = S. (C)

x(s,O)=s1,

y(s,O)=s, u ( s , 0 ) = s .

Draw characteristics m each case.

13 2

FirscOrdrr, Quui-Lineoar Eqalionr nnd T k Method of Charmsristics

13. Show that the solution of the equation y"' - x u " = 0

+

containing the curye x 2 y' =a2. u = y . do- not exist. 14. Solve the following Cauehy problems: (a) x'u,-y'u,=O, u+c' as y + - , (b) y u , + x u , = O . u=sinx o n x ' + y ' = I , ( c ) -xu, + y u , = I in 0 < x < y . u = 2x on y = 3 x , (d) Z x u . +(x+l)u, = y in x > 0. u = Z y o n x = I , (c) x u , - 2 y u , = x 1 + y '

inx>0.y>0.u=x2ony=l.

IS. Find the solution surface of fhc equation (u' -y*)u2 + q u , + XY = 0. with

u =y

= x, x > O.

16. (a) Solve the Cauehy problem ",+UUy=l.

u(O,y)=q,

where a is a constant. (b) Find the solution of the eq~ationin 16(a) with the data X(*,O)

y(s.0) = s * ,

=zs,

u(O,s')= I .

17. Solve the following equations: (a) ( y + u ) " z + ( x + " ) " , = x + y ,

(b) .xu(#* + q ) u , - y u ( u ' + x y ) u ,

=x'.

18. Solve the cquauon

xr.+yr,=z, and find rhc CWCE which satisfy the associated characteristic equations and intersectthehelix x ' + y ' = o ' ,

z=bm-'(:).

19. Obtain rhe family of curves which represent rhe general solution of the p-I differential equation

(Zx- 4 y + 3u)u,

+( x - 2 y -

3")"" = -3(x - 2 y ) .

Delemine the panicular member of the family which contains the line u=xand y = O . 20. Find the solution of the equation

y u . - 2 q u , =2xu with the eondiuon u(O.y)= y'. 21. Obtain the general solution of the equation

(r+y+5z)p+4Zq+(x+y+z)=0.

( p = z , , q=z,).

3.6 Evreisrr

133

and find the particular solution which passes thmugh the circle

z=o, x ’ + y ‘ = a ‘ . 22. Obtlin the general solution of the equation (Z’-ZYz-Y’)p+x(Y+~)q=x(y-z).

(p=r,,q=z,)

Henee find ulc integral surfaces of this equation passing &rough (a) the x-axis, (b) the yaris, and (c) the z-axis.

Chapter 4

First-Order Nonlinear Equations and Their Applications 4.1

Introduction

Nonlinear, first-order. partial differential equatmns arise in various mar of physical sciences which include geometrical optics. fluid dynamics, and analytical dynamics. An imponant example of such equations is the Hamiltonlacobi equation used to describe dynamical systems. Another famous example of the first-order nonlinear equations is the eikonal equation which arises in nonlinear optics and also describes the propagation of wave fronts and discontinuiues for acoustic wave equations. Maxwell's equations. and equations of elastic wave propagation. Evidently. first-order, nonlinear equations play an important role in the development of these diverse mas. This chapter deals with the theory of the first-order nonlinear equations and their applications. The generalized method of characteristics is developed to solve these nonlinear equations. This IS followed by several mathematical examples, and examples of applications to analytical dynamics and nonlinear optics.

4.2

The Generalized Method of Characteristics

The most general, first-order, nonlinear panial differential equations in two independent variables x and y har the form F ( x , y . u . p . q ) = 0. (4.2.1) when u = u ( x , y ) , p = u , , andq=u,. It has k e n shown in Section 3.3 that complete solution (integral) of (4.2 1) ISa tweparameter family of surfacesof the form f ( x . y , u , o , b ) = 0, (4.2.2) where (1 and b are parameters. Specifying a space curve. through which the complete integral must pass, generates a solution surface. The equatmn of the tangent plane at each point ( x . y . u ) of the solution surface is p ( r - C ) + q ( y - d 4 u - C ) =o, (4.2.3)

136

First-Order Noniineor &nations and TheirApplicati~s

where (<,I),<) are running coordinates on thc mgcnt plane. Since the given equation (4.2.1) is B relauon between p and q f a m y solution surfan. then is a family of fangent planes comspnding to different values of p and q. We can find an equation that represents the envelope of these planes by considering the intersection of (4.2.3) and the tangent plane at the same point (x,y,u) comsponding to neighboring values p + dp and q + dq ofp and q. (p+dP)(x - C)+(q+dq)(y- q ) - ( U - 5) = 0 . Thus, thc intemction of (4.2.3) and (4.2.4) leads to the result

( x - S)dp+(Y- q)dq=O.

(4.2.4) (4.2.5)

W e can rewrite this equation in tmns of quantities specific to the given equation by calculating the differential of (4.2.1) with fixed L y. and u, that is, Fpdp+ Fq dq = O . (4.2.6) Eliminating dp and dq from (4.2.5) and (4.2.6) giver ( x - e ) ~ ,-(Y -

=o

or -=-_

.-c

y-11

FP

FQ

(4.2.7)

The equation of the surface, for which the tangent planes are thc envelopes. is thus, determined by eliminating p and q from (4.2.1). (4.2.3) and (4.2.7). This surfsee represents a cone for nonlinear equations. called the Mongc cone. We next combine (4.2.7) with (4.2.3) to obtlin (4.2.8)

Both ( x . y , u ) and (<.7.C)he on the tangent plane. and hence, (4.2.8) represents a line on the tangent plane to the solution surface. Further, x-5. y - q , and "-5 are direction numbers of B line on the tangent plane, and hence, a fixed direction on each tsngcnt plane (for particular values ofp and q ) is determined by (4.2.8). This direction is determined by the vector (F,,F,.pF,+qF-). A s p and q change, these directions. known as the chomctcrisric directions, determine the family of lines (4.2.8) that generate a Monge cone at (x.y.u). To find differential equations for the characteristics as this point moves along a solution surface. we replace x-5, y - 7 . u - 5 by dr. dy, du respectively, so that (4.2.8) becomes

137

4.2 The Gcnrmhrd Ueihod of Chorocarirficr

The system (4.2.9) represents equations for the characteristic curves of (4.2.1). As in the case of the quasi-linear equation, equations (4.2.9) depend on the unknown solution u(x,y), but unlike the quasi-linear ease, they also depend on p and q. This means that the characteristic curves also depend on the orientation of the tangent planer on the Monge cone at each point. As this geometrical analysis suggests. there is a whole (Mongc) cone of characteristics, not just one characteristic, as in the quasi-linear case. By equating the ratios in (4.2.9) with dr, we obtain the differential equations for the characteristic cwves in the parametne form &=F?. 4 = F , . $=pF,+qF,. (4.2.10) dr dr This ISa system of rhree equations for five unknowns x.y,u,p. and q . Thus. for the nonlinear equation (4.2.1). the System (4.2.10) is not a closed set. We need two more equations to close the system, and hence. it is natural to look for

3.

?!E and Since p = p [ x ( f ) . y ( r ) ] and q = q [ x ( f ) . y ( r ) ] . we dr dr have, by Ule chain ale and (4.2.10).

equations for

dp,p,’+p dt df

I/

2, dt P.F,+P?FV

-ds _ - q, & + q 3= q.F, + q,F,. dr dr ’ dr Further. u = u(x,y) is a wlution of t k equation (4.2.1) which gives

C = F, +PF. dr

+P,F, +S,F, = 0.

C = F, +qF, + p,F, + q,F, dy

=0 .

(4.2.11) (4.2.12)

(4.2.13) (4.2.14)

We next use the fact that p, = q1 which can te considered an inregrobiliry condition for the solution surface and then. utihze (4.2.13>(4.2.14) in (4.2.11)(4.2.12) to obtain *=-(F,+pF.),

&=-(F,+qF.).

(4.2.15)

Equations (4.2.10) and (4.2.15) form a closed set of five ordinary differential equations for five unknown functions x,y.u,p, andq. They an known as chamcrcristicequntions for the nonlinear equation (4.2.1). In principle, these can be solved provided that all five unknown functions arc prescribed at 1 = O . Usually, as in the case of quasi-linear equation, the Cauchy data an sufficient to specify x,y,u, at 1 = 0 m terms of some parameter s on an initial curve C.It IS also necessary to determine initial values forp and q in tams of s at 1 = 0. Eliminating dr from (4.2.10) and (4.2.15) gives the Charpit cqu(lfions

I38

First-OrderNonfmor Equarronr md Their Applicoaonr

_&= _ dy = _ _ - du Fs F, pF, +qFq

+ = -A F,+qF. . (4'2'16)

F. + p F w

We formulate the Cauchy problem for (4.2.1) to determine the family of c w e s in the parametric form z=x(f.s), y=y(f,s), and u = u ( r , s ) , (4.2.17)

andthevaluesofpandqar ~ = p ( f , s ) and q=q(r,a). (4.2.18) where f is B running variable along a panicular curve and I is a parameter which specifics a member of the family. We seek a solution surface u = "(2,y) with the given initial data at t = 0 as

C : x(O,s)=x,(s), y(O.s)=y,(s), u(O,s)=t&), end determine the initial conditions for p(f,r) and q(f.5) at f = 0 as

(4.2.19)

P(O.~)=P&) and d o . 4 = q o ( s ) , (4.2.20) and a(.) an obtained from the following. A set of five quantities xo(s), yo(.). %(s). p,(s) andq,(s) cannot all be chosen independently. since we must observe the relation where p&)

-duo dr -&&I *, dr + l4y , ;dy,i ; --P ~ ( ~ ) ~ ~ ( ~ ) + q ~ ( s ) Y ~(4.2.21) (~). where po(s) and qo(s) me the initial directions which determine the normal to the solution surface and u o ( s )=u(xQ(s),yo(s)).The relation (42.21) IS called the sfdpcondition.The initial c w e xo(s). ya(s). u.(s) and the odenutions of the tangent planes pa(.) and q&) are referred to as the inifid m i p , since the initial conditions can be viewed ar the initial curve combined with the initial fangent planer attached. Since po(s) and q&) are the initial values of p(r,s) and q(f.8) at f = O on the curve C. these values must satisfy the original equation, that is, (4.2.22) F(%.ym.%.~o.qo)=O. Evidently, equations (4.2.21) and (4.2.22) determine the initial values pa(.) and q&). Thus the set of five equations (4.2.10) and (4.2.15) with the initial data (42.19)-(4.2.20) can be solved u) obtain solutions (4.2.17) and (4.2.18). It is necessary u) use the first two equations of the system (4.2.17) for finding the parameters f and s ~n terms of x and y . and the result is substituted in u = u ( r , s ) to obtain a solution surface u = u(x.y). The sufficient condition for expressing f and s in terms of x and y fromthe first two equations of the system (4.2.17) is that the Iacobian 3 is nonzero along the initial strip. that is. ,=Jo=x.

d(f.s)

1, Y~=I; Y8

F

F Y:[*'*

(4.2.23)

4.3 Complrrr h l g r o l s of Certain Spcciol Nonlinear Equorions

139

in which the fin1 two equations of the system (4.2.10) an used and where F, and F, in (4.2.23) arccomputedaltheinitial suip xo.yo,uo,po andq,. Thus, the above analysis shows that if the lacobian 3 # 0 along the initial suip, there cxlsts B unique solutim of the equarm" (4.2.1) Passjag lhrough the given initial curve (4.2.19). However, in general, the solution is no1 unique. but 11 is unique for a fixed determination of the roo&p and q of equations (4.2.21) end (4 2 22). However, if 3 = 0 along the initial strip, then the Initial-value problem has a solution I f it is a eharactcristic suip. Indeed. Ihe problem may have infinitely many soluuons. The SoIutions *(I,*), y(t,.), and u ( t . 3 ) of the problem for fired s give B sp- curve. and p(r,r) and q(r,s) determine a tangent plane with normal vcctor ( p , q. - I ) at each point of the space curve. The space curve together with its rangcmt plane IS called Ihc ehamderistiicsrrrp. Similarly, the Charpit equations are known as Ihe chommtistic sttip eqm'onr.

4.3

Complete Integrals of Certain Special Nonlinear Equations

Case (i): Nonlinear equations (4.2.1) involving only p and q. Such equations must k of the form F ( p , q ) = 0 . Thus. the Charpit equations (4.2.16) bccane

=du = P 64

d ^ , dy = (4.3.1) F, F, pF,+qFq 0 0 The solutions of the last two equations of (4.3.1) are p = eonstant = a , and q = constant 0" the characteristic EUIY~S. Thus, F ( n , q ) = O which leads to q=fM It follows from the fimt two equations of the system (4.3.1) that du = pdr+ qdy ,

whish gives the complete solution by integration as u = ax + f ( o ) y + b , where b is a constant of integration.

(4.3 2)

Case (10:Equalmr nor involving independent variables. Such equations an of the form F(u,p,q) = 0 . Thus, the Charpit equations are

& = d y = d u = d p = d q . Fn

F,

Pfn +qfq

-PF.

-4f.

(4.3.3)

(4.3.4)

4.3 Complete lniigmls of Cenorn S p m d Nonlinror Equorions

141

This represents the two-parameter family of planes. It can be verified by direct differentiation that (4.3.9) satisfies Clairaut’s equation. Finally. it can also be shown that the characteristic s h p s are d l swight lines. We illustrate the above cases by the following examples. Prornplr 4.3.1 Find the complete integral of Ihc nonlinear equation pl +qy - u = 0. (4.3.10)

The sociated Charpit equations am -dr = dr

*=

F,=2p,

&=F,=y.

d”=pF,+qF,=Zp’+qy.

-(F. + p F * ) = p. * = - ( F ,

The last equation gives q = constant solved forp as

+qF.) =O.

= a . Clearly,

p = (u-

the given equation can be

&.

Also, we have dr=dy= du -Pdr+qdy zp y zp1+qy-=’

and hence, du =pdx+qdy=(u-oy)7dr+odyYS. (k

du-ody -=&.

lntegraung this equation gives Z(u - ay)i = X + b.

where b IS a constant of integration. Thus, the complete i o t e p l is given by u(x,y)=L(x+b)’+ay.

(4.3.11)

Example 4.3.2 Solve the Cauehy problem p2q=1. u(x.O)=x. (4.3.12) This equation is of Ihe type in erne (i) with F = p2q - 1. Hence, dp = 0 and dq = 0 which give p = constant = L? and q = constant = e . Thus the given equation yields e =

2

Consequently. the complete integral is ’

u=ar+-$y+b

Using the iniual data. we obmn x=nr+b

(4.3.13)

142

FirrI-Order Nonlinear Equlionr and Their Appliervionr

which gives

L?

= I and b = 0. Thus, the solution surface is u(x.y)=x+y.

(4.3.14)

Example 4.3.3 Solve the Cauchy problem p* + q +" = 0, "(X,O) The parametric forms of the Charpit equations are

3

x.

dp,-p.

d"=Zp. &=I. dy=2p'+q, dr dr dr The parameme forms of the initial data are

dr

(4.3.15)

!Lq dr

x(s.O)=s. y(s.O)=O. and u(s.O)=s. Differentiating u(s.O)=s with respect to s gives p(s.O)=l. and then. substituting in the given equation s h o w that q(s,O)= -($+I). The equations forp and q can be integrated with these initlal conditions to obtain p=..p(-t). q=-(s+l)exp(-r). (4.3.16)

Substituting these valuer of p and q into the first Uva Charpit equations gives

*=

a'

da 2cxp(-1), &=I, 2crp(-21)-(s+l)exp(-r). dr dr Integrating these equations with the initial data gives x =(s+l)-Zexp(-r),

y =I,

u =(s+l)exp(-I).

(4.3.17)

Eliminating r and s gives the solution surface u ( 1 . y ) = ( x - l ) C +e-*y

(4.3.18)

Example 4.3.4 Find the solution of the initial-value problem p*x+qy = u ,

+,I)

=--s.

(4.3.19)

The parametric forms of the Charpit equations am -= da zpx. iLY, du=2p'x+qy, dp=p-p', !Lo. dr dr dr I f We differentiate "($,I)=-* with respect to s to obtain p(s,O)=-I. Substituting this in the given equation gives q(s,O) = -2s. Integrating the last equation of the Charpit system gives q(5.r) = -2s.

-

The equation for p can be integrated to obtlin A d 1 p ) ' = exp(r), where A is a constant of Integration. Using the initial data p(s.O)=-I, we find that A = -2 and hence. p ( " . r ) = L .

(&Z)

(4.3.20)

143

4.3 C o m p l ~ Inicgrolr i~ ofconain Special Nonlinear Equations

Substituting this solution for p in the fin1 equation of the Charpit set and integrating with the condition x(s.0) = s, we obtain x(s.1)

= s(e, -2y

(4.3.21)

The equation for y can be salved with y ( s , O ) = I and the solution is y(s,r)=enp(r), so that the equations of characteristics are x = s ( y - 2 ) ' . will intersect at (0.2). Finally, the equation for u can he expressed in terms of I and I as

They

du=2s(r*-c'). Integrating this equation with u ( s , O ) = - s gives the solution "(V)= %xp(l){exp(l)-2).

(4.3.22)

In terms of x and y, the solutmn takes the form (4.3.23)

.(..y)=Z.

Y-2

This shows that the solution u(x.y) is singular at y = 2. Eromple 4.3.5 Solve the mtial-value problem p* -391 =o Wlth "(X.0) = x a .

-"

(4.3.24)

The charactenme smp quatiom me

&

*=

= F, =2p.

F, =-6q.

"= dr

pF +qF = 2 ( p 2-3q')=

'

9 =-(F. +pa)= p, 9 =-(F, +qF.)=q. dr

Z u , (4.3 25)

(4.3.26)

The initial dam in the parametric form 8re given by and p&) become

z(0.s) = x,(s) = s, y(0.s) = y&) = 0, "(0,s)= &) = s * , (4.3.27) and to($) me the solutions of equations (4.2.20) and (4.2.22) which p(O,s)=p,(s)=Zs

and p:-39:=s2,

(4.3.28)

and hence q0(s)=7 s . (4.3.29) Next, we use po = 2s and qa = --s to solve (4.3.26). It turns out that

p(r.s)=Zse' and q ( ~ . s ) = - x ' . (4.3.30) Substituting these results in (4.3.25) and solving the resulting equations with the initial data (4.3.27) gives ~ ( 1 , s =4s(e' )

-I)+$.

y(t,s)=6s(e' -1).

and

U(I.S)=S*L*'.

(4.3.31)

We use the first two results in (4.3.31) to finds and I in terms ofr and y ar

Fir$!-O&r Nodmoor Eguotionr ond Thew A p p l ~ a t i m s

144

s=1(3x-2y) and e x p ( r ) = i ( 2 x - y ) . (4.3.32) 2(3x-2y) Substituting these recsult~in the third equation of the System (4.3.31) gives the SOlYUO"

surface

.(x.y) =

(.-$

(4.3.33)

If we use the initial dam p&) = 2s and qo(s) = s and solve the problem in a similar way. we obtlin a different solution surface "(X,Y)

=

(.+$

(4.3.34)

Example 4.3.6 Solve the Clairaut equation (4.3.35) z = px+qy+ j ( p , q ) = px +qy+(i + p2 +q')i with the initial data x(0.s) = x&) = m o s s , y(0.s) = yo(.) = osins, z(0.x) = q ( r ) = 0. (4.3.36) The characteristic sttip quatiom Bn

9dl= x + f' P

;i;=y+',

f

L-'

f'

dt

(4.3.37)

d p = O and * = O . (4.3.38) f dr We have to find the initial data p(O,s)=p,(s) andq(O,s)=q,(s) which must satisfy the original equation and the svip condition, that is, op,(s)eoss+aq.(s)sins+(l+ pi + 4):; -np,(s)sins+oq,(s)coss=O,

=0 .

(4 3.39)

(=to).

(4.3.40)

Clearly, equation (4.3.40) is satisfied if p , ( s ) = ~ o ( s ) c o s sand q&)=a,(s)sins.

(4.3.41)

fitting these nsulu into (4.3.39) gives (a' - l)d(s) = 1 which leads to o.(s)=f(n'-i)p

= ~ n ~ t a n t ,(o>I).

(4.3.42)

We mke the negative sign for a&) to solve equations (4.3.37) and (4.3.38). In fact, the latter equation gives the solutions

145

4.4 Examples of Applicmions lo Anolyficol DyMmics

f = ( I + $ +q2)i =.(i - 1 r 5 ,

(4.3.44)

which is used to rewnle the first two equations in (4.3.37) as

The solution of the first equation in (4.3.45) with

~ ( 0 . 5= ) ~ W J Sis

given by

ad (4.3.48)

or,qutvalently, (4.3.49)

n u s , the equation of !he charaeterirtic snip is x 2 + y 2 =(a’ -

l)[z-F].

(4.3.50)

a’-1

Thts represents a cone. I n general, any ~olullonof the Clairaut equation represcnu a developable swfme, that is, a mlcd surfme which can be deformed into a plane without stretching or tearing.

4.4

Examples of Applications to Analytical Dynamics

Example 4.4.1 (The Homrlron-Jocobi Equorion). In analytical dynamics. the Hamilton principalfunction S(q,,r) characterizes the dvnamid system and satisfies the celebratedHomilron-lacobi equrion

where q, are the generalized coordinates, r is the time variable, and H is the Hamiltmian of a dynamic4 system. Thin equation is a first-order, nonlinear. partial differential equation with n + l independent variables. and it plays a fundamental role ~n the development of analytical dynamics. Using the

146

Firsc0rd.r Nonlinear Equorionr m d TheirApplications

an

JS

andp=S,, we rewrite

conventional notations S(q,,r)=u(q,,r), - = - = p , , Jq, Jq, in the form equation (4.4.1)

(4.4.2)

F(q,,r.p,.p) = p + H ( q , , f , p , ) = 0,

where the dependent vanable "(4, .I) does not occw directly in the equation. In terms of the parameler T, the Charpit equations assaeiated with (4.4.2)are given by

*

dr

r-= :

=dH ap, '

(4.4.3)

Fn=I.

(4.4.4)

"

"

x= du

X P , F , + PF, = Z P ,

= -(Fq, + p,Fw)= -%,

!& dr

z+ dH

2

= -(F,

P.

(4.4.5) (4.4.6)

24,

+p

~ u= )

-%

27'

(4.4.7)

The second quation (4.4.4)with the given initial condition I(T) = 0 at r = 0 givw I= r. Thus, the independent time variable Ican be used as the parameter ofthc eharactenstics. Thus.the above system of equations reducer to the form dr

(4.4.8)

Jp, '

-dp, --% dr

(4.4.9)

Jq, '

du " ; i ;= X P ,

g

+P.

dp=-dH dr

JI

(4.4.10) (4.4.11)

'

The first two equations (4.4.8)-(4.4.9)constitute

of 2n. coupled, firstorder, ordinary differential equations, precisely the Homilron cnwnicoi cquorionr of morion which reflect symmetry except for a negative sign. The ~01utionsof these equations represent the characteristics of the Hamilton-Jacobi equation. They also represent the generalized coordinates and generalized momenta of a dynamical system whom Hamiltmian is H ( q , , p , . f ) . The last equation (4.4.11) gives p = -H which is used to rewrite (4.4.10)in the form B set

du = x p , 2 - H .

(4.4.12)

In principle. the Hamilton system of 2n equations can be solved for q , ( t ) and p,(l). If we Substitute these solutions in right-hand side of (4.4.10) gives a known function of 1. and then, u(q,,t) can be found by integration. A similar argument can be used to find p by integrating (4.4.11). Thus. the upshot of this analysis is that characteristics of the Hamilton-Jacobt equation are the solutions of the Hamilton equations. By using Hamilton's equations (4.4.8)-(4.4.9). the equations of motion for any canonical function F(q,,p, . I ) can he expressed in the form

={F,H)+~.

(4.4.13)

where (F, H ) is called thc Poisson braekt of two functions F and H. If the canonical function F does not explicitly depend on time 1, then, F, = 0 and hence (4.4.13) h o m e r dF (4.4.14) ;i;={~.~}.

In addition, if { F ,H}= 0 , then. F is a eonstant of motion. In fact, equation (4.4.14) really includes the Hmllton equations, which can easily be verified by Setting F=q, andF=p,. Further. if F = H . (4.4.14) implies that H is E0"Stant

More generally, the Poisson bracket of any two functions F and G is defined by

Obvmudy. the Poisson bracket is antisymmehle. It also readily follows fmm the definition of the Poisson bracket that {%P,

I = 4,.

{r..e) = 0 = {P.,P,).

(4.4.16) (4.4.17ab)

148

where

FirscOrdcr Nonlinear Equolronr onl T k i r Appbcarionr

a,, is the Kronecker delta notation. Results (4.4.16)-(4.4.17ab) represent

the fundamental Poisson brskets for the canonically conjugate variables q, and

P.. It is impatant to point out that the sum in B Poisson bracket is taken over all the independem variables. Panid derivatives of only two dependent vonoble occur in a given Poisson bracket, and these two variables an used in the notation for mat bracket. To heat time t similar to (he other independent variables. we replacs r by q.+, and write (4.4.2) m the form (4.4.18) G(q..q+ P , . P ~ + = ~ )P,, + f k t a ) .

This sllows us to write the Hamilton eauations in the f m

L”, dt

ap,

dt

(4.4.19ab)

Jq, ’

where G = G ( q , , p , )and i = 1 , 2,...,n + l . Evidently, if

G is given, (4.4.19ab)

represent a system of (Zn+ Z),first-order, ordinary differential equations for the unknown functions q, andp,. The main problem is to find a suitable transformation under which the symmevic form of the Hamilton equations is preserved. Such a vandormation is called a cononid tmnsfomtion defined as

follows. A m s f m a t i o n of the form q , = q , ( % . f i , , ) .P , = P , ( & . f i r ) . where i=1,2, ....n + l . is called cononicol if there exists a function

(4.4.20ab)

(?($,,A)

such that equations (4.4.19ab) transform into (4.4.21ab)

There are other mathematical expressions invariant under canonical vansformationr of the form (4.4.20ab). One such set is the integral invariants of Po in cd . A theorem of Poincar6 states that the integral (4.4.22) is invariant under the canonical transformation, where S indicates that the integral is 10 be evaluated over any arbitrary two-dimensional surface in the phase space formed by coordinates qt , qr.,,,,q n ,and p , . p 2,._., p n . It is noted that the position of a point on m y two-dimensional surface is completely specified by not more than two parameters. We assume that u end v are two such parameters appropriate t o the surface S , so that q, = q , ( u , v )andp, = p , ( u , v ) on this surface. It 1s well known that the elementary dqdp, vansforms to the element of srca dudu according to the IPlatiO”

4.4 Eromples of Applications 10 Anolyticol Dynmies

149

when the lacobian determinant IS given by (4.4.24)

The integral I , has the s m e value for all canonical coordinates. that is,

which can also k expressed as

Since the region of integration IS arbitrary, the two integrals in (4.4.26) ma equal. provided that the two inlegrands an equal. that is, (4.4.27)

This means that the sum of the lacobian determinants is invariant. More explicitly. (4.4.27) can be written BE

Each side of Uus equation ir in the form of the so-called Lagrange brocket of two independent variables u and (N, V ) =

u defined by

x( .**.*). ,

Ju

Ju - Ju Ju

(4.4.29)

More generally, if Fk and C,, k = 1,2. ...,n are B set of functions of a number of independent variables, and if u and u arc any two of these variables, then, the L a g m g e bracket is defined by

(u,u)= ~(?%.%-?%.?%), JU au dU

(4.4.30)

Obviously, the Lagrange bracket is antisymmetric. We now consider the Lagrange bracket of the generalized crordinatu (4.4.31)

150

and Their Applicotim

FirsCOrdrr Nonlinear &w&m

Since the q's andp's an independent emrdinates, hence,

?& = 0 and 4,

*

= 0.

Jq,

Similarly. we can show that (p..p,)=o.

(4.4.32)

) = + . h Jq - d y .JP, ? & ] JP, = S , ,Jqz .

(4.4.33)

Finally, it can be shown that ( 4'. P

The second term in the above sum vanisher because 4 ' s and p's are independent. but thc tint term is not zero because

&=a,,,

?&=a

and

Jq.

Jp,

(4.4.34ab)

"'

and hen-. (q..P,)=ZS,,S*,

=%.

'Iklhus, (4.4.31)-(4.4.33) represent the Lagrange brackets of canonical variables

and BIC oftennfcmd to BS ulefundnmeml L o g m g e bmckfs. In contrast to Poirson brackets. the sum in a Lagrange bracket is taken over all the dependent variables. Panial derivatives with respect to fwo of the independent variables occur m a given Lagrange bracket, and these two independent variables an utilized in the noration for lhat bracka The following observations arc in order. First, the conditions that the transformatton be canonical can be expressed either in terms of Poisson brackets or m terms of Lagrange brackets. Second, there exists a mathematical relation between the Poisson and Lagrange brackets which can bc found in Goldstein (1965). Third, both Poisson and Lagrange brackets are found useful in the transition from classical mechanics to quantum mechanics. We consider a conservative dynamical system, and write S = u Et where u is independent of lime I and E is an arbitrary constant, we see that S is a solution of the Hamilton-Jacohi equation (4.4.1) if u satisfies the timeindependent Hamilton-Jacohiequation 8" the form

-

( , ;; 1

H q

=E.

(4.4.35)

We now derive the Hamilton-Jacob, equation for a single panicle of mass rn moving under the influence of a co~scly~tive force field. The Hamiltmian of this system is given by

H = T + V = -!-(p; 2rn

+ p: + p:)+

V(x.y.z).

where V ( 1 , y . z ) ISthe potential energy ofthe particle and

(4.4.36)

4.5 Applicorions

10

Nonlrnror Oprrcr

I51 (4.4.37)

firrr-Order Nonlinear Eguorionr and Their Appliarims

152

W h e n n is B constant, we can apply the ease (i) ~n Section 4.3 to find the general solution of (4.5.2). It follows from (4.3.2) that the complete solution of (4.5.2) can be written as u(x,y) = o m + n m y +b. (4.5.4) where a and n are constants. Or. equivalently, (4.5.5) u ( x , y) = xneos8+ ynsin 8 + b, when cose=n. With n(x.y)= constant = n . we solve (4.5.2) as the initial-value problem for x(s, T), y(s. r), u(s. 2). p(& T). and q(s. 2) with the Cauchy data X(0.T)

= Xo(T).

Y(0, T)= Yo(?,

P(0. r) = P O ( % )and

"(0.?)=%(2).

do. T)=qo(r).

(4.5.6) (4.5.7)

when s and r are parameten. According to the general theory of first-order equations, the Charpit equations associated with (4.5.2) are

*=2q, ds

&=Zp, ds

k = 2 ( p ' + q 2 ) . *=O. ds

ds

*=O. dr

(4.5.8)

Evidently, the last two equations m (4.5.8) give p ( s . T) and q(s. 7 ) which are constant on the chmctensties. that IS, p(s, T) = constant = pJr) and q(s.r)= c o n ~ h n t =q0(7). (4.5.9) Since p and q are constant for fired 'T. the first two equations in (4.5.8) imply that characteristics (light rays) are snaight lines which can k obtained by direct integration in the form x ( s , r)= 2por + ~ ~ ( 7 ) . y(s, 7)= 2qos+ yo(r). By ~ntcgration.the third equation in (4.5.8) gives u(s, C ) = 2(pi

+ q i ) s + uo(T) = Zn's +%(7).

(4.5.10)

(4.5.1 1)

Along the charactenstics (or light rays). equation (4.5.1 1) has the same form

as (4.5.3) with s playing the role of a sealed-time variable and ug(r) is independent of time like variable S. Eliminating po. qo. and I from (4.5.10) and (4.5.1 I ) gives the solution

("-"J ="'{(.-Io)'+(y-yJ}.

(4.5.12)

mlha represents a cone with its vertex at ( ~ , , y , , ~ ~Or. ) . equivalently. takmg h e posiuve square IWt. .(~,y)=uo+~{(x-x~)2+(y-y~)'~.

(4.5.13)

This is a three-parameter family of characteristic strips of (4.5.2) and hence. represents the solution of the eikonal equation (4.5.2). However,the SoIutlon

4.5 Appbcotrons

10 Nonlinear

153

Optics

(4.5.13) satisfies (4.5.2) cvcvywhcrc except at the initial point (xo.yo), where u. and u, are singular Thus, (4.5.13) is called the singular solution which determines a cone with its vertex at (x,.yo.uo), identical with the Monge cone through that point. Physically, the level curves u ( x , y ) = constant characterize the wovefrontr, where V u = ( u , . u , ) = ( p , q ) represents the normal vector to the wave fmnts. It follows from (4.5.8) that 2 ( p , q ) represents the tangent vector to the characteristics or light rays. Thus, the light rays are normoi to the wave fronts. This IS m e even for n variable index of rcfradon.

NOW,we discuss several special rolutions of cikmal equation (4.5.2) with constant n

and the following three sets of initial data:

(i) x o ( r ) = O , y , ( r ) = O , andu,(r)=O

In this case. solutions (4.5.10)reduce to the farm x =2p,s, y = Zq,s, and u = 2n’s. Eliminating s from the fin1 two results gives x * + y * =cl(p;+q;)s* =4n’sZ.

(4.5.14) (4.5.15)

Therefore, the S O I U ~ ~ OBE~ +,y)

= “(2 +y 2 ) i .

(4.5.16)

The level curves representing the wave fmnts are o ( x . y ) = con~tantwhich represents a family of on centric circles. The Strip condition (4.2.21) is automatically satisfied. the given equation leads to pi +qi =n2. and hence, po and qo are constant. Thus. the characteristics representing the rays are given by a family of straight lines q o x - p o y = O which parser through the origin. This example also confirms the fact that rays are normal to the wave fmm. (ii) x o ( r ) = r , y o ( ~ ) = r% , (r)=m It follows from equation (4.5.2) that p a ( r ) = ~ C O 0S and q , ( r ) = “sin 0 , for some red values of 0. In order to satisfy the strip condition (4.2.21) in which I is replaced by r. the fallowing nsult n(cos0+rin 0) = (I (4.5.17) must hold for some valuer of 9 . Consequenlly. s o l ~ t i ~(4.5.10) n ~ and (4.5.1I) become

FirrcOrdcr Nonlineor Equations ond T h i r Applicofions

154

x(s.r)=2nscosO+?. "(3.

y(r,r)=2msin8+

r,

r )= 2 2 s +nr(eoso+sin 8 ) .

(4.5.18) (4.5.19)

Eliminating sand r from (4.5.19) Eiws

s-

(x-Y) Zn(cos8-sine)

wd r - ( ~ ~ ~ e - . - w (4.5.20) ( m e - s i n e)

Substifuung the values for s and

r in (4.5.19) leads to the solution

u(x,y)=n(xcwE+ysin8). (4.5.21) In particular, when LI = n , then. EOS 8 + s m 8 = I , which implies that either 8 = 0 or 8 = ? . Therefore. soIulion (4.521) leads to two continuously

diffmntiable solutionsgiven by u(x,y)=nX or

The first

"y.

two initial conditions in (4.5.23)

y* = 2 x + 1 which is

B

(4.5.22)

determine the initial curve

parabola with its veltex at ( - + . 0 ) and Its focus a1

(0.0). To determine the initial data p0(?) andq,(r), we need to satisfy the given equation (4.5.2) and the s0ip condition as

P:(T)+q:(?) = n 2 ,

P~(r)*~(?)+e(r)Y~(r)=u~(T).

(4.5.24) (4.5.25)

These equationsgive two sets of solutions: (a) p a = " , q o = O :

(b)

p,=ncosr, q,=nsinr.

For case (a). the EOIutiOn can be obtained from (4.5.10) and (4.5.11) in the form x(s,7)=2ns+x0(7),

y(s.r)= yo(r). u ( s . 3 =2n2s+u,(r).

(4.5.26)

From the first and the third results with the dam (4.5.23). we obtain the SOl"ti.3"

" ( 1 . Y ) = nr.

(4.5.27)

The level curves u = constant give x = consmt which is a family of smaight lines parallel to the y-axis as shown in F i r 4.1.

Figore 4.1

The level

EUWU

(4.5.27)

(wave fronts) for aolutlons

and (4.5.29).

For case (b), thc solutionsare found from (4.5.10) and (4.5.1I) as x = Znscor r+ x.(r), y = fnrsin T+ yo(r), u(s, r) = ?n'r+u,(r). After some algebraic simplification,we obtain the solution as

+y')i-I].

.(x,y) = "[(2

(4.5.28)

Thus, the wave fmnu repmnted by the level e w e s u(x, y) = constant ~n x* + y * =a*. They represent concentric circles as shown in Figure 4. I .

(4.5.29)

Emrnpk 4.5.2 (Eikonol Equation with n Poim Source).

We consider the initial value problem for the cikonal equation with a point source which moves along a smight line in the ( x , y)-plane wlth constant speed v . Without loss of generality, we mume that motion is Jet up along the x-axis and passes through the origin, when the eikonal u = 0 . We solve the equauon pa +q'

with the Cauchy data x ( 0 , r ) = x O ( 7 ) = r, y(0.T)=yo(7)=

=L 2' 0, u(0.r) = uO(7)=?.

(4.5.30)

(4.5.31)

156

FirrcOrder Nodineor Eqwtiom and Their Applications

where c is a constant.The conditions (4.5.24)-(4.5.25) give

p i ( r ) + q i ( r ) = + and

~ ~ ( 7 ) = ~ .

(4.5.32ah)

and hence, q 0 ( 7 ) = f l ( M ’-I);,

when M =

(4.5.33)

is the Mach number.

The condition (4.5.32a) is satisfied if we set po(r)=* 9 * ( T ) = si“8. where the new parameter B is defined by B = c d (

end

-?-)

which

has rcal solutions for B in (0.2%) provided the Mach number M > I . Thus. the solutions of the present problem can be obtained fmm (4.5.10) and (4.5.1 I) as

x ( $ , r ) = S +r,

y(s,r)

=%sin S=?(M’ e

cM

-1);.

(4.5.34)

u ( x , r ) = 2 ” + ? , p(s. r ) = Mc, q(s, r ) = L ( M ’ -1);. (4.5.35) 2 CM Eliminating sand 7 from the first two r e ~ u l tin~ (4.5.34) and substituting them in the first equation in (4.5.35) gives the solution

”

nK other choice of

B giver

B

negative qo. and hence. the solution is given

by

“(x,y)=l[x-y(M’ (4.5.37) 4.6

Exercises

I . Solve the following first-order p m a l differential equations (a) p q = u with u ( O , y ) = y ’ , (b) p + q = p q with u ( x , O ) = m w h e n a isaconstant. (c) r p ’ + y 9 = u with u(s,l)=-Zs wheresisaparameter, (dJ p q = q with u ( x , y ) = - y at x = O . 2. Obtain the complete integral of the following equations (a) p q - u ’ = O , (h) p * + q * = u ,

157

(4 p ' + q + r ' = O , (el pq = w

(d) p ' + q y - u = O ,

(9) x ' p ' + y 2 q * = 4 ,

(h) y p * ( l + x ' ) - q x ' = O ,

(0

x'p2+y1q!=u1.

(i) zq(u- p x - q y ) = 1-4'. cj) (p' + q ' ) x - pu = 0. 3. Solve the following Cauchy problems (a) pl - q u = 0. " ( x , y ) = I 0" y = I - r , (b)

pz+qy-p~q-u=O.

(C)

Zp'x+qy-u=O,

(dl z p q - u = o ,

4.

u(x,y)=x+l

ony=z,

"(&I)=-L,

"(o,y)=+.

(e)

pq=L u ( x . O ) = x ,

(0

pq=u. u ( x . y ) = I o n y = - x .

Find the solution of the equation in the parametric form xp+ qy - $4 --u = 0 ,

with the initial data x(r,s)=s, y ( t . s ) = z ,

u(r.s)=a+l,

at

r=O.

5 . Show that the complete integrals of the equation xpq+yq'

6

=1

are@) (u+b)' = 4 ( a r + y ) . and (b) c x ( u + d ) = 2 + e * y , where 4b.c and d are constants. Show that only mtegral surfacer of the nonlinearequation Zq(u - x p ) - Z y ( q + x ) = 0 ,

which are developable surface& are M ~ C S (u+ar)'=Zy(x+b).

7

Show that the complete integrals of the equation zxu

+ 42 = .(.p + yq)

and (b) x ( y + a)'= 4(u -&). where 4b.c and d are constants. 8. Find the solution of the equation m the parametric form p q = w u(x,y)=-y when r = O . 9. Show that no solution exists for the Cauehy problem

are (a)

a+ a*x = my+ bx'.

pq=u,

u(x,y)=-l,

y=-x.

Firsf-Order Nonlinear Equationr and 7'herr Applications

158

10 Show that the solution of the Cauchy problem u: +u,

=o,

u(x,t)=

"(X,O)

=cx

+-a).

I I Find the solution of the following Cauchy problems. (a) x p + y q = q .

u(x.y)=L' ony=x

(a)

-xp+ yq = a , (O< x < y). u =Zx on y = 3x, whercois aconstant.

(C)

( p ' + q ' ) x = pu, u(0,s') = 2 r .

12 Obmn the solution of the equation

(

r(p' +9')+ -(I - z')+ = 0, p = r, and 9 = z?). so that the solution surface contains the line X(0.s)

= s. y ( 0 . s ) = sin e, +(s) = case. o < e < E .

I3 Assuming u = u ( x + .I) = "(5). show that the equation

reduces to the ordinary differentidequation

14 Show that the complete solution of the equation

z'(p' + q * + I ) = 1. (P =z,. q = ZJ. is the family of cylinders of unit radius whose axis lies in the (x.y)-plane, that is, z' + (xcos8 + ysin B+ b)' = 1, where a = tan 0 and b is a constant of internation.

Chapter 5

Conservation Laws and Shock Waves 5.1

Introduction

C o ~ ~ e ~ a tlaws i o n describe the COOS~N&" of some baris physical quantities of a system. and they arise in all branches of science and engineering. In this chapter we study first-order. quasi-linear. partial differential equations which become con~ervationlaws. We discuss the fundamental mle of characteristics in the study of quasi-linear equations and. then, solve the nonlinear. inrtaal-value problems with both eontinuou~and discontinuous initial data. Special attention is given to discontinuous (or weak) solutions, development of shock waves, and breaking phenomena. As we have observed, quai-linear equations arise from integral conservation laws which may be satisfied by functions which are not differentiable, not even continuour. simply bounded and measurable. These functions are called weak or generalized solutions, in contrast to classical solutions which are smooth (differentiable) functions. It is shown that the integral conservation law can be used to derive the so called jump condition which allows determining the speed of discontinuity or shoek waves. Finally, a formal definiuon of a shmk wave is given.

5.2

Conservation Laws

A conservation law states that the rate of change of the total amount of material contaned in a fixed domain of volume V IS equal to theflux of that material across the closed bounding surface S of the domain. If we denote the density of the material by p(x. I)and the flux vector by q(r,t), then. the

conse~ationlaw is given by

$JP "

whae dV

= -J(q.n)&.

(5.2.1)

S

thc volume element and dS is the surface element of the boundary surface S,II denotcs the outward unit normal vector to S as shown in Figure 5.1 and the right-hand side measures the ourward flux -hence. the minus sign is used.

IS

160

J Figure 5.1 Volome V of a closed domain bounded by P surface S wlth surface element dS and outward normal seetor I).

Applying the Gauss divergence thcorcm and taking

g inside the integral

sign, we obtain

jv($+div

q)dV = 0.

(5.2.2)

This result is true far any arbitrary volume V , and. if the intcgrand is C O ~ I ~ ~ U O U it E . must vanish everywhere in the domain. Thus. we obtain thc diflrrential fan" of the conservationlaw p, +divq = 0 . (5.2.3)

In the presence of B EOU~CC(or sink) function f(r,l . p ) , the total rate at which p is created (or desuoyd) in !he given domain is ~ f ( x . 1 . p dV. )

(5.2.4)

Inscning this term in (5.2.2), we obtain the integral form of the conservation law L j p d V + j d i v q d V = j f(x,r.p)dV. df" Y

(5.2.5)

An argument similar to that used &fore gives the differential form of the conservation law comsponding to (5.2.3) as p, + div q = f ( x . I. P).

(5.2.6)

5.2 Conrerwtion lmvr

161

where the flux t a m , diu q usually arises fmm the vanspan of p in the domain, and the wuxc term f is often called a growth term in biological problems or a reocrion term in chemistry. The on-dimensional version of the conservation law is (5.2.7) P, +q. = fb.1. P ) . From a mathematical or an empirical point of view, 11 is reasonable to assume a functional relation between q and p as 4= Q(P).

Thus. equations (5.2.7) and (5.2.8) form Substituting (5.2.8) m (5.2.7) gives P,+.(P)P,

(5.2.8) P

closed system far pmdq.

=f(x*f,P).

(5.2.9)

where , ( p ) = B .

dp

Equation (5.2.9) is universally considered the most fundamental. tint-order, quasi-lincar inhomogeneous wavc equation. In particular, when f s 0. (5.2.9) r e d m r to thc so called kinemaric w v c cqunrion

=a.

P,+.(P)P.

(5.2.10)

This equation often arises in nonlinear wave phenomena when the effects of dissipation, such as viscosity and diffusion. are neglected. We next investigate the development of shsxks from the inrrml-volucproblem for u ( x , r ) ",+c(")",=o,

"(&a)=

r>o,

--<*<-,

f(Z).

--< x <-,

(52.11)

(5.2.12)

where c(u)md f(x)arc C 1 ( R ) functions of their argumenu, that is, they are smwth functions. The characteristic equalions associated with (5.2.1 1) are _dr= _ & = _ , du I e(.) 0 These equations give

& = O and g = c ( u ) . (5.2.13 ab) dr Clearly, the solution of (5.2 13 b) represents characteristics of equation (5.2.1 1). Along these characteristics

& = u, dr

dr

+u, = +)"&

+",

=0

This means lhal u is constant on the characteristics which propagate with speed c ( u ) . 'Ike dependence of eon u produces a gradual nonlinear distortion of the wave profile as it propagates in the medium. It also follows that c ( u ) is

162

Cauervolbn Laws

Md S k k Wovcs

constant on the characteristics. and therefore, the characteristics must be stmight lines in the (x. ~)-plancwith eonstmt slope I / c ( u ) . ~4uationsof these lines are given by x - I c(u)=constant = A . (5.2.14) whcrc A isaconstanlthatis, x = x ( r ) = r c ( u ) + A . If any one of thcsc characteristics inlersecu the x-axis (1 = 0 ) at x ( 0 ) = 4 , then, by the inioal condition, ~ ( 4 . 0 = ) f(5). Thus. the quation of a typical chsrscteristicline (see Figure 5.2)joining two poinls (5.0)and (2.1) is

.={+I

~(t),

(5.2.15)

F(S)=c(f(f;)).

Figure 5.2

Characteristic lines of differcol slopes.

Since u ( x , ~ is ) eon~tanfan thc charaeterirtiss,it follows from(5.2.15) and the initial condition that u ( x , r ) = u ( 5 + 1 F(4). I ) =.(& O)=f(f;). Thus. I f a solution of the initial value problem exirls for I> 0 . then. the solution ean be written in thc pmmmie form

(OZZS)

164

Cmservarion lows and S h o d Wows u ( x , I) = f ( x -cr).

(5.2.21)

Thts solution represents a wave moving with constant velocity e in the positive direction of the x-aris wirburany change of shape. In this case. the charaetensUes x = 5 + cr form a family of parallel straight lines in the ( x , r)-plane as shown in Figure 5.3. Physical Sign@ancc of Solution (5.2.16obJ. For the general nonlinear problem, the dependence of the wave speedc OD u produces gradual nonlinear distortion of the wave as it pmpsgstcS in the medium. This means that some p m s of the wave travel faster than others. When c'(u) > 0, c ( u ) is an increasing function of u . In this e m , higher valuer of u propagate faster than lower on-. On the other hand, when e'(u) < 0. c(u) is B decreasing function of u and higher values of u travel slower than the lower ones. This means that the wave profile progressively distorts itself, leading to a multiple-valued solution. and hence. it breaks. In thc linear ease. c is constmt, there is no such distortion of the wave. and hence, it pmpagates wirhout any change of shape. Thus, there is a striking difference between thc linear and

nonlinear solutions.

Figure 5.3

Parallel characteristic Iher.

At any compressive pan of the wave, where the wave speed IS a decreasing function of x , BS shown in Figure 5.4. The wave profile distom progressively to produce B triple-valued solution for " ( x , I) and hence. it ultimately breaks. It follows from T h m m 5.2.1 that the solution of the nonlinear initial-value problem exists provided that 1 + r F'(<)# 0. This condition is always satisfied for a sufficiently small time I. It also follows from (5.2.18 ab) that both u, and u, tcnd to infinity 89 Icr F'(<)+O. This means that the solution

i k Ax*x

5.2 Conservorim Lows

165

,= h

0

Figure 5.4

Distortion of wave proflle with irnem~slngtime 1,

* 1, >o.

develops a disconrinuity (singularity) when I+[ F’(C)=O. Thus, on any characteristic for which F’(C) < 0, discontinuity meun at time I given by I=--

1

(5.2.22)

F ‘( 5) ’

whichispositivcbecausc F ’ ( < ) = c ‘ ( f ) f ’ ( C ) c O .Ifweassume c ’ ( f ) > O , this inequality implies that f’(e)cO. Hence. the solution (5.2.16 ab) ceases to exist for all time if the initial data is such that f’(<)c 0 for some value of 5 . Suppose that I= c is the time when the solution fin1 develops a discontinuity (singularity) for some value of C. Then, (5.2.23)

Thus, the shape of the initial curve for u ( x . ~ )changes continuously with increasing values of I , and the solution bceomcs multiple-valued for t a r . Therefore, the solution breaks down when P ” ( t ) < O for some 5 , and such breaking is a strikingly nonlinear phenomenon. In the linear theory, such breaking will never %CUT. More precisely, the development of a discontinuity in the solution for t 2 r can also be seen in the ( x , l)-pIane. If f’({)c0, we can find two values of {=5,,5,(5, c&) on the initial line ( I = O ) , such !hat the characteristics through them have different slopcs I/c(u,)and l/c(u2), where u, =A<,). u2 = f(C,), and c ( 3 )> c ( u , ) ?hese two characteristics will intersect at a paint in !he ( x . r)-plane for some f > 0. Since the characteristics eany constant values of u , the solution ceases to be single-valued at the point of intersection. As Figure 5.4 shows. the solution “(x, t) progressively distom ilself, and, at any instant of time. there exists an interval on the x-ax~swhere n becomes triple

(SZZS)

167

Figure 5.5

Characteristics of (5.2.24) nlth the Initial condition for a = ] .

solution becomes double-valued and the slope in the ( x , t ) plane becomes infinite. From this paint onward, the solution is a discontinuous function of psilion and cornponds to the onset of a s h n k wave. Moreover. it follows that all characteristics. originating from (x, 0) where x > - I . intersect the charactcrirticr startingfrom thc painfs x, where x 2 I, at some point or other In pmicular. the two charaetcriaics initially at x = 0 and x = 1 intersect at the point ( x . f ) = ( l , I). The solution would k doublevalued at (I, 1). Figure 5.6 reprr~cnfsthe propagation of the initial parabolic pulse with n = 1.

t"

FLgure 5.6

Propagation of a pnmbdie pulse for

(I

=1

As t increases, the initial pulse distorts progressively. This progressive change in the initial wave palm 1s the result of the nonlinear tern in the quation. In the linear case, (u, + e us = 0, c = conrtant) with the same initial

168

Cons.~otmnLnws m d

Shock Wows

data (5.2.25) for L? = I , the initial parabolic pulse propagates with constant velwity e in the positive direction of the x-axis without change ofshop.

Exornple 5.2.2 Salve the initial value problem u,+uu,=O. x s R , r > O . xsR.

u(x.O)=x'. According to (5.2.16 ab), the soluuon is

u ( x , 1)=

5 = .-15'

/(<) = 5 2 ,

Hence.the second quatian becomes

+ 5 -1 = 0. 5 , we obtain

15'

Solving this quadratic quntion in

*

<=&[-I

far r i0

(5.2.27)

Wus. the solution IS "(X,

t)=? =($)[-1*mZ,,

=

21'

(I +2xt

f m Z ] , I f 0.

(5.2.28)

This solution must satisfy the initial data

2 =+, O)=

lim u ( x , t). ,+o

With the positive sign before the radical in the sdution (5.2.28). the above limit does not exist With the negative sign before the radical. the limit exists by using the L' Hospid rule twice with fixed X . Thus, the solution is "(x, r ) = ~ [ l + 2 x t - ~ t] #O. ,

2r'

(5.2.29)

This represents a solution only in the region 1+4xt 20 which is the region ktwccn two branches of the hyperbela

XI=

-1. in the (x, t)-plane.

ErampIe 5.2.3 Solve the initial value problem u,+uu,=l, X E R , f > O , w h a t s is B parameter. me characlerisuc equalionr are dt= _dr= _du _ = I

The initial data at T = 0 are

U

I

(5.2.30) (5.2.31)

u(s,Zs)=s,

dr.

(5.2.32)

5.2 Conservation Lnws

169

x(O,r)=s. t(O,s)=Zs.

u(O.s)=a

(5.2.33)

Thus. the solutions of this system (5.2.32)-(5.2.33)are x( r, s) =

2, sc+s (5.2.34)

(. Eliminating plane as

7

7 , s )= 7 + s

from the first two results gives the characteristics in the

(x.1)-

(I - s)' = 2(" - s ) + Z . (5.2.35) where s is a parameter. This represents a family of parabolas for s = 0.f 1. f 2.. ... We leave the eon~rm~uon of the diagram for parabolsr; sr; an exercise for the nadcr. We fin1 use (5.2.35) to find s and. then. use the second result of (5.2.34) to find 7 and I in terms of x and I . This gives

(5.2.36) Substituting there results into the third quation in (5.2.34) gives the integral surf- of the quation (5.2.30) sr; (ZX

+.I)=-

- Z I + I*)

2(t-1)

(5.2.37)

Thus, the initial-value problem has the solution (5.2.37) everywhere in the (x. !)-plane except the ssaight line f = I , Ercnnpk 5.2.4 (Initial-due problem with discontinuous initial data). We consider the initial value problem for equation (5.2.11) with disconunuow initial data at I = 0.

(5.2.38)

ad (5.2.39) and 4 are constants. T h e n are two cases: (i) u, > y and (ii) u, < y. Cae (i) u, > u1 with e, > c,. In this case, breaking will mar almost immediately. and this can be seen from Figure 5.7. The muluplovalued region stms right at the origin and is whar

Y,

170

Conservolion laws

Md S h a h Wave$

bounded by the characteristics x = q and x = c , ~ . This cornrponds to a centered compssion wave with overlapping characteristicsm the (x, t)plan~.

Figure 5.7

Compression wave with overlapping eh~melorialiea.

Ca~e(ii) y c uz with e, < c2.

In this case, the initial condition is expansive with c, c c 2 . A continuous solution can be found fmm (5.2.16 ab) in whch all values of F ( x )in [q.c l ] are laken on the characteristics thmugh the origin 5 =O. This cornsponds to a centered fan of charweristics x = cf when c, S e S c2 in the ( x . r)-plane so that the solution has the explicit fam (seeFigure 5.8) x

c=f '

Figom 5.8

c~c;
Ccnlemd expansive wave.

Thus. the complete solution is given by

(5.2.40)

5.2 Conservotion l a w ,

171

c,, xSe,r

(5.2.41) x

&-pie dara

z c,r

5.2.5 Solve the initial-value pmblcm with discontinuous initial u~+uu.=O, X E R , r Z O ;

(5.2.42) (5.2.43)

Aceording to the p m c b i c solution (5.2.16 ab), we obtain the solution (5.2.44)

The solution and the characteristicsare drawn in various regions. m shown in F i g u n 5 . 9 . h O S g S l . the solution is u(x.r)=l-r.Thirisnotdcfincdat I-r I=].

The characteristics are straight lines inkmeting the x-axis at (5.0). and u is am propagated along the characteristics. As shown in Figure 5.9. the characteristicsme drawn in constant on a given characteristic line. me values of u on r = 0

Figore 5.9 Characterlstlw and the gnpbleal Iypmentatleo of solulioa.

I72

Conservotion l a w s

ond Shock Waves

vmous regions in the (&I)-plane. The value of u on B characteristic line

originating from a point (4.0)

is

I - < , w h e r e O < ~ < l , and these

characteristics all intersect at (1. I). This means that u is multiple-valued at (I, I), and hence, the solution breaks down at this point. It also follows h m Figure 5.9 that characteristics originating from (4.0). with

<
intersect

those originatingfrom (4,O) with < > I . As u hasdifferent values on these charecfendies, it follows that it is not defined in the quadmt I L 1. I 2 1 . Figure 5.9 also suggests that the most likely weak solution is one which is discontinuous across some curve Originating fmm (1.1) with u = 1 to i s left and u = 0 to its right. In conservation fom. the given equauan em bc written as

",+('"Z')

r

=o.

(5.2.46)

where the flux is iu'. It also follows from this equation combined with shock condition that the slope of the shock path is 2. Thus the equation of the shock psthis I=Zx-l,asshowninFigurcS.lO

Figure 5.10

5.3

Weak o r generalized solution.

Discontinuous Solutions and Shock Waves

It should be pointed out that the nonlinear consswation equation p,+q.=o

(5.3.1)

has been solved under two basic assumptions: (i) there exists a functional relation q = Q ( p ) , and (ii) p and q are continuously differentiable. When breaking occurs, questions arise about the validity of thesc assumptions. To examine the development of discontinuities or shocks. we still assume that

173

5.3 Discontinuous Solutions and Shock W o w s

q = Q ( p ) but allow jump discontinuity at x = s(r). We also assume xI and 1%

suchthat x , < s ( f ) < ~ ~ a n d * = U ( r ) . Without any source (or sink), we assume that the integral form of rhe conseworion equation (5.2.5) holds for the one-dimensional case , and has the form (5.3.2)

d~pd.=q(x,.r)-q(x,,r). dl i

or ~ [ ~ p & + ~ p & ] + q ( ~ ~ . l ) - q ( ~ , . l ) = ~(5.3.3) .

This implies that j p , &+ ip(s-, I ) +

2.

*- i p(s*.r) ++,,

r ) - q ( x , . r) = 0 , (5.3.4)

where p ( s ~ . l ) , p ( s * , r )arc the values of p(x.1). as x + s

from below and

above, respectively. Since p , IS bounded in x, < x < s -

and in s* c x < x , , respectlvcly. integrals In (5.3.4) must vanish BE xI +s-andx, + s f . Consequently. in the limit. q(s-. r)-q(s*, I ) = u{p(r-,1 ) - p ( s * * 1 ) } .

(5.3.5)

In the conventional notation of shock dynamics, condition (5.3.5) em be written a3

[41=

w.

(5.3.6)

where [ q ] = ( q - - q * ) a n d [ p ] = ( p - - p * ) denote the j u m p in qandp. respectively.arrossthediowntinulty x = s ( l ) , BS showninFigure5.11.

-~ Figure 5.11

Diseontinaity at r = s ( f ) .

174

Comervation Lows a d Shmk Wows

Result (5.3.6) is usually called the jump (shock) condirion which gives a relation &ad (right) of and behind (lefl) the discontinuity and the speed of the discontinuity O(r) (reciprocal of the slope). Thc discontinuity in p that pmpagalesalongtheanvox = s ( r ) i s ~ o w n a s t h + ~ h o c k w n v ~ , a x=s(t) nd is simply called the shock. In gas dynamics. (5.3.6)is known as the RonkincHugmior condition for the speed of the rhoek wave. Thus, the basic problem can be written as

*+%=o dr & -O[p]+[q]=O

atpoints ofconunuity.

(5.3.7)

at points ofdaeonbnuity.

(5.3.1)

This leads to a nice CnreJpondcnce

+u*

$ 4 1

(5.3.9 eb)

between !he drfferential equation and the shoek condition. We can now generalize the concept of the solution for equation (5.3.1) to include discontinuous solutions. This kind of extension of a solution is not a purely mathematical exercise as the notion of a shock IS of great significance in applied ares, such as fluid dynamics, gas dynamics, and plasma physics. From a physical pomt of view, a shock represents M idealization of a thin rransirion region separating two regions when solutions of the basic flow equations can readily be detmnined. Now,it is possible to find discontinuous solutions of the equation (5.3.1). In any continuous part of the solution, equatfon (5.3.1)is still satafied, and the assumption, q = Q(p) EM be retained. Thus, q- = Q(p-)and q* = Q(p') hold on the two sides of a shoek, and the shock condition (5.3.6)can be written as

u(F -P*)= Q(p-)-Q(p+).

(5.3.10)

This leads to a problem of fitting shock discontinuities into the solution (5.2.16 ab) so that (5.3.10) is satisfied end multiple-valued solutions can be a"&

5.4

Weak or Generalized Solutions

We formally examine the possibility of generalizing the concept of a classical solution so s to include discontinuous and nondiffcrentiable funcuons. We now conrider the general pmcedure necessary to define what is usually called a weak sofurion of the nonlinear, initial-value problem p,+q.=o. xerf. 1 > 0 , (5.4.1) P(X,0)= f ( x ) .

I E R,

whcn q ( x . r ) is B continuously diffcrenhable function on R.

(5.4.2)

5.4 We& or Genrmltzed SDlurions

175

A classical solution of the penial differential equation (5.4.1)is a smooth

function p = p ( x , r ) which satisfies (5.4.1). W e assume that p(x,r) is B classical solution of (5.4.1).We consider B class of test functions @ =@(x,f) such that

4 E C and it has

a compact support in !he (z.f)-plme.

We e m

chwse an arbitrary rectangle D = ( ( 1 . 1 ) : n < x < b, 0 < 1 < T] where @ = 0 outside D, and on the boundary lines x = a , x = b and t = T,as shown in Figure

5.12.

t'

Figure 5.12

D o n a h D with eonpact support of

4.

Equation (5.4.1)is multiplied by $(x, 1) and intcmud over D to obtain

JJ(P, + d )Oh 'fl= 0.

(5.4.3)

Integrating both terms by pans gives

or &T

-J+(x.O)~(X)&-JJP@.

bT

dlh-JJq@, d f h = O .

.o

.O

Thus !he final form of UK above equation is

I

(p d + q $.)&df+J$(x, O ) f W h = O .

(5.4.4)

DUlXl

This holds for all test functions @(x,f) in the half plane. Result (5.4.4)does not involve my derivatives of p or q . Indeed. it nmans valid even if p and q or !heir daivatives me discontinuous. Thus, we have proved that, if p(x,r) is a classical solution of !he problem (5.4.l)-(5.4.2),then, (5.4.4)holds for all test functions @ with compact support

Conservation Laws and S h t Waves

176

in the half (x, t)plane. The functions p ( x , l ) , which satisfy (5.4.4)for all test functions $, are- called weak or g m e m f k d sofurionsof the problem. This leads to the following:

Definition 5.4.I (Weak Sofurion). A bounded measurable function p(x, I ) is called a weak solution of the initial-value problem (5.4.1)-(5.4.2) with bounded and mcasurablc initial data f(x), provided that (5.4.4)holds for all test functions $(x, t ) with canpact support in the half plane. This is a more general definition of a solution that does not q u i r e the smoothness property. To show that the concept of a weak solution is, indeed, a generalization of a classical solution, we pmve the following: Theorem 5.4.1 If (5.4.4) holds for all test functions $ with compact support for f > O and if u is smooth. then, p is a classical solution of the Initial-value problem (5.4.1)and (5.4.2). PlooJ Since $ has compact support for by p m to obtain

I

I

> 0 , then. we i n t e p t e (5.4.4)

+ 4.) 4 dx dr = 0 .

(5.4.5)

nu-

This is me for all test functions $, and hence, p,+q,=O f o r x ~ l 1, > 0 . (5.4.6) W e next multiply (5.4.6)by 4 and integrate the resulting equation by par13 over the region D to find

II

( p 0,+ q $ , ) d r d l + ~ $ ( x , O ) p ( x 0)dx , = 0.

(5.4.7)

n",a

Subtracting (5.4.7) from (5.4.4)giver J[P(X,O ) - f ( x ) ]

C(., 0)dx =o.

(5.4.8)

This is me for all test functions m(z.0). Since f(x) is continuous, (5.4.8) leads to p(x, 0 ) = f ( x ) . ?his shows that p(x. I) is the classical solution of the initial value problem. We next consider only those weak solutions p(x, I ) which are ~ntinuavsly differentiable in two p m D,and D2 of a domain D. but with a jump discontinuity (shock) along the dividing smooth C Y N ~ r between D,and Dl.as shown in Figure 5.13. Suppose that the equation of r is x=x(I). Since p(x, I ) has a jump discontinuity along this curve, x = $ ( I ) , p(x, r) is smooth

5 4 W e d or Gmncnrliud Solurions away from

171

r and limits p-(f) =p(s-.

+, f) from the left and

1) and p'(f) = p(s

the dght exist. The speed of the shock wave is i ( f ) which is the reeiproeal of the slope in Figure 5.13. For my test function @, result (5.4.4) becomes

0=

ll(P@,+ q@.)

+s@..) dr dr.

dt = I l ( P @ ,+ q h ) dr df + J l ( p @ ,

(5.4.9)

&

4

" Figure 5.13 Smoolh curve discontinuity.

Since p(x. I ) is smooth in

4d

r

across which p(r,f) has a jump

i n 4. for n = I , 2, the divergence hhcorcm

= I@(-pdr+qdf).

a.

Because @ = 0 on the boundary dD, these line integrals vanish everywhere, uccp~0"

r. nus ~@(-p*+qdf)=~Q(-p~*+q~df), 'I

I@(-pdr+qdf)=-rjO(-pfdr+qfdr). 4

Therefore, equation (5.4.9) gives

j@(-[P1 * + [ q l d f ) = 0.

(5.4.10)

This result is me for all test funeuons 4. and therefore,the inlegrand must vanish, that is. on each point on r,

I78

C 0 m c w l b ~Laws

Md S k k Waves

i [~1=[41.

(5.4.11)

This is the s m c jump condition (5 3.6). Thus. a weak ~alutionleads to the jump condition across a jump discontinuity. finmple 5.4.1 Solve the nonlinear Csuehy problem u,+uu.=O. X S R ,t>O,

(5.4.12)

with two different sets of discontinuou~initial data (i)

u(x.~)={:,

":.,

The conservationlaw assoeiated with equation (5.4.12) is Y,

+(

$).

(5.4.13)

= 0.

In case (I). c(n) = Y, c(u(x, 0)) is a decreasing function of x, and condition (5.2.11) is violated. Thus. there is no continuous solution of the initial-value problem. However, then exists a discontinuoussolution i v e n by

.};::

(5.4.14)

" ( =*t) ={; ;

which satisfies (5.4.12) for x f U 1 and (5.4.11) on the line x = U t . Hence. (5.4.14) is called a generalized or wrok solution. Thn ~olutionhas B jump discontinuity dong the line x = Ut, where U is the speed of the shock wave

The initial data, charseteristics,and the wcak solution are shown in Figure 5.14. In w e (it), e(u) = u. c(u(x. 0)) is M increasing function of

X.

and condition

(5.2.17) is setirfied. Hence, there is a continuous solution of (5.4.12) (except at x = I = 0) with imtial data (ii) given by 0. x < o "(X.l)=

t I 2, o < x < r X>t

(5.4.15)

5.4 Weak or Genrmliud Soluiions

Figore 5.14

Initid dat8, characteristics, and SOlotio".

179

P

direontinnous

A solution of the farm (5.4.15) is called an rxpursivc 01 refrocrive wave. Often this solution in the sect01 0 < x < t is referredto as a centered simple wwc. The boundary lines x = 0 and x = t of the centend simple wave are weak waves

which propagate along the characteristics. as shown in Figure 5.15. This represents a sclution of (5.4.12) for t Z O which can be verified by direct substitution

(3(X)

+ - - =-1+J=o. , 1' t' me initial data. characteristics. and the solution are shown in Pigun 5.15.

Figore 5.15

Initial data, characteristics, and the continoow solution.

On the other hand, a remarkable fact involvcd in this nonlinear Cauchy problem with discontinuous data (ii) is that them is also B discontinuous solution given by

(5.4.16)

Obviously, this represents a solution everywhere in the (x. I)-plane, except the line of discontinuity I= Zx. The jump condition gives the shock speed

The characteristics and disconunuow solution are drawn in Figure 5.16.

Figure 5.16

Cbsraetrristlrs and the discontinuous solution.

Thus. there are at least IWO solutions of equation (5.4.12) with the initial data (ii). and hence the solution IS nor unique. The main question is which one of these solutions represents a physically meaningful solutmn. Obviously. another mathematical criterion is q u i d for determining a unique weak solution. It seems that criterion for determining a physically meaningful weak solution is closely related to the search for a criterion for an acceptable discontinuous solution. It is also clear from the above analysis that. in general, a discontinuous solution arises whenever characteristics of a nonlinear equation intersect in the (1.I) -plane. It em be pmved that there is a unique weak discontinuous solution of the Cauchy pmblsm which satisfiesthe following inequality on the curve of discontinuity &) > (I> a(.*). (5.4.17) This is called the Lax cnrropy cn'ren'on. Mathematically, this criterion means that the wave speed just behind the shock i i greater that the wave s p e d j w t ahead of it. In other words. the wave behind the shock catches up to the wave

5 5 fi#rn,cr

181

ahead of it. This entropy criterion 1s a special case of the second law of thermodynamics that entropy increases LICIOSS (I d o c k . Geometrically, the criterion for an unique weak discontinuous solution can bc stated as follows: The C ~ ~ O C I W ~ S I ~ Corigimting S on either side of the discontinuify cuwe, when contimed in the direction of mcreming I , intersectthe cuwe of discontinuity Clearly, solution (5.4.14) satisfies the entropy inequality (5.14.17). whereas solution (5.4.16) does not satisfy (5.4.17). Therefore. the former is an acceptable discontinuous solution. but the latter is not. This leads to a formal definition of a shock wave. A discontinuity which satisfies both the jump condition (5.4.1 I) and the inequality (5.4.17) on i u curve of discontinuity is called a shock. The main problem is to investigate whether every nanlmear initial-value problem has B unique weak ~olnliondefined for all I 2 0 with only shock as diiconlinuty. The prmf of existence and uniquemrr of a we& sol~tionIS f a dy difficult and IS beyond the scope of this bwk, but is provided by L a x (1973).

Exercises

5.5

Find the solution of the initial value problems (a) u ,+u u ,= O ,x e R, I > O , u ( x . O ) = ~ i n x , x s R (b)

u,+uu,=O,

X S R, t > O , u ( x . O ) = x , x s R

Show &st quation u, +xu, =o gives an infinite n u m k of conservation laws

a u" + d ( l

=o,

""*I)

at ax " + I wheren=1.2,3. .. Hence,deduce rhc integral conservation law

where x , and x2 are two fixed p i n e . Show that the solulion of the initial value problem u,+uu,+ou=O.

when

0 and

XER. f > O ,

u(x,O)=bx. X E R .

b are constam, is abxev-4 (o+b)-bcrp(-of)

Find the solution of the initial-value problem

5 5 Exercrrrr

183

xso

11. Find the weak solution in t 2 0 of the initial-value pmbkm u,+uu.=O,

X E R , r>O.

where oO. 12. Show that the solution of the initial-value pmblem u,+u'u,=O,xsR,t>O,u(x.O)=x,

s

13. solve the equauon

x s R , t>O

u,+uu,=O,

for two cases: I,

*so

0.

x2a

Examine both cases in the limit, as (I + 0. 14. Solve the initial-value pmblem u,+uy=O, x s R , t>O,

xsR

5 5 Exercrrrr

183

xso

11. Find the weak solution in t 2 0 of the initial-value pmbkm X E R , r>O.

u,+uu.=O,

where oO. 12. Show that the solution of the initial-value pmblem u,+u'u,=O,xsR,t>O,u(x.O)=x,

s

13. solve the equauon

x s R , t>O

u,+uu,=O,

for two cases: I,

*so

0.

x2a

Examine both cases in the limit, as

(I

+ 0.

14. Solve the initial-value pmblem u,+uy=O, x s R , t>O,

xsR

Chapter 6

Kinematic Waves and Specific Real-World Nonlinear Problems 6.1

Introduction

This Chapter deals with the theory and applications of kinematic waves to several nal-world problems, which include wftic flow on highways, flmd waves in rivers, glacier flow. roll waves in an inclined channel. chromatographic models, and sediment VanspR in riven. The general ideas and essential features of these problems are of wide applicability. Other applications of conservation laws include various chromatographic models in chemistry and movement of pollutants in waterways. The propagation of traffic jams is almost similar to the shock waves that cause noise pollution near airports and spacepons. Kinematic wave phenomena also play an imponant role in traveling detonation and combustion fronts, the wetting water fronts observed in soils after rainfall, and clanking observed in shunting trains. All of these problems are es~entiallybased an the theory of kinematic WBVCE developed by Lighthill and Whithsm (1955). Many basic ideas and imponant features of hyperbolic waves and kinematic shock waves are found to originate from gas dynamics. So, specific nonlinear models which describe Riemann's simple waves with Riemann's invariants and shock waves m gas dynamics are discussed. Considerable attention is also given to nonlinear hyperbolic systems and Riemann's invanants, generalizcd simple waves, and gemdized Riemann's invariants.

6.2

Kinematic Waves

Classical wave motions me described by Newton's w o o d law of motton together wtth some reasonable assumptm relating a s u s s to B displaccmcnt (as in gravity waver), to a strain (as in nondispersive longitudinal and Vansvene waves), or to a curvature (as in capillary waves and flexural waves). I n contrast with the case of dynamic waves,, a class of waves IScalled kinematic waves when an appropriate functional relation exists between the density and the flux of some physically observed quantity. Kinematic waves are nor at all waves in the classical sense. and they are physically quite different from the clarrieal wave motions involved in dynamical systems. They describe, approximately. many

Kinrmnlic Waves a d Specr+c Real-Worki Nmlinem Problems

186

importvt real-world problem including uafflc flows on highways. flwd waves in rivers, roll waves in an inclined channel, and chromatographic models in chemistry. Lighthill and Whitham (1955) first gave B general and systematic treatment of kinematic waves with applications. In many problems of one-dimensional wave propagation where there is a continuous dismbution of either material or some state of the medium, wc can defineadensiry p(x.1) per unit lcngth and a f l u x q(x.1) per unit time. Then. we can inhoduee B flow velocity "(x, I) by u = q/p. Assuming that the material (or state) is conserved, we can stipulate that the time rate of change of the total amount tn any arbihary interval x , L I L x2 must be balanced by the net influx across x, and x I . Physically. this states that the quantity in a small length segment changes at a rate equal to the diffemnee between inflow and outflow. Mathematically, this can be formulated as ~3P(I.I)~=g(r,,t)-9(x~.t)=-2(%)*.

(6.2.1)

or,equivalently.

ic-?+% 1* = o .

(6.2.2)

If this result is to hold for any arbitrary interval x , L x L x2, the iotepand must vanish identically. so that

"+&=o, (6.2.3) JI Jx provided p and q are sufficiently smooth functions. As stated in Chapter 5. (6.2.3) is called the conscrvotion I m , kinemtic wave equation, or the equation of conrinuiry. Based on theoretical or empirical grounds, we assume that there cxisu a relation between 9 andp, so that we can write (6.2.4)

4=9(P)-

Thus. equations (6.2.3) and (6.2.4) form a closed system since there are two equauons with two unknown functions. Substituting (6.2.4) in (6.2.3) gives

*+,(p)*= JI

Jr

0,

(6.2.5)

Where

(6.2.6)

187

6.2Kinemnc Waver

Similarity, multiplying (6.2.3) by c(p) leads to the equation

&++)

m

4 =0 ax

(6.2.7)

This means that q or p is constant in waves propagating with velocity c(p) given by (6.2.6), and hence. e(p) is called the wove propagolion veloeily. Mathematically. equation (6.2.7) has only one system of characteristics given by dr

;i;='(P)*

(6.2.8)

and along each of thesc charasteristics the flow q or the density p IS con~tant. The wave velocity e given by (6.2.6)is the slop of the flow-density e w e for fixed X. In terms of the mean flow velocity u = q/p. the wave propagation velocity is given by c = 4,(p) =

Thus. e > u when

*

qUp) = "+P& dp dP

(6.2.9)

> 0, Ihat is. the flow velocity increaseswith density as in

dp

flwd WVBYCSin dvm. and c < u when

*

d 0, that is, it decreases with density dP as in w f f c flow in highways. Further, the following observations are in order. First. there is one important difference ktwecn kinematic waves and dynamic waves. me former hss only one wave velocity at each patnt. while the latter possess at least two velocities (forwards and beckwards relative to the medium). Second, kinematic waves arc nondispenwe, but they suffer from a change in form due to nonlinearity (dependenceofuh wave speed c on the flow 9 camed by the wave) exactly as do traveling sound waves of finite amplitude Consequently, continuous wave

The law of motion of kinematic shock waves can be derived from the conservation laws. as was the law governing confmuous kinematic waves. If the density and flow assume the vducs pl and 9, on one side, and p2 and q2 an the other side of the shock wave which propagates with velocity U , then, the quantity crossing 11 per unit time can be written either as 9, -Upmoras q2 Upz.This gives the velocity of the shwk waves as

-

"=O (P*-PJ

(6.2.10)

189

6.3 Tmffi Flow Pmbkrns

6.3

Traffic Flow Problems

Traffic flow on B highway is one of the most common real-world problems. Based on the works of Lighthill m d whitham (1955) and Richards (1956). we consider the traffic flow on a long highway under the assumptions that cars do not enter or exit thc highway at any one of its points and that individual CM an replaced by a continuous density funeuon. We take the x - a m along the highway and amme the traffic flows in the positive direction. This problem can be desenbed by three fundamental wffiic variables: wffie density p(x,I ) . which is equal to the number of e m per unit length at position x and at time 1, thc mffic flow q(x. 1 ) . the number of c m passing a fired point in unit time. and the uaffic veloeity u(x. I ) which represene the velaeity of a car. We use the theory of lunematic waves to formulate the problem in terms of a first-order. nonlinear, partial differential equation on the basis of conservation of cars and expe"mcntal relationships between the car velocity and traffic density. So. the equations governing the traffle flow M

"+?Lo, JI 2.x q=q@).

(6.3.1)

(6.3.2)

where q is some function of p as determined by theontical or cxpcrimentsl findings of mffie flow phenomena. The functional relation (6.3.2) seems lo be reasonable in the sense that the density of CM surrounding a given car indeed contro1s the speed of that car. In fact. this relation depends on other facton including speed limits, weather conditions, and mad conditions. Several specific relations have k e n suggested by Haight (1963). On the basis of observations of traffic flow, we make a basic simplifying assumption that the velocity of B car at any point along the highway depends only on the traffic density. that is, u = U ( p ) . and hence, the traffic flow q=pU@). Clearly, u(p) must be a monotonically decreasing function of density p . If there are no cars on the highway (corresponding to very low densities), then, the car would mvel at a finite meximum speed umu. that is, U(0)=urn. = u r n . As the density of CM increases. the velocities of the cars would continue m decrease, and henee, U(p)= 0 as p +,.p = p.. when pmis the maximum trafficdensity conesponding to what is called bumper-to-bumper

traffic. Also. ~tfollows that

! ! = U'(p)SO. The decreasing feature of mffilc dp

velocity is shown in Figure 6.l(a).

K k m t i c Wmes onZ Specific Real-WorldNonlhror Problmml

190

On the other hand, the important feature of traffic flow Q(p) = pU(p) can be

inferred from

U ( p ) . Clearly,

Q(p)+O,asp+O.andp+pms

=pm

( ~ ( p ,= ) 0). his means that the traffic flow ~ ( pis) an increasing function of density p until it attains a maximum value Q- = qn for some p =p,, in O
&)

(b)

F i y r e 6.1

(a) Veloelty-density carve *ad

(b) Flowdensity curve.

The pmpagauon speed c = 9 of density waves carrying continuous changes of dP

flow Uvaugh the streams of cars is the dope of the tangent to the curve at that point. This slope is smaller provided that the mean speed decreases with increased density. Hence,we can Write

c=dg=d(p")="+p'".

(6.3.3) dP

< 0. and c = u only at a very low density. dp Usually, under light traffic condrtions, there are few c m on the highway traveling at high speed. On the other hand. under heavy traffic conditions, there ace many cars on the road. and these cars travel slowly. So again. traffic flow would be small. Evidently, c < u if

6 3 Tragic Flow Problems

191

Substituting q = U(p) in (6.3.1) gives the first-order, nonlinear, partial differential quation for traffic flow in the form

*+

.(p)& = 0, Jx

(6.3.4)

Q(d=U(P)+PU'@)

(6.3.5)

Jt

* .(PI=

represents the velocity of t k density WBVS. Since W(p)< 0, e(p) < U @ ) , that IS, the velocity of these waves is always less than that of CUE. and drivers experience such waves and warned of disturbances ahead. Fngure 6.1(b) shows that c@) = Q'(p)> 0 far all p in [O.p,), zcm at p = pu. and then, negative in (pn.pm].All these mean that waves propagate forward relative to the highway in [O,p,]. are stationary at p=pw, and then, travel backward in [p,,,~,]. Further, discontinuous waves are likely to occur on any segment of highway when the traffic IS very kavy in front and light bchind. For waves on which the rraffiic flow is less dense, cars travel forward faster than. and hence tend to catch up with. those on which the flow is more dense When this happens. a group of continuous waves can coalinto discontinuous (orshock) waves. According to the theory of characteristics of the first-order nonlinear equation. lhe characteristicequations for (6.3.4)are dt= _ dz _ = _ dp (6.3.6) I 4P) 0 with the solution p=constant 0"

d"+P).

(6.3.7)

Since c(p) is ons st ant when p is constant. the characteristics are straight lines in the (1.r)-plane. and the characteristic velocity c(p) = P ( p ) is the slope of the tangent to the a w e at a point. Actual observational data of traffic flow indicate that typical results on a single lane highway are pm= 225 cars per mile. pu = 80 cars per mile and qw -1590 car per hour. Thus, the maximum traffic flow qx occurs at a low speed u = qrjpn 20 miles per hour.

-

Next. we salve the nonlinear traffic flow quation (6.3.4)with the initial and boundary conditions P(X. 0 ) =Ax). x

E

R,

(6.3.8)

Kinanotic Wove, and Specific Real-World Nonlmem Problems

192

where f(x) vanishes ouuidc

B

finite domain of x and p(x.

1)

+0, as 1x1 +-.

This initial-value problem has already been solved in Section 5.2 and hence. we state !he result without proof. The solution is given by P(x*t)=f(S). x =

5 +1F(5),

(6.3.9ab)

whae F(S)=c{f(S)]=c(p).

(6.3.10)

The velocity e(p) = Q @ ) is the slope of the tr&k flaw e w e drawn in 6.l.(b), so that the density waves move forward or backward relative to the highway depending on whether p c p,, or p > p,, .

We consider an initial density curve p(x, 0) = f(x) in the form of B hump which has a maximum p. at some point x , as shown in Figure 6.2 so that pm>poandc(p,)ce(pm). Thus, thc point on the hump with density pm wvcls slower than the point with density po.As time progresses. the density profile continues to steepen at the baek and flatten at the front. eventually, leading to B multiple-valued solution which is physically inadmissible. I n other words, a venical segmentdevelops at the baek for 1%>z t l . The vcnical segment cornsponds to a point in the ( x , t)-plme 1which the density is discontinuous and, therefore. at which at least one of the panial d d v a t i v e p, and p, no longer exists. Thus the dirtmion of the initial density profile and the development of diseontrnuity as a shock are remarkable features of nonlinear traffic flow

:-pF+*

PO

0

Figare 6.2

=I

s

T

Maltlple-Valued density profile at t = t , >>t, > t , >O.

phenomena. However, the solution inemprating the discontinuity must satisfy con%ervation laws, and hence. the total number of e m tn any interval ( x , , x,) nprescntcd by the integral of p with respect to x from x , to x2 (> x , ) must be unchanged. Thus, the shock must be inserted at that value of x which leaves the

6.3 Trait Flow Problems

193

total area under the curve unchanged. that is. at the paint S such that the a m ABC= area CDE. This simple result that the two areas cut off by the vertical line through S are equal is known as Whitfmm'r mle. So, the Whitham mle of equal area can be used to dctmninc the position of the shock path for certain "0"h"ear equalionr. We consider a situation in which traffic initially bccomcs heavy further along the highway. As already indicated in Section 5.2, the solution represents a compression wave. It is convenient to assume the initial data, BE shown in Figure 6.3(a), (6.3.1 1) where p, andp, are constants. Osp, U,. The density wave for the lighter traffic mvel with velocity c(p,) = Q ( p l ) .which is w a t e r than the velocity c(p,) = Q ( p , ) of the heavier traffic density wave, that is, e(p,)>e(p,). AsetoffharacterirtifousingUleinitial dara(6.3.11)isshown in Figure 6.3.(b). In any situation where the traffic becomer denser further along the road. characteristics intersect. At any point where two characteristics intersect, p=p, and p=p2,but. it is physically impossible that traffic density is multiple valued. A jump occurs B C ~ S Sa curve as a shack. The behavior of the 'hump' in the density p in (x.1)-plane just treated indicates that

(4

(b)

Figure 6.3 (a) Initially heavier trsme ahead ( p 2> p , ) sod

(b) intrrsretlng eh~mcteristics.

a shock should be inserted between two regions of eonstant density. Since the overlapping stam at the origin, we Iwk for B solution of the form shown I" Figure 6.4, two regions of conitant density p separated by a constant shock vclnity given by (5.3.6). It C M be shown that equation (6.3.4) and the initial data (6.3.1 I ) arc satisfied on both sides of the shoek and the condition (5.3.6) is

Kinematic Waver and Specific Real-World

194

Nonlheor Problem

satisfied on the shoek. Evidently, p = p , =constant satisfies equation (6.3.4) and(6.3.11)for x < O a t f = O ; similarly, p=p,=constant for x > O a t I = O . The shoek condition (5.3.6) is satisfied if (6.3.12) that is. if the shock propagates at the constant velocity given by (6.3.12). The shaek path is a straight line through the origin in the ( x , 1)-plane. Thus the

traffic density p(x, I)satisfies all equations and initial data and is single-valued for all x and Iexcept across the shaek.

Figure 6.4

Graphical representation of rolotioa.

There is another c s e associated with the initial data which cornsponds to p, > p2,and hence, If, < If?, and c ( p , )< ~ ( p ~Physically, ) . this ease represents a situation m which the traffic initially kcomes light funher along the highway. Using the initial conditions (6.3.1 I), characteristics are drawn in Figure 6.5. It is imponant to point out that there are no characteristics within the infinite sector AOB, and hence, no solution far the traffic density p(x. I). According to Section 5.4, there are two regions, each of canslant density p . separated by a fan cenarcdat the origin. The solution is given by /p,, p ( x . l ) = p,,

[po.

to Ule left of OA

totherightofOB

l(6.3.13)

constant,p, <pa
6.3 Trafic Flow Problems

195

4t

Figom 6.5

The solution with

P

fan centered at the origin.

The solution in the sector AOB is the characteristic form. equation (6.3.7). of the solution of equation (6.3.4). Across OA and OB, p(x, I)is continuous, but

p. and p, ~ndiscontinuous ( p , and p, arc nonzero in the sector AOB but zero elsewhem). Thus. the problem is solved for both cases: (i) p, < p , and (ii) p, >p2. However, a remarkable faet involved in this problem with case (81) IS that there exists another discontinuous solution which satisties all equations and initial conditions. This solution consists of a shock dividing two regions of con~tant density p . All characteristics on the left of the shock an parallel. and so are those to the dghl of the shock as shown in Figure 6.6.

Figore 6.6

Characteristics and the discontinuous solution.

For case (in). there are 81 least ovo solutions: (a) the fanentered solution and (b) the shock ~ l u t ~ o Both n . satisfy a11 equations and the initid conditions, but the solution is not unique. The main question is which one of these solutions

196

Ki,!zmorie Waves a d Specific Red- World Nonlinear Problem

represents a physically meaningful wlution. "hi$question Section 5.4 by inucducing UK Lax entmpy entenon (5.4.17).

WBS

answered in

Exemplc 6.3.1 (The Green Signal Problem in Fmnr of the Trafie). We consider a sueam of cars stopped by L red signal at x = 0. The road is jammed initially behind the signal and there 1s no fraffiic ahead of the signal. As soon as the sipnal turns s t e m . the stream of cars starts mwing across I =O. % initial sate of the trafficis given by p(x,O)=p-!f(-x)=r'

xco

1.

(6.3.14)

0, x > o where p- is the maximum density and H ( x ) is the Heavisidc unit step function. We assume that the traffic flow q(p) is quadratic in the region 0 c p c p . and zero otherwise,that is,

Pi

o
q = B I - L p.

(6.3.15)

PSI

It is noted that the maximum flow rate is

1,

whneh OCCM at p = l p m .The

velaity of the m f i c flow is u = 4,that is.

,);-

u = "* (I

(6.3.16)

where urn= ( q m / p m )is the maximum speed when p = 0. The traffic flow qtmtionfmtkdensity p(1.r) is given by

P,+Cln)P,

=o.

(6.3.17)

whae

c(P)=d(P)=%(l-~). Thus, the characteristicequationsfor (6.3.17) an dt dr dp _=-=I .id 0

(6.3.18)

6.3 Tmfiic Flow Problems

191

(1-.]:

dp =O.

= c ( p ) = urn

(6.3.19sb)

Thus the density p remains constantalong a characteristicgiven by

(6.3.20)

f = urn(.)I:Any characteristicthat int-eu

UK positive x-axis at x = xl) > 0 has the slope

This gives a family of characteristics BE the right-leaning straight lines. as shown i n Figure 6.7, x = urnt + xo. (6.3.22) On the other hand, any characteristic that intersects the negative x-axis at x =.re< 0 has the slope

(6.3.23) As shown in Figure 6.7. equation (6.3.23) gives a family of characteristics rcpresentrng the left-leaning straight lines x = -u.r+xo. (6.3.24)

x=u",l

x=--U",t

J?ALizx p=pm

Figure 6.7

0

p=o

Characteristics due to density.

P

discontimoos Initial

Kincmric Wwer and Spmfic Real-World Nodineor Problaru

198

In the fanlike region -"-I < x < u,r. all characteristics must pass thmugh the origin and must be slraight lines. Hence, &=?. It follows from (6.3.20) that dr r :=urn

(I-2)

(6.3.25)

Solving for p gives (6.3.26) At any instant. density varies linearly in x from pa at x = -umr to zero at x = Y J Thus. the solution (6.3.26) f a p(x,r) for all x a1 any time r is drawn in Figure 6.8.

x=-umr

Figure 6.8

0

x = q

Tmme density after the signal

I u m s green.

Example 6.3.2(The Traffic Signal Problem). We consider the same problem as in Example 6.3.1 with the traffic flow q = 1 ( 2 M ) - p ) p , and 10

hence. the vclaeity of the traffic density waves is c(p) = !(IMl-p).

We have

to find the traffic density when the vaftic signal turns green at I = 0 , so that the initial density is given by (6.3.14). The typical value of p. varies from 200 to 225. According to the Iheory described above, the solution of the initial-value problem is

f(1.r)

= f(C)=p.

H(-5),

x = 5+c(/(C))=

5 +~{100-/(5)}1.(6.3.27ab)

199

6.3TrOgc Row Problems Using pm= ZW,we obtain t k solution as

These two results can be combined to obtain thc Oaftic density for any I > 0 (6.3.29) However. the solution for p(x.t) is nor known in the interval -Mt

< x < 601.

The nonexistence of the solution in this interval is somewhat unusual. It is possibly due to the fact that the initial condition of the traffic flow IS discontinuous at = 0,but the use of the con~ervationlaw (6.3.4) requires thal p(1.r) bediffamtiable.

<

Exmple 6.3.3 Obtain the Vaffie density p ( x , t ) governed by (6.3.4) with the given traffic flow q ( p ) = L ( Z W - p ) p

10

and with the initial state of the

tram flow as p(x.0) = f ( z ) = I

- x.

In this problem. the velocity of the density WBYSI

(6.3.30) IS given by

(6.3.31)

The soluuon of the initial-value problem is given by (6.3.9ab). that is. p(x.t)=f(<).

=<+rc(f(<))3

where c { f ( { ) ] = 1 ( l W - l + < ) = ~ ( 9 9 + < ) .

Solving for

(6.3.32ab)

< from (6.3.32b)

gives (6.3.33)

This result can be used to eliminate

< from (6.3.32a) so that the solution

befomes

p(..*)= ( I - <)=

0 ( I + ; )

(6.3.34)

200

Kinemlie Waves ond Spedflc Red-World Nonlinear Problems

Wis traffic flow model is based on the finborder approximation and. henee, the original assumptions that q = Q ( p ) m d u = U ( p ) are not good approximations. For a better approximation, it is reasonable to assume that q and Y are functions of both p and its gradient p.. Hence, we can assume that q = Q ( p ) - vp., u = U ( p ) - v(%).

(6.3.35ab)

when v is a positive wmmt in the context of wffic flow because driven are supposed to reduce their speed to Bnouot for heavy traffic ahead. This introduces two additions1 effects: (i) diffusion of the waves and (ii) response time defined by the tfme lag m the response of the driver and his cm to any changer in traffic wnditions. To incorporate the first effect of diffusion pmduced by the driver's response to conditions ahead, we substitute (6.3.35a) in the traffic flow equation (6.3.1) which gives a new parabolic equation in the form

P,+.(P)P, 'VP".

(6.3.36)

This equation is known as the Burgen cquafionwhich will be discussed in great detail in Chapter 8. Physically, the second-order term introduced by diffusion will smooth the traffic flow. In other words, the effect of diffusion tends to eliminate any discontinuity involved in the solution of the original hyperbolic I d f i e flow equabon (6.3.4). To incorporate the second effect. the following equation

D" = u, +"US =

-;[. U(P)+"($9 -

(6.3.37)

em be introduced for the acceleration of the car when the coefficient 7 is a measure of the response time. Equation (6.3.37) is to be solved combined with the conmation law

P,+(Pu).=o.

F a the case of small "and

(6.3.38)

(6.3.37) em be approximated by the original fint-order result, that is. by u = U ( p ) . 7,

Before we solve the nonlinear problem, it would be helpful to examine the linearized version of equations (6.3.37) and (6.3.38). These equations can be linearized for small perturbations about the equilibrium state p = p o and u = u o = U ( p o ) . sothat wecan introduce p=p.+P(x,f)endu=5+S(x,t). We substitute these results in (6.3.37) and (6.3.38) and retain only finborder terms to obtain

~ ( i+% , i,) = - [ i - U ' ( p , ) f i + ( i ) fi].

(6.3.39)

6.3 TmBc Flow Problems

201

p, +u, px+PO iz =o.

(6.3.40)

The propgation vcloeity of the density wave is represented by %=Po q P o ) + u ( P O L

(6.3.41)

end hence, Ll’(p,)=-(u,-e,)/p, which is used in (6.3.39)-(63.40) to eliminate E , and lhe final form of lhe equation for P(x, t) is

It is noted that the first term on the right-hand ride represents diffusion, whereas the second term arises fmm the effect of the response time 7.

If

Y and

r an very small. (6.3.42) reduces to a simple first-adcr equation.

p, +I?, p, = 0.

(6.3.43)

~ h solution s of h i s equation is p = f ( x - c o I ) representing waves traveling with speed ca. Equation (6.3.43) suggests that -a --c

-.a

m

(6.3.44)

OaX

This approximation is utilized in the right-hand side of (6.3.42). so that it dwcs to the form

]-

( ~ , + c ~ P . , ) = ( v - ? ( u , - 2c ~alp ) ax‘

(6.3.45) ’

It can bc inferred from this equauon that the solution would bc stable or unstabledcpcnding on whether v>?(uo-c,)’

01

vc?(uo-c,)’.

(6.3.46ab)

We examine the stability of the plane wave solutions of the original equation (6.3.42) without any appmximauon. This equation admits solutions of the form b(x. 1 ) = a exp[i(b - a)]

(6.3.47)

provided Ihe quation v(w -uo k)’ +i(w -e$k)- v X’ = 0

(6.3.48)

is satisfied. Thus, the solutions would be stable if d ( w ) < O for both roots of (6.3.48).

Finally. we observe that the right hand side of (6.3.42) io a wave operator so that it corresponds to factoring BS

202

Kinematic Wows and Spreific Real- World Nonlrnior Problems

1+, Jx

(;+c+g)(&+.g)A

When $I = v o + e .

c.

(6.3.49)

=%-c.

(6.3.50ab)

Evidently, one wave propagates with the speed c+ and the other with the speed c- < c+. Thus. for small r , the quation (6.3.49) can be approximated by

b, + co0,= 0 (6.3 51) provided c. c co
Flood Waves in Long Rivers

Another application of the theory of kinematic waves dealing with flood waves in long rivers will be presented here in some detail. We consider the propagation of flood waves in a long rectangular river of constant breadth. We take the x-axis along the river which flows in the positive x-direction and assume that the disturbance is approximately the same across the breadth. In this problem, thedepth h(x.r) of the river plays the role of density involved in the traffic flow madel discussed in Section 6.3. We denote the volume flaw per unit breadth and per unit time by q(x. 1). An argument similar to that of Seetion 6.2 gives the kinematic wave quation for the flow in a nver as *+d4=0. at Jx

(6.4.1)

Although the flow in a river is fairly complicated. we assume a simple functional relation q = Q(h) as a first approximation to increased flow as the water level riser. Thus, equation (6.4.1) becomes *+c(h)*=O. m ax

(6.4.2)

where e(h)=@'(h) and Q(h) is essentially determined from the balance between the gravitational force and the frictional force of the river bcd. One such result for Q ( h ) is the famous ChPry formulo given by Q(h) = h u ,

(6.4.3)

6 4 Mood Waver io

203

long Rivers

where u is the average velocity of the river flow and it is pmponional to &, so that "=a&.where a is a constant. The propagation velocity of flood WBVCS. by (6.2.6). is given by

e(h) = @(h) = d(,h)= u + h + = 2 a&=Ju. (6.4.4) dh dh 2 2 This clearly shows that the flood waves propagate one and a half timer faster than the sfnam velocity. A more general mull IS u = ah" for a. con~tanta so that the volume flow is given by

Q(h)= hu = ah*'. Thus the propagation velocity of flood waves is c ( h ) = @(h) = ( n + l ) u .

(6.4.5) (6.4.6)

This also confirms the fact that klwd waves propagates faster than the fluid velocity for all positive n. When n =%, (6.4.5) is called the Manning formula. When the kinematic wave theory, described in Section 5.3, is applied to flwd waves in rivers. kinematic shock waves play a major mlc in the forward regions of flood waves. The general solution may be taken from Seeuon 5 3 with shocks fitted in as discontinuities which satisfy the condition

"-(u'4-%4)

(4-4)

(6.4.7)

The process by which kinematic waves may steepen into shock waves with a considerable change in flow in a relatively shon distance has not been very clearly described m the literature. However, the possibility that such B wave propagates down a river, with different, uniform flows upstream and downstream of it. has been visualized as a flood wave model. Such a wave IS called the monocli~lfloodwave (orthe slendy wave profile). I n fact, this is B progressing wave whose profile tends to different constant states upstream and downsueam, with lower depth downsueam, joined by a Steadily falling region, as indicated schematically in Figure 6.9.

It has been suggested by Whitham (1974) ulat discontinuities described by the jump conditions so hydraulic theory are in reality the Nrbulent borer familiar in water wave phenomena as hydraulicjumps or bre&rs on B beach. Although the above nonlinear model bared on a first-order appmximation, gives a fairly g wd description of the flood wave phenomena. Seddon (1900) raised a number of questions about the formulation of my river flow model bared solely on the C h h y relation or its extensions. Fint, rivers do not have a uniformly sloping bed, nor do LEY,m any way, approximate this situation.

Figure 6.9

Monorlhd flood wave.

Second, the river bcd constantly varies with time. since its material is readily handled by the flow. To improve the above model. it is reasonable to assume that the volume flow depends on both h and x so that q = q(h. x ) . This relation can be combined with (6.4.1) to give a tint approximation for unsteady flows which vary slowly. Then, h(x. f) satisfies the equation

ah+a,,ah=-a,. Jx dh Jx ax Multiplying (6.4.1) by e

=($)

(6.4.8)

= c(h, x ) . we obtain

I-,

$+c(h,x)($)=O.

(6.4.9)

This means that q is constant for waves haveling past the point with velocity e(h, x). Mathematically, the equation has only one system of characteristics

6 = e , and the flow q is constant along each of these ehsracteriotier. dr Th-e wave veloeity c = c(h. I)is the slope of the density-flow curve for fixed x . This nsult for flood waves is known as the Kleitz-Seddon low. given by

6.5

Chromatographic Models and Sediment Transport in Rivers

In chemistry, chromatography is dereribcd as an exchange process between a solid and fluid. In general. the exchange process may involve panicles or ions of some substance or 11 may be heat exchange between the solid bed and a fluid. The general problem is extremely complicated. However, it is possible to develop a simple one-dimensional mathematical model for single-solute

6.5 Chromotogrophic Models and Sedhmr Tronrpn zn Rivrrr

205

chromatography based on the assumption of local equilibrium. The concentration of solute per unit volume in the fluid phase is denoted by e(x.r). and the concenwtion of solute in the solid phase by n(x, 1). A simple argument of the balance of influx and outnux leads to the basic conservationequation E d ’ + E ” ~ + ( l - E ) L o .

JI

JI

Jx

(6.5.1)

when E is a consmt fractional void volume in the bed and u is a ~ o n s t l l t interstitialvelaeity of fluid through the bed. The equilibriumequation is

.=f(4.

(6.5.2)

whne f ( c ) is assumed to be positive for physical considerations,but f ( c ) . in general. depends on temperature.Oflen, f ( e ) is called an nbsorpzion isotherm.

Substituting (6.5.2) in (6.5.1) gives [E+(I-E)f’(e)]

Inhodueing

v=(v)

$+Eu

$4.

(6.5.3)

in (6.5.3) yields

[I+ v f ’ ( , ) ] d ’ + , d ’ = 0 . JI Jx Finally, in terms of a new function defined by g(e)=e+ (6.5.4) ~ s s m c the s f m

(6.5.4) Y f ( e ) , equation

”d’

p,(c)d’ + Jx = 0. (6.5.5) JI The function g(c) is called the column isotherm. This equation is somewhat similar to that of the wffic flow equation. So. we may skip the detailed method of solution and state the salient features of the solution. It is convenient lo intmduce nondimcnsional variablw defined by x.

=?’ X

t . 2

t’

(6.5.6ab)

characteristic length for the column. hopping the asterisks. equation (6.5.5) becomes where I IS

B

g’(c) d”+d’=o. (6.5.7) JI & This nonlinear chmmatographie equation can be solved subject to the initial and boundary data c(=.O)=O(x).

where

0 and

am mkiwry functions.

c(O,r)= V(I)

(6.5.8ab)

206

Kinematic Wows and Spectflc Real-World Nonlinear Problems

It is often convenient to express the initial and boundary data in the param&C form

.=(,

c=#({)

t=r,

c=v(r)

at t = o ,

(6.5.9)

x=o.

(6.5.10)

81

The chmtcristic equation of (6.5.7)is (6.5.11)

d'=g'(c). da

that g"(c)= v / " ( c ) . For convex isotherms. /"(c)
so

and hence.

Next, we solve a particular model, called the Longmuir isotherm model. dcsnibcdby

ae

n=-=-=NKC (l+KC)

when

(I+ XC)

(6.5.12)

f(4

N is the saturation eonscnuation. x is a rate constant, and a = NK. In

this case. the characteristicequation is

d'=g'(c)= I + vf'(c)=I dx

+ v "

(1 + m)'

(6.5.13) '

We consider a particular case of (6.5.9)-(6.5.10) representing distribution of solute over a finite segment of the x-axis so that

t=r, w

c=o

81

x=o,

B

linear

(6.5.15)

h c, is constant.

The graphs of the Langmuir isotherm and thc characteristics in the plancandraullinFigurc6.10(a)and6.10(b).

a <.=)

tb)

F i y r e 6.10 (a) The LPngmuir Isotherm and (b) ChamcIerislics in (x, I) plane.

I)-

(1.

6.5 Ckromatogmpkic Mode& ond Sedimeni Trampor1 m R w m

207

It follows f r m Figure 6 1O(b) that all characteristics originating f r m the raxis have slope g’(O)=I+ v a , whenas thohe from the x-axis for < > a me less steep with a slope g’(c.). These two sets of characteristics yield two constant states, c = O.md c = c a respectively. On the other hand. the characteristics originating from the interval 0 S 5 S II OD lhc x-axis have slope between g’(0) and g’(c.). and hence, they represent a / m or expamive wme. Thus, the solution consists of two constant states separated by the expansion wave. For any point in the expansion wave region,wc obtain

In the limit, as a + 0. the characteristics originating from the interval 0 S S a tend to translate to the left, but their slopes remain unchanged. I n fact. these characteristics all originate from lhe origin, that is, the simple waves become centered there. In reality, this represents the elution of a uniformly absorbed column, which is B Riemann problem. In this case ( a --t 0).we obtain

<

( 6517) from any point in the simple wave region, so that function of

(4)

(KC)

can be expressed

as a

only

(m) = -1

+.G (II);

(6.5.18)

This result was recognized in the theory of chmmatogmphy as early as 1945. We next consider lhe saturationof a clean column with the conditions e(x,t)=O

or t = O

/or X > O

Obviously. the characteristicsoriginating from lhe x-axis ( r = 0) have slopes g’(O)=(l+ va), and they arc steeper than those originating from the r-axis (x=O), rineetheslopesofthelattcrhavcslo~ g ’ ( ~ . ) = l + ( v a j / ( l + ~ ~ ~ ) ‘ . to point out that there is a charactenstic cornspanding to every concentration between 0 and co generating from lhe origin.h characteristics PA, PO, and PB conesponding

The characterisues are shown In Figure 6.11. It is imponant

to conconhatiom 0.

:[ I A:-- If]

- -1 +

va

1

,and

ca , rcspcctivcly. intersect at

Kinrmtrc Waver Md Sprepc Real-World Nonlinror Problem

208

P. Hence. tkn m thne values of e at P showing the muluplc-vduedners of the concentration function c . This physically unseeeptlble situation miser from the fact that the higher concentrations move faster than the lower ones. The eoncentration w o w propagating with a sloping front leads to n compression

f

B e =e, x A

Figure 6.11

Figure 6.12

c=O

Overlapping characteristic limes.

Distortion of the waveform.

wave which gets progressively s t e e p BS the higher concentration overtlkes the Iowcr, until the wave ovenops itself BS shown in Figure 6.12. Ultimately. the waveform breaks to give a multiple-valued solution for c(x. 1 ) . In other words. the solution develops a discontinuity. In cenmt to linear waves, Ule brcaking phenomenon is typical of nonlinear waves. Another example of application of kinematic waves was given by Kyneh (1952). me pmess of sedimenmtion of solid parucks dispersed in a fluid is one of great practical importance. It is based on an assumption that the velocity u of a panicle is B function of only the local concentration p of panicles in its immediate neighborhwd. It is also assumed that the particles have the Same size

6.2 Chmmoto~rophicM d e l s a d Sedimenr Trompon in Rivers

209

and shape. The panicle flux S = pu npresenb the number of particles crossing a horizontal section per unit m a per unit time. Therefore. the panicle flux S at any level determines or is determined by the panicle concentration. As p increases from zero to ib maximum value 0.. the velocitv u decreases coniinuourly from a finite \.due u to zero The v m a t m of S IS very complicated, but a simple model 13 useful for undcntandinp the key feature$ of the nrohlcm We denote she hoehl ufanv level a h o w the bonom of the column of dispersed panicle by X . Based on the argument that the accumulation of panicles between two adjacent layers a1 x and x + dr is the differenceb e t w e n the inflow and the outflow, the continuity equation is given by

.-

ap -=_ dt

as

(6.5.20)

ax'

or,equivalc"tly, (6.5.21)

dp+c(p)dp=o, at dx

uane 4 P ) = -S'(P) = -[.(P)+P

U'(P)].

(6.5.22)

Equation (6.521) is the kinematic wave equation, and i s mathematical analysis would be similar lo that of Seetion 6.2 Clearly, the characteristicsof (6.5.21) are given by &

-= .(p).

(6.5.23)

As p and. hence, c(p) is constant,the characteristicsarc straight lines. ?he basic equation for the sediment Iransp~Rof solid particles is similar to the kinematic wave equation. So, the r e ~ u l tof~ the kinematic wave theory developed in Section 6.2 can be applied to sediment trampon phenomena. Without any funher discussion of mathematical analysis. we simply mention some key featuns of this problem. Any discontinuous changes in the panicle concentration can =cur in two ways: (i) a discontinuity of the first kind in the panicle concentration is due to a sudden finite change of concentration at a ceRsin level, and (ii) B discontinuity of the second kind is due to a very rmoll change in the partlcle eoncenfration.If p, and u, represent the concentrationand the veloeity of panicles above the dixontinuity and p2 and u2 denote the same quantities below the discontinuity, the velocity of the shock waves U is obtained from the equation P d 4 +U)=P& so that

+U).

(6.5.24)

In a diagram in the (p, S)-plane. the velocity

(I

represents the dope of the

chord joining the points (p,,S,)and(p,,S,). On the other hand. the discontinuity of the second kind gives p I -p, = dp where dp IS small EO that (6.5.25) becomes O = - g = c ( p ) . This means that the wave velocity e is dP equal to the Shock wave velocity between concentrations p a n d p + d p . Evidently, a small change dp. if maintained. is propagated through a dispcrsion of concentration p with velocity c just as sound waves propagate through air with B definite velocity. Fmally, for dispersions where the concentration increases downward toward the bottom, the condition for the formation of a firstorder discontinuity can be expressed m the following equivalent ways: (a)

the characteristics in the ( x , t)-plane would intersect,

(b) the velwity of propagation c(p) increases with concentration, M (c)

the flow-concentration curve represented by S = S(p) in the sediment transport phcnomenon is concave to the p-axis.

If e(p) increases with p. small coneentration changes from denser regions below move faster upward than those in thc less dense regions above and, subsequently. overtake them. This means that the concentration gradient increases until a first-orderdiscontinuity is developed. On the other hand, if c ( p ) decreases with p . the concentration gradient decreases, and hence, any discontinuity will be disseminated.

6.6

Glacier Flow

A glacier is a huge m a s of ice that flows over the land under its own weight. Mathematically, a glacier can be hated, crsentially, as a one-dimensional flow system, which continuously either gains new material by snowfall or loses it hy melting and evaparauon The following discussion on waves 20 glaciers is based on papers by Nye (1960. 1963) who developed a modem theoreticalanalysis of glacier flow. We consider the model of B glacier (or a large ice sheet) which rests on an inclined plane of slope a and flows down the plane in the x-direction. We also assume that the glacier is of unlimited extent in the direction normal to X . and a = a(.) is a slowly varying funeuon of X . The thickness of the glacier is

6.6 Glarier Flow

211

assumed as h = h ( x , I) such that (h,), << 1. Then. the surface slope B ( x , I) downhill in the x direction 1s given by p ( x , I) = a- h,. (6.6.1)

We denote the volume of ice by q ( x , I) at a given point x and in unit time and in unit breadth. This quantity q ( x . I ) is called aflow. The consewation of ice volume giver the equation

ah+*=+, at Jx

(6.6.2)

1).

where a(*,I ) is the rate of accumulation of ice at the surface, that is, Ule rate of addition of ice to the upper surface by snowfall and avalanching. measured as thickness of ice per unit time. The negative value of LI corresponds to melting or evaporation of ice from the upper or lower surface of the glaaer. Assuming that q is a given function of x and h end, then, following Lighthill and

Whitham (1955). we multiply the equation (6.6.2) by c=($)

to obtain the

equation fm the glacier flow

d4+,d4=.,. (6.6.3) JI Jr If o=O. q would be constant in the characteristics, that is. there exist kinematic waves of con~tantflow q traveling with velocity e . In general, c differs h m the average velocity of the ice iself which is given by u = 1.To examine the instability of the compression region of a glacier flow, we assume that q = q ( x . h. a) Multiplying quation (6.6.2) by c =

c(2)+(z]x,* (+

(2)..

giver

c

(6.6.4)

Or. equivalently.

J:

Jx

(6.6.5)

Using (6.6.1). that IS. replacing a,by -hx,, we obtain

"'+,d4+,'h=,,, Ji Jx Jxdr

(6.6.6)

Kinematic Wovrr Md Specific Real-World Nonlinem Pmblems

211

where

K=($)=,~

is called the diffusion coefficient. Substituting

from

(6.6.2)gives the equation for q(x, I) in the form (6.6.7)

In terms of perturbations q1 and o=o,(x)+o,(x,t)

0 , . we write q=qo(x)+q,(x,I) and where qoanda, are the steady state values. Since

‘40 = a,.we obtain the equation dr (6.6.8)

This is the equation for the finite perturbation q,(x, I)produced by a change a,(’. I) in the rate of accumulation. When K = O . the equation repnsenu the quation for q1( x . f ) . In view of the diffreultiesin collectingdata on flow curves for glaciers, we consider the shearing motion of thc two-dimensionalsteady flow down B eonstant slope. We suppose that u(y) is the velocity of the layer st a height y from the ground, and ~ ( y )is the shearing E ~ E I . The appropriate smss-strainrelation for ice is assumed as (6.6.9)

where n = 3 or 4 . Further. the appropriate law for ice slipping in its bed is V u ( O ) = Tm(0),

wherern=L(n+l)-Z w h e n n = 3 . a n d r n . . 5 w h e n n = 4 . I f

(6.6.10)

aistheangle

of the slope and p is the density of ice, the shearing sfless satisfies the equation

_dr= _

(6.6.11)

gprina withthesurfacecondition r ( y = h ) = O . Thus, the solution ofequation (6.6.11) dy

r(y) = (h - y ) g p sin a.

(6.6.12)

Using this value of r(y). we next solve the equation (6.6.9) with (6.6.10) obtain

to

213

6.7 Roll Waver and T k i r Sfability Analysis

u ( y ) = i ( g p sins)^ h " + l . H [ h " + ' - ( h - y)"']. P (n+O The steady flow q, per unit breadth is given by q,(h) = f.(y)

dy = i ( g p sin a)'h"*l Y

+o"h"*2 (n+Z) .

(6.6.13)

(6.6.14)

It is convenient to take q , ( h ) = A h N whcreA is a constant and N is approximately in the range 3 5 N 6 5 . Thus. it turns out that the wave propagation veloeity is

e = d4- = N(+) = N v,, dh

(6.6.15)

where u, =% is the average steady flow velocity. This shows that the kinematic wavcs travel *bout three to five timer faster than the average steady flow sped.

6.7

Roll Waves and Their Stability Analysis

We consider an incompressible viscous fluid flowing down an inclined channel that makes an angle n with the horizontal under the action of gravitationalfield g. We consider only two-dimensionalunsteady flow derc"bed by the velosily field (u. v ) . The fallowing Figure 6.13 exhibits the pdmary flow configuration.

Fibvre 6.13

Flow configuration in an inclined channel.

Kmemalrc Wmrs md SpecificReal- World Nonlincor Problem

214

The unsteady two-dimensional flow is governed by the Navier-Stokes equations and the conunuity equation u,

+ u u, + v u y = - i p x + gsin a+ v 01

v, + u v.

+ v v3 = -1,

-gcosa+

Y

u,

V'

v,

y + V? =o, w h p and

Y

(6.7.1) (6.7.2) (6.7.3)

me the density and the kinematic viscosity of the fluid.

With the free surface elevation y = h ( x , r ) , the kinematic free surface condition IS given by on y=h(x,r).

h,+uh,-v=O

(6.7.4)

The dynamic fm surface conditions described by the tangential smss and the normal s m s with pressureon the free surface are given by on y=h(x,r),

6,=p(u3+vr)=0

(6.7.5)

-P+Zv vy+ g h cora = 0 on y = h(x, r ) . (6.7.6) P The bottom boundary conditions a u=v=o ony=O. (6.7.7) It can be shown that the primary flow field satisfying (6.7.1) - (6.7.7) erisu and is given by Y

)-; (;).

= U(y) = uo(2

P = p,(y)=gp(h-y)

u = 0,

c o w h =4,

(6.7.8ab) (6.7.9)

whae

U, = I ( g h ' s i n a ) .

(6.7.10)

2"

Based on the hydraulic approximation (that is, v is small), we integrate the continuity equation (6.7.3) with r a p t to y from y = 0 toy = h ( x , I) and. then. combine the result with (6.7.4) to derive

?;I

dx+ u h,

or,equuvalenuy,

+ h, = 0.

(6.7.11)

21s

6.7 Roll Waves and Their Srobilisy Analysis

(6.7.12)

Similarly, integrating (6.7.1) with respect to y from y = O to y = h and using (6.7.3). (6.7.4). and (6.7.7) gives $j~dy+$i”’.y=-ldipdy+ghsina-F. 0 02x0

(6.7.13)

when F npnsenls the fictional force term given by F = v [ ~ ~ ( ~ , O , r ) - ~ ~ ( ~ , h , r ) + g ( x . h , r ) ] - ~ p ( x . h(6.7.14) .r)h..

In view of the hydraulic approximation. the velocity component v

and hence, equation (6.7.2) has ee appmiimale form

iz

gcosa=o.

This C M be integrated fmm y = 0 t o y = h p(w.t)

IS small

(6.7.15)

to obtain

= -gp(y-h)cosa.

(6.7.16)

w h a t p(x. h , I) = 0 . Consequently,(6.7.13) reduces to rhc form

$2.

dy

+$d i dy =-1.2

gcosa d ( h ’ ) + g h a n

Jx

a- F. (6.7.17)

Using the pnrnary flow field (6.7.9). the flow rate q is given by q = j u dy = I gh’sin 0 3”

We define i7 BS he avemge value of

a.

u so that q =

(6.7.18) I , and therefore,

i7 = (g s i n a ) h’. I 3v

(6.7.19)

We next write the integral involved in the second tmn of the left-hand side of (6.7.17) as ~ u ’ d y = n ~ h = q ,

(6.7 20)

when a is a eonstant parameter. Using the primary flow field (6.7.8a) m (6.7.20) gives a =

6 = 1. The primary steady flow is simply caused by a balance

betwen F a n d g h s i n a , that is. F = g h s m a which gives, by (6.7.19).

216

Kinemoric Waver ond Specific Rd.World Nodineor Problem

F = 3 v g h-'.

(6.7.21)

In view of (6.7.m (6.7.20).and (6.7.21).equations (6.7.12)and (6.7.17)reduce 1"

*+?Lo Jt

( 2 + + ) + i g c o s a ZJx' when q = h

u." h e

two equations

(6.7.22)

Jx

h' )-- & - h s.m a - x , h

hydraulics. Neglecting the Id-hand side of (6.7.23). so that equation (6.7.22)becomes ah

-

(6.7.23)

are known as the mll wave equorions in = l ( g h 2 sin a), 3"

ah

(6.7.24) -+U(h) -SO a t a x It follows from this analysis that roll waves can be described by the kinematic wave equation (6.7.24). so that these waves propagate with the velocity U ( h ) . A famous example of such waves is shown in Figure 6.14 and a beautiful photograph of this phenomenon (see Figure 6.15) was mksn from a book by Cornish (1934) who described his observations of roll waves with numerical data and several interesting photographs of roll waves in a long rectangular open conduit at Merligen which feeds water from a mountain to Lake Thun in Alps. These photographs clearly show the exact periodic smcture of the waver. although in this particular situation, the waves seem to be unusually long and shallow. The photograph given in Figure 6.15 confirms the fact that these waves exhibit a periodic structure of a Series of discontinuous bares separated by smooth profiles. Such waves frequenlly occur in sufficiently steep channels. for example. spillways in dams or in open channels, such as that of Figure 6.14.Observational data reported by Cornish also reveals that roll waves travel faster than the mean flow velocity. The Froude number F=(U'/gh) associated with this phenomenon always exceeds the critical value 2. Subsequently. h i d e r (1949)made an impanant contribution to the subject of roll waves, and investigated how to ~onsmctnonlinear solutions of mll-wave equations,with appropriate jump conditions, to describe the roII-wave patterns. He has shown that smooth solutions can be pieced together through

Figure 6.14

Roll w a v e in

B

-, sleep channel.

LIZ

Kincmalic Waver md Spec@ ReoCWorld Nonlmem Problem

218

Substituting (6.7.25)-(6.7.26)into (6.7.22) and (6.7.23) and retaining only terms linear in q and ii leads to !he following equations

(r&-e)q+h, i i = O . -ikc(

(6.7.28)

uQq + h,E ) + ik(ut q +2goho P)+ ikg h,q m a = g q sina-?~+%

h, Wc next u s (6.7.28)to express can be wrioen as

4

,,,

(6.7.29)

ii in terms of q, so that equation (6.7.29)

-(u0- c ) } + a { ~ i-

tk[-c{~~

2

(go ~ -c)}+g ~ 4cma]

= gsina+?[(U0

4

- c)+ Us].(6.7.30)

Or, equivalently.

i k [ - c 1 + o ( 2 e - u ~ , ) u ~ + g l ~ ~ ~ a ] - ( 3 v ~ ’ ) ( (6.7.31) 3u~-~). The critical stak cxisu for valusr of e which are given by e -3?7*

=ZOO and c’ -0(2c

-ro)uo -(8 h,)cosa =O.

(6.7.32ab)

in krms of ule critical Froude number F, = (u;/gh,)as

4(1-~)F,=eosn

(6.7.33)

for various values of a . I n particular. when a=?,

25

the growth rate of the

instability fan be obtained a8 F > F.

=5 wsa.

(6.7.34)

This result can also be derived from the stability analysis of the original equauonr (6.7.1)-(6.7.3) In addition to above discontinuous roll-wave solutions. we conclude this section by mentioning Dns~ler’icontinuous roll.wusve soluuons which describe periadic, progressive waves flowing over a sloping bonom in terms of cnoidal waves. as shown in Figure 6 16. For further mathematical details, thc reader i s referred u) Dressier's (1949)paper.

Figore 6.16

A typical continuous roll-wave solution.

6.8 Simple Waves and Riemann's Invariants In one of his famous papers. Ricmann (1858) made a remarkable discovery which laid the foundation for all subsequent work on the theory and applications of nonlinear plane sound waves and simple waves. Based on his general formulation. we discuss the mathematical analysis of the theory of simple waves and Riemann invanants. To develop the governingequations for finite-amphde wave motions in gas dynamics, we consider the continuity equation and momentum equations of an unsteady empmssibls fluid without M y forces in the form Dp+p(v.")=o.

(6.8.1)

DI

pDu=-vpp. DI

(6.8.2)

where p is the density, u = (u. 1). w ) is the fluid velocity vector. p is the pmsrm, and the total derivative is

E 2 + u . V .

(6.8.3)

DI at It follows from thermodynamics that density is a function of pressure and entropy. that is, p =p(p, s), where s is the mtropy. Hence. (6.8.4)

For isentropic flow, s

IS Constant, that IS. h = O .

Thus equation (6.8.4).

written in terms of the total derivative following a fluid element. become

Dp=lDp DI

c'

DI'

(6.8.5)

220

K m e m o t ~Wows a d Spec~jic Reol-Wodd

Nonlinear Problem

whne c is the vrlociry of sound. so that c1

A.

(6.8.6)

dP

Substituting(6.8.5) into (6.8.1) gives

__ Dp+pv-.=o. Df

(6.8.7)

c'

For one-dimensional unsteady isentropic flow, equations (6.8.7) and (6.8.2) komc

T (++.* Z ax) +pdu,o, ax

"$)+ 2

(6.8.8)

= 0.

(6.8.9)

[$+(u+c)$]+d[$+(u+c)s]=O.

(6.8.10)

p($+ Adding (6.8.8) and (6.89) gives

Subtracting (6.8.8) fmm (6.8.9) yields

[$+

(u

- c)$]-

$[$+(n- ~$3

= 0.

(6.8.I I)

In general and in principle, a solutionof thesc equations gives u = u(x. I ) and

p = p ( x , t ) . Howevn,inmducing P = e . we rewrite (6.8.10) and (63.11) to PC

obrain

d("+ P )+ (" + &("ax L(=-P)+("-c)

+ P)= 0,

d (.-P)=O. ax

(6.8 12) (6.8.13)

A d i n g to the theory ofchanrctnirticsfor first-order,quasi-linearequations. equations (6 8.12)-(6.8.13) sipify that u+P=constantdongacurve C, suchthat * = u + c ,

(6.8.14)

u-P=constantdongacurvc C. such that f i = u - c .

(6.8.15)

These me b l y remarkableresults first obtained by Ricmann (1858). Physically. the curyes C, and C. in the (x, f)-plane 815 r e f e d to as positive and negative

211

6.8 Simple Waves ond R i e m n ' s InvoMnlr

wavsbrs respectively. In general,these wavelets do not travel at constant speed.

but. everywhere. they move with the local speed of sound relative to the fluid velocity. More precisely. a curve C, is the locus of a point ( A I)which always moves forward (that is. in the posirivc x-direction) at the local wave speed c relative lo the local fluid velocity u . Similarly, B curve C. is the INUS of a p i n t in the (x.t)-plane which always moves backward, that is, in the negalive xdirretion. To gain further physical understanding of the Ricmann theory. we examine an initial-voluc problem. If, at time I= 0 . all the fluid except that in a certain slab of the x-axis is undisturbed, that is. u = O a n d P = O a t I = O , while the disturbance within the slab may be large but. of course. with constant enmpy. The main question is how disturbances propagate ahead of and behind the slab in the subsequent motion.We draw a Figure 6.17 in the (1, I)-plane. At I= 0 , the disturbance is confined to the slab between the points B (for back) and F (for front). To investigate the subsequent development of the flow. we consider a large number of C, curves shown on the figure including those designated as C," and C,' which miginate at I= O from the points B and F,and. similarly, with the C. curves.The slope of the% curves is not known in advance because it depends on how u and c change. However. some imporwt information about them is described below. First. equation (6.8.15) indicates that u + P takes a constant value along each C- curve, but different for different members of the C- family. However. C: and all curves ahead of it originate fmm the region ahead of F, whcn u = P = 0 . The constant value of u P on each of these can only be zero. Thus, we conclude that u = P ahead of C.' Throughout this

-

region ahead of C!, which inereaxs prognosively wider than the region ahead of the slab, we examine how dirnrrbances propagate. Similar arguments reveal the

B Figore 6.17

O

F

Rirmsnn's inilial-value problem.

222

Kinemtic Wmes and Specific Real-World Nonlinaor Problcm

fact that u = P behind C!. Analogous arpmenU for the C+ curves indicates that u = -P behind C," and also ahead of C,'. It follows from the above ansly~isthat C,' must lie ahead of C!, and the region ahead of C: remains undisturbed with u = P = 0 because there are no distumances in this region. Similar comments apply to the region behind C.', that is, this region is also undisturbed with u = P = O Figure 6.17 also indicates that, after some time ' = I r , C[ interests C: at a point A and the region C[ACf remains undisturbed with u = P = O . Evidently, during the time interval O < I < I,, the disturbances become disentangled. and thereafter, he y propagale as two simple waves - one forward and one backward with an u n d i s W region in between. So. the upshot of the Riemann analysis is that the disturbance initially confined to BF becomes disentangled when r = r d . into a pair of simple waves propagating in opposite directions (one traveling to the right and the other traveling to the left). separated by an undisludxd region.

To give a more general formulation of the p~~pagatio" of simple waves, we rewrite the equations (6.8.8)-(6.8.9) for one-dimensional, unsteady, isentropic flow in terms of density p as

dp+"dp+p&=,,

m

ax

ax

(6.8.16) (6.8.17)

In mohii form. this system of equations IS (6.8.18) w

h (I,A. and I .we mahices given by

It is of interest to determine how a saluuon evolves with lime I . Henw, we leave the time variable unchanged and replace the space variable x by some arbitrary curvilinear coordinate so that the semicurvilinear cooidmate

<

transformationfrom (x. I ) to

(<, r ) can k i n d u c e d by <=<(GI),

T=I

(6.8 20)

If the Iacobian of this transformation is nonzero. we can transform (6.8.19) by the following correspondence rule

6.8 Simple Waver ond Rienonn'r hvarionrr

223

a - a g a + ar a a g a + _a m m ag m ar at at dr' a = _ag_ +a ar a ag a ax ax ag ax ar ax ag

(6.8.21ab)

This mule tTaniformi equation (6.8.18) into the form (6.8.22) m i l equationcan be used to dewmine a(I/ag provided that the determinant of its cocftircient matrix is nonzero. Ohviously, this condition depends on the nature of the curvilinear coordinate curves e ( x , f ) = constant. which has been kept arbitrary. We assume now that the determinant vanishes for the particular choice 6 = q . so that

(6.8.23)

In view of this, aU/,/aq will bccomc indctmninate on the family of curves q=eonstant, and hence, a(I/,/aqmay be discontinuous across the curves q = eonstant. 'Ibis implies that each element of a(I/aq will be discontinuous m s s any of the curves q =constant. Then, it is necessary to find how thex discontinuities in the elcmcnts of aII/aq are related across the curve q=conrtant. W e next consider the solutions U which are evcrywherc continuous with discontinuous derivatives alllaq across the particular w i v e q = con~tant=qo.Since (I is continuous. elements of the matrix A me not disconunuour acmss q = qo, so tM A can he determined in the neighborhood of a point Pon q = qo.And since a/& repnxnts differentiation along the curves q =constant. a(I/dr is continuous everywhere, end hence. it is continuous across the curve q = qo at P. In view of the above facts, it follows that differential equation (6.8.22) a5 = q = qoat P gives

thecurve

(Zl+$A),[g],

[/I,

=O.

(6.8.24)

where = / ( P + ) - f ( P - ) denotes the discontinuous jump in the quantity f across the e w e q = qo. and / ( P - ) and / ( P + ) n p m n t the values to the immediate left and tmmcdists right of the curve at P. Since P is my arbmiuy point on the curve, J/aq denotes the diffwentiation normal to the curves q=constant. so that equation (6.8.24) can be regarded as the compatihility

Kinemric Waves ond S p c i / c ReaCWorid Nonlinear Problems

224

condition satisfied by JU/Jq on either side of and normal to there C Y N ~ Sin the (x. +plane. Obviously, (6.8.24) is a homogeneous system of equations for the two jump quantities [au/,latl]. Therefore,for the existence of a nontrivial solution, the coefficient determinant must vanish, that is, (6.8.25)

(6.8.26) so that these e w e s have the slop

(6.8.27) Consequently, equations (6.8.24) and (6.8.25) can be expressed in terms of 1 the form

80

Ll

(6.8.28)

IA-UI=O.

(6.8.29)

(A-2)

&!

=O.

where 1 represents the eigenvalucr of the matrix A and [JUlJq] arc pmponional to the comsponding right eigenvatm ofA. Since A is B 2 x 2 m a h i , it must have two eigenvalus. If these are real and distinct. integrauon of (6.8.27) leads to two distinct families of real curves r, and r2in the (x. t)-plane:

r,: $=A,, r = ~ , z . (6.8.3Oab) r, are called the ehomcnrirtic cumes of the system one of these families of curves r, may be ehoscn for the

The families of

CUN~S

(6.8.18). Any curvilinear coordinate curves q=constant. The eigenvalucs A, have the dimensions of vcloeity, and the 1,associated with each family will, then, be the velocity of propagation of the matrix column vector [aU/,latl] along the EUWS r, belonging to that family. In this particular CBY. the eigenvalucs (6.8.29). that is, by the equation

1 of the matrix A are determined by

so that

1 = 1,=u*c,

I=

(6.8.32ab)

1.2.

Consequently. the families of thc characteristic curves (6.8.30ab) become

r, ; d"=.+,,

and

r,(r=1.2) defined by

r, : %=,-,.

(6.8.33~

f

In physical t e n . there re~ultsindicate that disrurbanccs propagate with the sum of the velocities of the fluid and sound along the family of curves In the seeand family r,, they propagate with the difference of lhe fluid velocity u and the sound vclaity C .

r,.

[3

The right eigenveetors R, I

are solutions of the equations

( A - 1 , l ) R, =0,

I=

I, 2.

(6.8.34)

Or. equivalently.

This rc~ultcombined with (6.8.29) gives

w h a t a ISa constant. In other words. across a wavefront m the chsranenstic ewes

-lJ~lJd-[dujdql, 1 CIP and moss a wavefront in b e

r, family of (6.8.37)

r2family of characteristic curves (6.8.38)

where c and p have values appropriate to the wavefront.

K i ~ m n t r cW m e Md S,mqic Rcol~WorfdNonlrneor Problems

226

To introduce the Riemann invariants explicitly, we form the linear combtnaion of the eigenveclors (kclp.1) with equations (6.8.16)-(6.8.17) to obmn fl(p,+pu,+up,)+ P

u +uu +'p (

(6.8.39)

=O.

' P . 1

8

We use &lap = *c/p from (6.8.37)-(6.8.38) and rewrite (6.8.39) as

+

(6.8.40)

k l [ p ,+ ( u i c ) p , ] + [u, (ut c)u,] = 0.

In view of (6.8.33ab), quauan (6.8.40) becomes

on

du+ldp=O P

r,,

r=1,2.

(6.8.41)

or,qwvalenUy, d[F(p)*n]=O

OD

r,,

(6.8.42)

Whcn

F ( p )=

I'dd A P

(6.8.43)

dp.

Integration of (6.8.42) gives F(p)+u=Zr

on

r, and

F(p)-u=Zs

on

r,,

(6.8.44ab)

when 2r and 2s arc constants of integration on r, end r,, respectively. The quantities rand s are called the Ricmonn invotiantr. As staled above, I is an arbitrary constant on characteristics r,.and hence, in general. I will vary on each r,. Similarly, s 1s comtantoneach r, bul will vary an r,.11 is nstwal to inuduce rands as new curvilinear ewrdinates. Since r is eon~tanlon r,, s can be treated as the parameter on r,. Similarly, I can be regarded as the parameteron r,. Then. d . = ( u f e ) d r o n r , impliesthat

*=(.+,)fi

r,,

(6.8.45)

*=(u-c)fi on r,.

(6.8.46)

1

dr

on ds dr

is a con~tanton r,. Therefore, the derivatives m equations (6.8.45)-(6.8.46) are panial dcnvative with respect lo sand r, so that we can nwrite them as I n fact. r is a constant on

r,,and s ax

-=(U+C)dS, at

Jr

(6.8.47)

227

6 8 Simple Waver and Ricmmn’s lnvoriona

(6.8.48)

These two first-order panid differential equations, in general. can be solved for x=x (r,s),an d f=f(r,s), andthen, by inversion, r a n d s canbeobtainedas

funcuons of x and f . Once this IS done, we use (6.8.44ab) to determine x(x, f ) snd p(x, r) in terms of I and s as “ ( X . f)

= r--s, F(P) = r + s .

(6.8.49ab)

W h e n one of the Riemann invariants I or s is constant throughout the flow. the cornspanding solution is tremendously simplified. The solutions are known as simple wave mofions representing simple waves in one direction only. The generating meehanirmr of simple WBWS wilh their propagation laws can be illusuated by the so-called pisfonproblem in gas dynamics (see Example 6.8.2). We close lhir section by reducing the two first-order equations (6.8.47) and (6.8 48) to a single second-order linear hyperbolic equation. Thts can be done by equating x,, and x,, which eliminates x from the system (6.8.47)-(6.8.48), so that WE obtain the single equation ( u + c ) ~ , , + ( u , + c , )=(u-c)r.+(u,-c,)r,. ~,

(6.8.50)

The coefficients of this equation are functions of r a n d s . and they can be determined explicitly. First, we find that u, = 1 and u, = -1 from (6.8.49a). Second. from (6.8.49b). we calculate F ( p ) p , = F’(p)p, = I , and hence, from result (6.8.43), F’(p) = 5 so that p, =p, c, =c’(p)p,

= E , Finally, we determine lhat

= p . d ( p ) , c,=c’(p)p,=p.c’(p).

(6.8.51ab)

These results a n used to simrJify (6.8.50) &*her as

(

. (

:

~ c r , , + I + ;dp) P dc r = - I + - -

:)I,.

(6.8.52)

Or. equivalently,

(6.8.53) Thesoefficientof (f, + f , ) depends only on p . and hence, by (6.8.49b). it is a function of (r+s), so that we can write it as f ( r + s ) . More explicitly, we wile

Kinimolic Waver and Specifre Real-World Nonlincor Problems

138

iC( ),

- I+P& e

=f(r+s).

(6.8.54)

Consequently,equation (6.8.53) reduces to Be second-der hypcrbolie equation (6.8.55)

t"+f(r+s)(t,+r,)=O. Once we can solve this equation for t = f ( r . s ) , it can

be used

to

solve the

system (6.8.47)-(6.8.48) for x = x(r, s). Esmnpb 6.8.1 (Riemann'r Invariants for L? Polytropic Gas). A plytropic gas is usually characterized by the pressure-density law p = kpr when k and y are constam. In this case, the velocity of sound c is calculated by using the formula e' = kyp'-'. It follows fmm this result that

In view of the% results. equations of motion (6.8.16)-(6.8.17) assume the form c, + u c,

+(?)

c ", = 0.

(6.8.57)

1 cc,=o.

",+UU,+

(6.8.58)

(7-J

We divide (6.8.57) by ( y - I) and (6.8.58)by 2, then. add them, and subtract the former from the latter to obtain the followingequations (Riemann, (1858)):

-+ ( c + u ) e = 0. ax

at

whae

I

and I

BIC the R i c m m

e

r=-+--U,

at

l

e l r=--+-u=e~n~tant y-I 2 s=

I

y-l

2

ax

e

l

7-1

2

S=---".

y-l 2 Equations (6.8.Mlab) signify that

C

-u)

*-(c

= 0,

(6.8.59ab)

inva"mts given by

u = constant

on 0"

(6.8.6Oab)

dr

-=c+u.

r

&

-= c dr

-",

(6.8.61) (6.8.62)

6.8 Simple Waver md Rirmnn'r Invclrionls

Furthermore,

u = I - - s =consmt,

and

129

e = L ( y - l ) ( r + $ ) = constBnt

confirming the fact that u and c are separately constant. Consequently, the charanerisoCs$=c+u=cmstant

which repnfents two families of straight

lines in the (x. I)-plane. Mathematically. the characteristics exist because the governing quasi-hear equations (6.8.57) and (6.8.58) are hyperbolic. Physically, they representthe socalled wovehrs propagating in both directions with speed c while also connected by the fluid with local spepd I(. If. BS often happens. one of the Riemann invariants is o on st ant everywhere (that is, it takes the same consmnt value on every characteristic line of ~ t sfamily), the resulting solution is said to form a simple wave. This example simply illusrrates Ricmann's onginal general approach to the problem. However. ~tshould be painted out that the Ricmann invariant can be derived directly from (6.8.44ab) by wine: (6.8.43) combined with the sound

The solution of the equations of motion in the above simple wave assume the fOn .(x,t)=

f({), 5=.-(u+c)r.

(6.8.63)

where u ( r O ) = f ( x ) represents the initial data. If f ( x ) > O , u + c will be everywhere positive. and hence. the family of characteristics carrying the simple wave has positive slop in the ( x , I)-plane, and the wave is said to be forword proprrrsinp. The sped of propagation of the forward progressing wave is given by

"+e=~(y+l)u+c,.

(6.8.64)

where a slightly different form of (6.8.61), that is, e =l(y-l)u+constant is used, and c, is the round speed of a onedimensional compressive wave which cannot remain continuous indefinitely, whereas a similar expansion wave can.

Differentiating (6.8.63) combined with (6.8.64) gives

Evidently, both u, and u, tend to infinity BS (6.8.67) which is positive provided that /'(C) c 0 . corresponding to the compressive pan of the wave profile. The discontinuity m u i s first at the inrtvlt (6.8.68)

The solution develops a discontinuity in the farm of shock wwcs. The gas is 1 rest. From the constant value in this simple wave of the Riemann invariant, it IS endent that the greater the speed u the greater the value of c and. therefore. of the pressure p . Hence the largest velocities and pressures move most rapidly through the gas. The peaks of velocity or P ~ C S S Utend ~ ~ to overtake the uoughr, thus. leading to the breakdown of the uniqueness of the solution whenever the initial profile contains a compressive pan. This has already been illusuatcd in Chapter 5 (see Figure 5.4.) a similar situation observed in the case of a general. nonlinear Initial-value problem. As time increases, the initial velocity (or pressure) profile is progressively distoncd, and the part of h e wave profile that was initially eXpmsiVe (velocity increases in the direction of motion) tends to elongate. Ultimately. the compressive part of the wave profile develops multiple-valuedncss and, then, breaks at the point when it h o m e s vertical. This simple analysis confirms that, in inviscid flow. a shoek wave develops in onedimensional unsteady flow as exhibited in photographs included in Figure 6.18

(21

(bl

(6)

Figure 6.18 Development of shock waves at successive instants. (From Lighthill (1956)).

"(I-")

(6.8.74)

w,+-w=o (r+$

Indeed, this is a rclcgrnph equation with a variable crrffkicnt. Following Riemann anslysir (1858). we invoducc the Riernonn junction R(r, 3.p. 6) which must be a function of ( r . s ) . and look for the Ricmann function as a function of a single variable (6.8.75)

More explicitly, we set R ( r , s , p . o ) = g ( r ) so that R satisfies (6.8.74) and hence, g ( r ) can be determined. We also assume that R satisfies some additional conditions z = O for r = p . JS

z =Ofor r = u , and R ( p . u , p . o ) = l . Jr

(6.8.76)

We next substitute R(r,s,p. u ) = g ( r )in (6.8.74) to gencrate the ordinary differentialquauon for g(r) in the form

r ( l - z) g " ( r ) + ( l - Z r ) g'(z)-n(l-

m) g ( r ) = 0.

(6.8.77)

mIhri IS the ~pergcomcrticequarion r(l-z) g"+[e -(I + a+ b)r]g'-& withn=l-n,

g =0

(6.8.78)

b = n and e = l .

The first two conditions in (6.8.76) are automatically satisfied by R ( r , s , p , u ) = g ( r ) ,and the last condition in (6.8.76) needs g(O)=l. The solution of (6.8.78) IS the standardhypergeometric function

R(r,s,p,u)=F(I-n,n,I,-r),

(6.8.79)

where F ( q b, c, z)is the hypergwmetric function defined by

F(n,b. c, z ) = I

+( f)+oob(b+l)L+.... z

++I)

(6.8.80)

21

For the special case in which n is an integer. a mnrfomation r = ( r + s ) - ' ( t , +I.)

(6.8.81)

can be used to solve the onginal quation (6.8.71) which becomes

r" +(

")

r+s

(r,+ r,) = 0

(6.8.82)

233

6.8 Simple Wows and R i c m n ' r lnvorionfr

This shows that. if a solution of (6.8.71) erirfs for any n, then. the use of the mansformation (6.8.81) leads us to find a solution for n+ I . In particular. when n = 0, equation (6.8.71) reduces to the canonical hyperbolic equation ;,=0, (6.8.83) which gives a solution by direct integration as I(,. s)

(6.8.84)

= @(.)+ V(4.

where @and arc arbitrary functions. Using (6.8.81). the general solution of (6.8.82) with n = 0 , that is, of (6.8.71) with n = I, is given by

.(.,.)-

V(r)+v'(s).

(6.8.85)

(r+4

Similarly, for n = 2, we obtain (6.8.86)

.(r, $ ) = ( r + r ) - ' { @ " ( r ) ~ + " ( ~ ) } - 2 ( r + s ) - ' { @ ' ( ~r ')(+s ) } .

In general, when n is an integer, the solution of (6.8.71) is given by

where @(,)and ~ ( s )are arbivary functions. This result is also valid for fractional values of n and can expressed in t e r n of fractional derivatives with n = m+ a, m B nonnegative integer and 0 S a c 1. Consequently, (6.8.87) lakes the form

where the fractionalintegrals (see Debnath. 1595) are defined by , D , < ' - " ' f ( r ) = L i(r-x)-*

W-a)

f(x)dr.

(6.8.89)

L)

Result (6.8.88) is valid for n Z 0 and for n < 0. Then (6.8.88) also holds if m is s e t e q u a l t o m a n d a=". Wc consider another special C B S ~in which n is half an integer. 11 IS convenient to handle this c ~ s ewith Y and e instead of I and s as independent variables where U=I-S

and c = e ( r + s )

(6.8.90ab)

Using these results. we change the variables horn r, s to u, e so that quation (6.8.71) d u c c r t o

234

Kinemtic Wmcs and Specific Real-World Nonlinear Pmblrmr

This is known as the Euler-Poisson-Dorbowequation. If m is an integer and Zn=rn-l.then,equation (6.8.91)becames f"#

-(yr (,= +y

[<)

(6.8.92)

= 0.

This can k mated as a wave equation in m space variables for a problem with spherical symmeuy To check this situation, we take e = +x!' + ..+xml , and write

41:

(6.8.93) end then, assumethat I = f ( x , . xt , . . . x , . u ) = I ( e , u ) . With these assumptions, we obtain VZI =fee

+(?)

(6.8.94)

f<.

men, equation (6.8.92) reduces to the form (6.8.95)

This is a wave quation where the waves uavel with the speed

i(y - I).

f i m p k 6.8.2 (The Piston Problem in n Polytropic Gos). me problem is to determine how B simple wave is produced by the prescribed motion of a prrton in the closed end of a semi-infinite Nbe filled with gas. This is a one-dimensional unsteady problem in gas dynamics. We assume that the gas is initially at rest with a uniform state u=O.p=p,, and e=c,. The piston stans fmm rest at the origin and is allowed to withdraw from the lube with a variable velocity for a time 1,. after which the velocity of withdrawal remains constant. The piston path IS shown by a dotted cuwe in Figure 6.19. In the (x, I ) plane, the path of the piston is given by x = X ( I )with X ( 0 )= 0 . me

fluid velwity Y IS equal to the piston velocity X(t) on the piston x = X(r), which will be used as the boundary condition on the piston. Thsinitidstateofthegasi~givenbyu = u u , p = p o . m d e = c , at f = O in x 2 0 . The characteristic line rothat bounds it and passes through the ongin IS detemined by the equation

Kinematic Wove$ md Spe+

236

Red-World Nonlbeor Problem

(dxldr) = 5. when 5 is a parameter with 5 = co on ro.The r, characteristics ared.w&finedby ( & / d r ) = u + c , s o t h a t < = u + c . Hence,eliminating c from (6.8.97b) gives

" = ")(
(6.8.100)

-co).

Subsmuting this value of u in (6.8.97b). we obtain

= It follows from e = co that

(")<

Y+l

+

2%. y+1

(6.8.101)

and (6.8.97b) with the initial data p = p o and

e' = ykp'.'

+

p

e.1""'

(6.8.102)

<

With = x / t . nsulu (6.8.1W) through (6.8.102) give the complete sduuon of the piston problem in terms of x and r .

Finally, the equation of the characterisuc line r, is found by integrating the second equation of (6.8.99) and using the boundary condition on the piston. W h e n a line r, inkrseeu the piston path at time r = r, then. u = X(r) along it, and the quation becomes

{-

I

x=x(r)+ e + " + ' X ( r ) ( r - 3 .

(6.8.103)

Note that the family r, represents straight lines with slope (dxldr) increasing with velocity U . Consequently, the characteristics arc likely to overlap on the piston. that is, X(r)>Ofmany r. If u increases, so do c, p.and p . so that instability develops. It shows that shocks will b f d in the compressive pan of the dsturbana. Ex@mple 6.8.3 (Shallow Worer Waves). We consider the propagation of whose wavelength is large- compand with the depth h ( x ) . as shown in Figure 6.20. In terms of the horizontal velcdy compnent u = ~ ( x1), and L e freesurface elevation q = q ( x , t ) , shallow water equations ilre given by surface waves.

u,+uu.=-zrl,,

(6.8.104)

[ 4 q + h ) ] , =-7,.

(6.8.105)

238

Kinemolic Waves and Specific Real-WorldNmlimor Problems

characteristics C, : e = u f c , respectively. It may also be observed that the I

Jg(rl+h)

distinct because c = #0 since we assume that 11 > -h. that is, the free surface never coincides with the bottom. I n panieular, for a fluid of eonstent depth h. if, = 0, and the quantity m in (6.8.108) and (6.8.109)1s zero.Thus the functions (u f 2 e ) are constants on the two families of characteristics C, an really

two families of characteristics C, :

fi= u f e . Funher, one of the two families dr

ofcharactensues C, consists of suaight lines along each of which uand c are constant. Consequently, the corresponding moflon can be referred to as a simple wave. As in the care of tk initial-value problem for sound waves treated in Section 6.8, the shallow water wave problem can easily be handled. The only difference is that. in the present situation, the solution will develop what is called a bore or hydraulicjump. In other words, the compressive pan of the water waves will always break. that is, the development of bore or hydraulic jumo in water is very similar to t k development of shock waves in gas dynamics.

ExampIe 6.8.4 (Nonlinear Wnvc Equorion). We consider the nonlinear wave equauon in the form

".

-c'(".)"=

=o,

(6.8.110)

where. as indicated, the velocity c depends OD the slope of u . W e transform this equation into B system of two equations by inmducing u, = v, and u, = W . so that (6.8.1 10) d u e s u) the system of coupled equations v. w, = 0, (6.8.lll)

-

u,-c'(w)w,

=o.

(6.8.112)

We next rearrange the terns of the second equation to write the system in the maoix form

Acewding to (6.8.29). the cigcnvaluc equation for

1 is (6.8.114)

So. the cigenvalues

1M 1= f c ,

(6.8.115)

6.9 The Nonlinror Hyperbolic System ond Riemonn's

239

bvononts

and thc comsponding clgenvectorsare ( e r I). In consistent with the notation used before, we obtain two farmlies of chamctenstics as C,:

A=d"=c;

C.:

t

A=*=-c. I

(6.8.116)

We multiply (6.8.1 1I) by c and, then. add the result with (6.8.112) to obtain

(", + e v.) - .(W,

+ c w.) = 0.

(6.8.117)

Similarly. we multiply (6.8.79) by e and SubVact the result from (6.8.112) to

(u, - e u,)+ c(w, - e W ' ) = 0. Wereplace &)by

(6.8.118)

fi in(6.8.117)and(6.8.1l8)tofind dvfc(w)&+=O

OD

C,.

(6.8.1 19)

W e then Write c(w)& = dF(w) in the 1st equation, so that d [ u f F ( w ) ] = O on C,.

(6.8.120)

This enables us to mboduce Ricmann invariants r a n d s as before by Z r = u + F(w), zs= u- F ( W ) .

(6.8.121sb)

We can now continue the same mathematical analysis following equauons (6.8.44ab), and so,the rcst is left to the reader s an exercise.

6.9

The Nonlinear Hyperbolic System and Riemann's Invariants

One of the simplest nonlinear or quasi-linear) hyperbolic system of evolution equations describing wave propagation has the form

(6.9.1)

U=(::]. and A=("'a*,

"")

4,

with^,,=^,,(^,,^,)

I n this system, the dements a,, of the matrix A depend explicitly on u, and u2,

but only implicitly on x end I through u, and u2. Following the discussion of Section 6 8, the real eigenvaluei are the solutionsof the equation

240

Kinemtic Wove9 Md Spmfic Real-World Nonlinror Problem

IA-UI = 0 .

(6.9.2)

The system will be called strictly hyperbolic if the eigenvalues I , and A, are real and distinct. We denote the left cigcnvectors t, and 4 cornspanding to the eigenvalucs I , and I , , so that

L , A , = I , L,. i=1.2.

(6.9.3)

Multiplicationof (6.9.1) by L, from the left combined with (6.9.3) gives

~ (dr~ + ax~ , ~ ) u i ==l , o2 . , This shows that the operstors

(6.9 4)

- represent differentiation dong ($+ A4

the

respectivefamiliesof chamctcr*fic curves C, definedby

&=a,.

c,:

(6.9.5ab)

d

We represent the differentiation along C, b y d and along C- by to d5 dv uansform (6.9.4) into the form

along

c,,

(6.9.6)

along

c..

(6.9.7)

65

4%l

dv Wewrite l , = ( t , 1 . t , r ) a n d 4 = ( 4 , , ~ toreduse(6.9.6)and(6.9.7)tothe r) form du du 4,-+t,.'=o

d5

along

c,,

(6.9.8)

d5

4,%+&=0 along c.. (6.9.9) dv dv 4 and 4 depend OD u, and u, through the matrix elements a,, =n,(u,.u,)of A. If equations (6.9.8) and (6.9.9) are not exact. they can always be made exact equationsby multiplyingby the integrating factor p . We next multiply by p and integrate them to obtain

We note that the eigenveetors

4, du, Ip

+I. 4%

du, = r(v) along C,,

4,du, + I p & du,

=s(C)

dong C..

(6.9.10) (6.9.11)

6.9 The Nonlinear Hyperbolic System ond Rrcmonni lnvorionrs

241

and s a x arbitrary functions. llese functions r( 7) and ~ ( 4 are ) called Rirmonn invorinnrs of the hyperbolic system (6.9.1). However. it is oot necessarily the same constant an each C,. In other words, r . in general. will vary on C.. Similarly, s is constant on each C-, but will usually vary on each C,. In fact, the two families ofcharacteristics are given by the equations whae

I

c,

:

44,; c.

:

'=A

r dr ' (6'9'Izab) Furthermore, 4 7 ) = constant and s( C) = consmi are shown in the Figure 6.21 In which the initial data for u, and ut are given BS follow^:

+..

0)= +@)* .,(+. 0)= 4 (XJ,

U2(Xo.

0)= fi*(=J,

(6.9.13)

0)= & ( X I ) . (6.9.14) It can be shown that, in general, a solution can be found everywhere in the ( x , r)-planc at which the solution remains smwth. However, the families of characteristics C, themselves depend on the solution. hence, it is oecessary to detmnine the solutionand the charannistities simulmeoudy.

xo Figure 6.21

%(XI.

0 Characteristics C,.

242

Kinmofic Waves and Specific Red-World Nonlimar Problems

An intensting special case exists when one of the Riemann invariants s(e)

is identically constant, that is, $(<)ss,. (6.9.1 I) can be integrated to obtain

= 411) along C,

/&)+/;(uJ

wd

wh-

f,*(u,)

=I.

In this ease. equations (6.9.10) and (6.9.15)

so alowc. =I@ 4,4.

(6.9.16)

4 . 4 and f ; ( r )

This demonstrates the fact that everywhere along a charactenstic C, specified by q = qo =constant, u, and u2 must be constant, because they are the solutionr of the nonlinear system of simultaneous equations

/;(4+/;(4= 4%).

(6.9.17)

f;(%)+/;(%)=%.

(6.9.18)

However. solutions of this system may wt be unique due to nonlinearity, and this fact is assoeiatcd with possible discontinuous solutions as shwk waves. This is a typical feature of nonlinear partial differential equations as explained earlier in various other problems. The solutions u, and 5 .we thus constant along the C+ characteristic specified by setting 1)= 1. Hence, 1,= I , ( u , , y ) ,it follows that the characteristic must be a straight line. W h e n one of the Riemann invariants r o r s is identically constant, the comsponding solution is said to form a simple wave. Simple waves npresent the simplest noncon~tantsolutions of a nonlinear hyperbolic system (6.9.1). T k impnant fact that the system contains neither dispersive nor dissipative terms means that no physical mechanism exists which can prevent the formation of a discontinuous solution or that can permit the propagation of mveling waves. Excmplc 6.9.1 (Riemann’s Invariants of solve the linear initial-value problem #m-c’$,=O,

Wove Equation). We

L? Linear

XER,

with the Cauchy data #(x,O)=f(x), #,(r,O)=g(x)

(6.9.19)

f>O

for

XER.

(6.9.2Oab)

We can factorize (6.9.19) in the form

(”-.”(”+.”) h a a t a x #=O

(6.9.21)

6 9 Tho Nonlinear Hyperbolic System and Rimonn'r lnvadono

243

Introducing two new functions u(x.1) and u(x.1) definedby u=@>

and

v=e4,,

(6.9.22ab)

we can combine (6.9.21) with (6.9.22ah) to obmn two first-order equauans

(Le -Jx J

)

(u+u)=0.

(6.9.23)

E 3("-")=o.

(6.9.24)

+c-

These equations imply thal the quantities u f u remain constant along two

families of characteristic given by C, :?!! = fc. Further, u + u and u - u arc df two Ricmann's invariants so ulat u+ v = eonslant along a CUNC

C+such that

2=e , dr

u

- v = constant along a C U N ~C. such that

= -e. f

I" other words, alongc, : q = x - c t ,

u+w=r(g) u-

(6.9.25)

<

along c. : = x+et.

u =.(<)

(6.9.26)

where r ( 7 ) and $(<) are arbitrary functions of q and<. rcspeclwely, to be determined from the given initial data u ( x . O ) = u o ( x ) and v(x,o)= v,(z). Along the characteristics C+ : x - cr = 7 and C. : x + cf = 6, we obtain u + v = 47)= %(7)+ 4(7).

(6.9.27)

(<)=d<)u.(<).

(6.9.28) Salving these equations gives lhe point of intersection of the lines C+and C. on the (x.1)-plane. Clearly, both u and v are funcuonr of and 7 only, and and 7 only. by definition (6.9.22ab). hence, 0 is a function of Consequently, we can write u-u=s

<

@(c)+

<

@(..I) = Y'(q)= 0(x+cr)+ P(x-cf), (6.9.29) where 0 and Y are arbitrary funelions to be determined from the given Cauchy data (6.9.20ab). It turns out lhat

Kincmotie W o v e and Specific Real-World Nonlinear Problem

244

where p ( x ) = $ , ( x , O ) = u , ( x ) . This is the well-known D'Alembcrt solution of the wave equation. As indicated in Chaptcr 3. this method of solving the wave equation by using the characteristic C U W ~ S C, IE called the method of chnractcristics.This method has also been used to solve quasi-linear or nonlinear equallo"%

The wave equation (6.9.19) with (6.9.22ab) can k expressed in the mamx fmm (6.9.31)

It has k e n shown that ulc values of u and v at the point of intersection of the two characteristics C, and C. can k found in terms of their initial values u ( x , O ) = u , ( x ) and u(x,O) = v,(x). Furthermore, the phase velocities of the waves

fi=f c are found to determine the families of characteristics

C,. It

turns out that thcse phase speeds are just the eigcnvalues of the matrix A. To prove this. we denote the eigenvalue by 1 and the corresponding right eigenvcctor by R. Thus, AR=m. (6.9.32) It follows from this equation that A = A , = f c ( r = 1,2) and the cornspondmg right cigenvectors are given by

(6.9.33)

To investigate the eigenstates ffipffircnted by 9.we write 0,= $ 4 .where $ = $ ( x . t ) is an arbimry scalar function of x and 1. Substituting this in quation (6.9.31) gives the equation for $ ( x . t ) in the form (6.9.34) Thus, solutions of (6.9.31) corresponding to the eigenvalucs I = f c. are

u ~ = $ + ( r - c i ) R +and

U.=&(x+ct)R..

(6.9.35ab)

They represent waves propagating to the right and left with constant velocities f C . respeetmly. According to the superposiuon principle, the general sOIUtlOn of the linear wave equation is

6.9 The Nonlinear Hyprrbolic Syrtcm and Rremam's lnvnrionfs

245

U(x,f) = 0,+ U. = @+(x-cr)R+ + C ( x + c r ) R . .

(6.9.36)

W h e n the initial condition O(x.O)=U,(x) at f = O is specified, equation (6.9.36) leads to @+(x)R, + @.(x)R. =U&).

(6.9.37)

In principle. this equation can be solved, and hence, h ( x ) can be uniquely determined,since R, an linearly independent. Note that the solutlon represents the superposition of two suitably combined cigcnstates of the matrix A mvolved in the equation. To obtam equations (6.9.25) snd (6.9.26) involving the Riemann invariants of the System (6.9.31). we intmducc the left eigenvectors L which satisfy the equation U=U

(6.9.38)

so that L, =(I, TI). Multiplying the equation (6.9.31) by gives

L*($+l*$)U*=O.

4 from the left (6.9.39)

The differential operators in (6.9.39) represent the total de"vative along the characteristics C, : x - c r = c and the characteristics C- : x + c r = q . c m p o n d i n g to the eigenvalws A, = c and I - = -e respectively. Thus, along the eharaetsristics = x - cr =constant and r ) = x + d =constant.

4u*=o. Integrating these cquatlonr and using the eigcnvccton tho same equations as (6.9.25) and (6.9.26).

(6.9.40)

4 = (1.

T 1). we obtain

Example 6.9.2 (Srrady Two-Dimensional Supersonic Fluid Flow). The equations of motion for two-dimensional, irrotational, isentropic flow me (c2-u*)us - u u ( u ~ + u , ) + ( e 2 - U ' ) g = 0 ,

u. - u,

=0 ,

(6.9.41) (6.9.42)

u an fluid velocities along thex-axis andy-axis, respectively, and e is the l a a l sound speed. I n matrix natations, the above equations can be eipnssed in the form A, Ux+B, U, =0, (6.9.43) when u and

Whae

Since A;' e r i m provided c2 -u'

#

0. we multiply (6.9.43) by A;' h m the

left to obtain Uz+AU, =0,

whae A=(-?

ZUV

(6.9.45)

y]. el-vz

The eigenvalues of the problem are the solutions of the equation I A which has the crdicit form

that is. (6.9.46)

which gives two m e

I,, = -

(6.9.47)

Clearly, the roots arc real provided that 4 ' = u' + v2> c1 and hence. the equatlom (6.9.41) and (6 9.42) are hypcrbalie. me local Mach number is defined by M = q/e which is greater than unity ensunng that the flow is supcrronrc. In supersonic flow, the characteristics in the (x,y)-plane are defined by the eqUatl0"S

* = I , , r=1.2. (6.9.48) & These characteristicsare d l e d the Mach lines and can bc determined provided the mavir A;' is nonsingular. This simply m a n s that A, # 0 or c' - ua # 0. If this matrix is singular, then, u = c , that is, the x-component of the fluid velocity is equal to the loeal sound speed.

I I

247

6.9 The Nonlinear Hyperbolic $xmn and Rirmom'r lnwrionrr

The left eigenvclues corresponding to the cigenvalues A,, given by (6.9.47)

ae

t, =(I. A,)

4 =(I, A,).

and

(6.9.49ab)

It follows fromequations (6.9.8>(6.9.9) that the characteristics in the (u. u)plane an given by

r,: * + ~ , d v = o : r - : * + A , *=o, JC

where

4

and

can be Wriftc"

Jt

JV

(69.50ab)

as

arc pmetterr. In terms of these pammcterr, quation (6.9.48) a9

C ' *-A *'

&=O:

* - A *=O. (6.9.51ab) - Jrl 'drl (6.9.50b) and (6.9.51b) leads to the

C :

'a4

Combining (6.95%) and (6.9.51a)

01

reS"It

(6.9.52)

_ du= _ _dy _

du

&

Or. equivalently.

dy . dv = -1.

(6.9.53)

& du

This is the condition of orthogonality of two curves in the (x,y)-planeand (u.v)-plane when they are represented in the same coordinate plane. More precisely. C. and characteristics and C, and characteristics are

r,

r-

mutually orthogonal. The system of quation (6.9.1) or (6.9.45) is called reducible provided the elements of !he cocftilcient matrix A depend explicitly only on the elemcnu of U.The name originates from the fact that the syrlem can be reducedlo a linear system in the new independent variabler; u and v by interchanging the dependent and independent variables. Such a tansformation exists provided the heobian J = u, v, - u, v, is nonzero. Thus it turn8 out that

d u = J d y . du=J& ax JU a'

*. a~

dv=-J- -=-Jd" a JX

JU

(6.9.54)

JU

This tansfmation is called the hodDgmph tmnsfomrion. Por more delzlr, the readerirreferredtoCollrantandFricdrichs(1948).

248

Kincrnorzc Waves and Spmfic Real-World Nodineor Problems

and Potential in n Tmnsmisrion Line). We recall the equations for the current /(I,r ) and potential V(z,1) from problem26 in Erereire 1.12 as LI, + V . + R I = 0, (6.9.55a) fiaolple 6.9.3 (Equations for the Current

CVc+ 1.

+ GV = 0.

(6.9.55b)

In matrix notation, these become A, (5 +A2 Ur+A,U = O

(6.9.56)

W k R

We multiply theequation (6.9.56) by A,.' to obtm

U,+ A Us+ BU = 0 ,

(6.9.57)

Whcn

lle eigenvalue equation IA - 1I I = 0 gives two eigenvaluss Iu = k c , where c = (Lcj;.

clearly, the velocities ofthe two waves Brc C.: * = I , = - c .

C,: * = A , = c , t

(6.9.58ab)

f

The left eigenvalors I,, corresponding to I , , are given by

4 =(cL, I)

and

4 =(cL, - I ) .

(6.9.59ab)

FQ a lossless transmission line. R = G = O , equation (6.9.57) becomer the classical wave equation, and the Riemann invariants become Cl+

01-

v = r( q ) v = S(t)

c,,

(6.9.m)

along c..

(6.9.60b)

along

However. in general, II and G are nonzero quantities, and the equations comsponding 10 (6.9.8)-(6.9.9)which are obtained by multiplying(6.9.57) by 4 and & * m e

249

6.9 7'hz Nodineor Hyperbolic Syrrmm and Riemonni lnvorionlr

(d) 1" - V , = -(Rc I -

V)

along C..

(6.9.61b)

In general, lheie equations am not integrable, but lhey can be integrated in some special cases. For the Heavisidc distortionless cable, _ = _ = ons st ant = k , equations (6.9.61ab) become L C $(d,+V)=-;(cLl+V)

along C,.

(6.9.62a)

;q ( C L I - V)= CLI - V ) along C-. L (!

(6.9.62b)

These equauans can be integratedto obrain c L t + V = A( 7 ) cxp CLI

- v = s ({ )exp

(-: (-!

4 q)

along C,.

(6.9.63a)

along C..

(6.9.63b)

Adding and subtrsetlng leads to the results

V=' 2 [

A(q)exp {:( L

-B({)exp

)

(MI --q

.

(6.9.65)

We can identify the equations for the characteristics as < = x + c t and q = x - c r , and these results can be used to reorganize the exponential factors in (6.9.64) and (6.9.65). so that the final form of the solutions becomes I

=& V='

e x p ( - y x ) [f(x-cr)+g(x+ct)]. exp(-?x)

[f(x-n)-g(x+et)],

(6.9.66) (6.9.67)

wkre f and g are arbiwry functions of their argumenls.

Evidently, the cumnl and the potential represent the waveforms f + g end f - 8 , respectively, which propagate along the Heaviside Vanmission line without change of shape except for the attenuation factor e x p ( - F x ) .

25 1

6.10 Gmemlized Simple Wmrs and Generaliced Riemm’r lnvoriantr

hyperbolic. then. all the eigenvaluer A, are real and distinct, and there are n linearly independenteigcnvalues R,. Hence. the family of characteristics C, is given by

C,:

r=1.2 ,...,n.

-=A,=-(*))/($), &

*

dt

(6.10.4)

Jx

Or, equivalently, *dt+d”.dz=O

m

ax

alongeach C,,

so that du, = 0 and hence,

u,(x,t)=consfBnt

alongeach C,.

(6.10.5)

Since A, =A,(“,), it follows that each member of the n families of characteristicsrepresents svaight lines along which U =constant. On the other hand, if (6.10.4) is written in the form (6.10.6)

it follows that u, is the solution of the nonlinear equation (6.10.6) and this solution and hence, U = U ( u , ) may or may not evolve as B discontinuous solution at a point depending on the initial data and on the form of A,(”,). In B linear problem in which the matrix A is independent of U . 1 and R become specific functions of x and t, and there ere n wave m o d e whose phase velocities an given by the cigenvalues A,,A *.....A”. In panicular, if A is a constant matrix, the linear superposition principle enables us to obtain the general solution in terms of an arbitray set of functions $,(x-A,t) in the form (6.10.7) when $, can be uniquely determined h m the initial condition U ( x . 0 )= f(x)

for all X. If we now set r = k and consider the WI generalized simple wave, equation (6.10.2) must be proportional to the corresponding kth right eigenveetor of A withelements R:’,$’ ,...,R!’SO that (6.10.8)

These equations represent a set of n first-order. o r d i n q diffcrsntial equations which determines the nature of the soluuon (Iacross the sosalled generalized

252

Kincmasc Wovm nnd Speilic ReaCWorld Nonlmeaor Problem

I,-sirnplc WOW. Intcgraung (6.10.8) gives ("-1) lincarly indcpendea relations between the n elements of U.There (n - I) invariant relations along members of the kth family of characteristics C, are called the genembrcd Ll,)-Riernonn invariants that me denoted by

Il,)(U)= constant

for

I=

1.2.....(n - I).

(6.10.9)

These relations hold throughout the generalized I, -simple wave, and they cat be used to determine the nature of the soluuon provided it remains differentiable. Each gcncrnlizcd hl,)-R(lcmann invariant definer a manifold I" the (u,,u, ,....u,)-space on the rnth of which

d

hLl must satisfy the condition

Jii = 0 which can be expressed as

Weconsiderthcgeneralmd Ah,-simplewaves with a parameter (6.10.8) as

Ry

5

to

rewrite

d",=&,

% RIIl = R'; !!!l= (6.10.11) _..

These lead to the results

*=R?). for i=1.2 ,...,n . dS Substituting these results into (6.10.9) gives an equivalent condition (V.Jik))R1" = 0,

(6.10.12)

(6.10.13)

w h a t i = l , 2 , ...,( "-1) and V" represenvlthegradientoperatorwithreipestto u,.u, ,...,u". It is noted here that Lax (1957, 1973) utilized condition (6.10.13). which requires the mutual orthogonality of V" J;,) and R"). fin1 to introduce

UX idea of generalized hl,)-simple waves. In the rest of this section, we follow Jeffrey (1976) to discuss briefly the quadinear hyperbolic system of equations with spatial derivative in the form of a onedrmmsional divergence

3L4+dc(U)=/qU),

at

dx

(6.10.14)

Jg, 28, Jg,... -

a#, J,,,

A=

a,,"

J8, Jg, - ... -

Ju,

..

Ju,

..

...

Jun .

axm ... agm - agn

au, au,

(6.10.15)

154

Krnnmmtic Wows m d SpecQ5c Reel-World Nonbneaor Problem

In conclusion, we m a l l the generalization of B classical solution (see Chapter 5 , Section 5.4) to a weak solution to resolve the difficulty involved 10 the breakdown of differentiability in quasi-linear hyperbolic systems. In the simplest physical prohlem described by the nonlinear scalar equation

?!+d

(,(a)]= 0 at ax the fundon u = U(X,I) is called the weak solurion if, for I > 0,

j j [ U @ , + f (40.l

&h=O

(6.10.18)

(6.10.19)

0..

forevery ICII function )(*,I) E C' which vanishes ouUide some finite region of the half plane I > 0. I1 has already been shown in Chapter 5 , Section 5.4 that both discontinuous classical end weak solutions satisfy the generalized RankineHugoniot condition, end a piwewire C' weak solution is also a piecewise C' classical solution. However. a weak solution is no1 uniquely determined by the inttial data. So, the Lax cntiopy criterion is used to aieenain a unique and physically realistic solution. It is important to point out that weak solutions suffer from a major weakness that a variable change in the original conservation equation leads to different weak solutions. For more information shout weak solutions and s h a k WBYCS. we refer to Smoller (1983) and Le V q u e (1990).

6.11 Exercises If h(x, I) 1s the depth of the water level above impcnncahle rock and if the horizontal seepage velocity u and the slope of the water level satisfy the Darcy law, u = - - k h , . when k is a constant, show that the mass conservation equation is given by

h, = Kh: + K h h,. In Example 6.3.2,the discontinuous initial data can be approximated by continuous data in the form

Find the solution far p(x.I)= f ( { ) , with the value of x in terms of 5 and I. Express the solution p in the terms of x and I , and draw the grsphof p(x,O).

3.

(a)

Solve the traffic flow equation

(6.3.4)

with the flow

q(p)=$(ZW-p)p. end with data representing a viangular hump. whose

center is at the origin given by .rx.0)-n.,={

ZS(3-lx)).

,

IxISl

1.

xl>l @) Ensminc the time evolution of the density profile at three different

I ,and l ( l + B ) .

times t = I ( I - E ) .

15

IS

O < B < I . Show that the solution

15

( 4)

brcaks down for I = L.Draw the graph of the solution p x 15

4.

Show that any solution of the wffiic flow equation P, +cg P, =aP. when e, and 0 are eonstanls, has the form p(x, I ) = exp(n1)f ( x

-s t )

5 . Find the solution of the wffic flow model

P,+.(P)P.

whac c(p)=60-!p.

=o,

with initial data

p(x, o) I[“0{+;(1-lxl)}. 150, 6.

Is

‘1,

xJ>1

Obtain Ule solution of the traffic flow equation p, + c(p)p. = 0. p ( ~ 0=)SOH(-x), when .(p)=60+2p

where I . For a traffic flow model with flow q = A p ( p . - p ) - r p , A,& and I( are constant^, show that the traffic density satisfies the equation P, +A[P(Pm-P)]. = KP,.

Using the nondimensional variables

256

Kinematic Wows ond Specific Real. World Nonlinear Problems

where I is a characteristic length scale aswiated with the initial data, show that the Burgers equation, dropping the asterisks, is given by u,

+"",

= VU",

I/

where v = - I is the dimensionless parameter that is the ratio of the

d i f i s i m lenglh and the characteristic length t . 8. For a traffic flow model with the quadratic flow q ( p ) in the region 0 < p < p . and zem otherwise. that is,

where p . is the maximum traffic density. show that p(x. 1 ) satisfies the quaticm

=o.

P,+.(P)P,

9 . Theeoncentration C(x,1) of a cloud of uniform sedimenung patides in a viscous fluid sausfies the equation (Kyneh, 1952)

where u. and a are constants. If the bottom at x = O is impermeable and the initial concentration is C, for d l x < O , where C, is a constant. discuss the changes of C(x,1) for all I < 0 and t > 0. 10. Apply the method of ehmcteriitier to solve the System of equations

&+&!=o,

m ax

&!+?Lo

m

dr

with the initial data u(z, 0) = e' and u ( x , 0) = e-'.

Show that the Riemann invariants are Zr(~)=ZcoshS: and Zs(q)=Zsinhq. Hence. showthat Ule solutions are (u. v ) = cosh(x- I ) f sinh(x

+1).

257

6.11 Exrrcircr

The shallow water equationsare u,+uu,+gh,=O,

h, +uh. +ha, = 0. where u = u(x. 1 ) is the fluid velocity and h = h(x. I)is the height of the free surface above the honronral bottom. If u = u(h). show !hat

"=*E.

u=*z(c-c,),

dh

what=@;,

e , = ~ , a n d h , i s t h e e q u i l i b t i u m v a l u e o f h.Findthe Riemann invariants of this system Far one-dimensional enisentropic flow. !he Euler equations of motion are p, +up,

+ c*p u, = 0,

",+nu,

+A& = 0 ,

s, + u sr = 0 , where p = J(p, S),u is the velaity, p is the density. p is the pressure. and S is the entropy. Show that this system has three families of characteristics given by

C.:fi=u. I

when c*

C + : f i = u + c , and C.:fi=.-c, dr

t

=I"),

=constant.

dp Show that the second-orderequation a $"+Zb $ * + c $- = 0 ,

[:i].

where a, b, e are real functions of $* and in the matrix form (6.9.31)where A =

0,. C M be reduced to the system

Show that eigenvalues A are the roots of the quadratic equation o d ' - Z b l + c = O . Henee,show!hat!h~lefteigenvenonare

Kinemtie Waves Md Specific Real-World Nonlincor Problem

258

where the indices I and 2 in L corresponds to the indices 2 and 1 in I , respectively. 14. If a. b, and e are constanls and a' = 4b, then. the linear telegraph equation u. -c' u,, +a",

+ bu = 0

can be factorized into two first-wderequations. Direuro the characteristics of

the two first-oderequations. 15. An observer a1 a position x =x(t) moves wi!h !he traffic. Show that the rate of change of the Wffic density measured by !he observer 1s given by

[

dp = .-"Y) dp, dt

dp

ax

where u is the velocity of the car. 16. Show that equations (6.8.106) and (6.8.107) can be written as Uc+ A l l 2 = B , when

I , , = u f e and the corresponding left eigenveftorsan l , = ( l , Z ) and 4=(1,-2).Derivetheequationsforthe dinctionaldcrivativcs of (uf Ze) along the C, characteristics. Hcncc. find the% equations when ti, = gh = eonrtant = rn. Show that eigenvalues are

17. Show that !he nonlinearwave equation c2

(+.I@,, - 9, = 03

where c is an even function of

ex,can be wntten as

Us+ A g = 0, when

Hence. show that the eigenvaluer A,,, and the corresponding left eigenvectors 4 , me given by A,,=Tc

and

l,x=(e.fl).

259

6.11 Exrrcires

Show that the Riemann invariants are .(?)and$(<)

da

given by

and m ( u ) - u = s ( { ) along * = + c ,

m ( u ) + v = r ( q ) along -=--c.

whcrc m(u) isdefinedby m ( u ) = l c ( u ) d u

18. Express (Jeffrey and TaniuU 1964)equations (6.8.16)and (6.18.17) in the matrix form U,+ A (5 = 0. where U and A an given by (6.8.19).Show that the eigenvalucs A , , and the corresponding left cigcnvectors are given by I, = u f c and

=

(;,-

1

fl .Hence, show that

da

u+m(p)=r(q)

along C, : - = u + c .

u-m(p)=-s(<)

along c . : - = u - c , t

f

da

where m(p) isdefined by

Discuss the simple wave solution and discontinuity at the cusp of the of chsractenstlcs.

envcl-

19. Using Example 6.8.3 in a fluid of constant depth (€Ir =O), show that the Riemann invariants r a n d s an

c,

u+2c=r(q)

al0"gcharacteriSticJ

u-Zc=s({)

alongcharaclcristics C- : f i = u - c .

Hence. derive

:d"=.+c,

Use the 1 s t two rcsul6 to obtain the E " l ~ ~ - P ~ i ~ ~ equation ~ " - D for ~ ~ b ~ " ~ r(r,s)

in the form Z(r-s)1,,+3(r,-r,)=O.

Use Exercise 16 to show that equations (6.8.106) and (6.8.107) can be writtenin the conicrvatm form

Show that the Tricomi equation U"+Z"

=o

has the matrix form

U, + AU3 = 0. W

k

Solve the eigcnvalue equation I A - U l = O

equation is hyperbolic, x < o , r=O, or x < o .

parabolic.

to show that the Tricomi or elliptic according as

The equations of motion for the unsteady isentropic compressible fluid flows are

*+div(pu)

= 0.

md

d"+(..V).+lVp=0. r P w h e n u = ( u , u , w ) istheflowfield, p = p ( p ) . m d c ' = *

isthcsquan dp of the local sound speed. Express the above equations in the matrix form U, + A Ux+ BU, + C p = 0. W

k

261

6.11 Exercises

.'/p

u=

,

I(

0

0

0

O

U

O

0

w

o

o

0

o

w

0

0

u o o

B=

A=

U

.'/p

0

"

0

C=

Show that the equation of motion has the conservationform

W

h

PU

PW

P +P W 2

23. In the one-dimensionalcase of the system of equations in Exercise 22, show that [he mafrir form of the system is U,+ A l l . = 0,

whereU=[)

and A = [

'1.

C'lP u 24. Reduce the second-order nonlinear wave equation for the displacement y(.,r) ofasfring

to the lint-order system of equations u, =c'(l+ u y

",

u, = ",

Real-World Nodbeor Problem

Kinrmotic Wove$ and Spe&

262

where u = y, and u = y ,

Express this system in the mahii form U,+ A U, = 0,

25. Show that the "onuniqueness of !he methd of reduction can be seen for the equation 0"" + C"= +" = 0 to the system

U,+ A U x+ B = 0 , when the two following equivalents u b s t i o n s are used. (i) v = u , . w - u , and(ii) u=u,, w = u , + u = .

I

Show also that the equation A - U I= 0 giver in both eases the genuine eigcnvalues I = i -,

for (i) the redundanteigenvalue I = I , and for

(ii) the redundant eigenvaluc I =

i,

26. Traffic velocity is related to traffle density by u = u ( p ) = urn exp(-p/p,). where urn is the maximum traffic velocity and pa is a reference traffic denaty. Solve the initial-valueproblem p,+c@)p,=O, xaR. f>O.

when e ( p ) = q , ( p ) = q P " ( P ) ] . Plot x

dp and for fired 1.

BgBlnSt

( p / p , ) for O < L , PO

Chapter 7

Nonlinear Dispersive Waves and Whitham's Equations 7.1

Introduction

Historically, the study of nonlinear dispersive waves started with the pioneering work of Stokes in 1847 on water waves. Stokes first proved the existence of periodic wavetrains that are possible in nonlinear dispersive wave systems. He also determined that the dispersion relation involves the wave amplitude. The dependence of the dispersion relation on the amplitude produces significant qualitative changes in the khavior of nonlinear waves. It also introducer many new phenomena in the theory of dispersive waves. not merely the correction of linear results. These fundamental ideas and the results of Stokes have provided a tremendous impact on the subjects of nonlinear water waves, in particular. and on nonlinear dispersive wave phenomena m general. Stokes' profound investigations in water waves can be considered the starting point for the modem theory of nonlinear dispersive waves. In fact. most of the fundamental concepts and results for nonlinear dispersive waves originated in the investigation of water waves. The study of nonlinear dispersive waves has pmeeded at B very rapid pace with nmarkablc developments over the past three decades. This chaptcr is devoted to general mesrments of linear and nonlinear dispersive waves. The initial-value problems of linear dispersivc waves and their asvmDtotic analvsis are brieflv described. Included are Whitham's theom of nonlinear dispersive waves, Whitham's averaged variational principle, and Whitham's nonlinear instability analysis and 81s application to water waves.

..

7.2

Linear Dispersive Waves

We consider a dynamical problem so that B small disturbance + ( x , t ) with referenee to the undishlrbd stable m e is governed by B linear p d a l differential equation with constant eocffieients (7.2.1)

264

Nonlrnear Dispersive Waves and Whiihm's Equorionr

where P i s a polynomial, and 1 and x = ( x , , x l , x , ) = ( x , y , z ) areindependent time and space variables. The term F ( x , l ) represents the action of external forces on the dynamical system and is usually referred to B givenforcing t e n . We seek a plane wave w b t i m of the homogeneous equation

(7.2.2) in the form #(=.r)=Ae=p[i(r.x-wr)].

(7.2.3)

where A IE the amplitude, *=(k,.k.k,)~(k,l,m) is the wavenumber vector and a, is the frequency. Such B solution exists provided that the dispersion reIBU0"

P(-ia,,tk,.ik2,ik,)

I O(m,M )

=0

(7.2.4)

is satisfied. Thus, there exists a direel relationship between the governing equation and b e dispersion relahoo through thc obvious c m p o n d e n c e (7.2.5ah)

This correspondence allows us to obtain the dispersion relation from the governing equation (7.2.Z),and conversely, quation (7.2.2)can be consrmcted from the given dispersion relation (7.2.4). If, in particular, the dispersion relation (7.2.4) can be solved explicitly in terms of real mots given by m = I 2 ( ~= ) R(k,).

(7.2.6)

then then may he several possible mots for the frequency, corresponding to different m&s of propgalion. "he phase velocity ofthe waves is the velocity at which B surface of con~tlnt phase moves It is defined by the relation C=-

0 . Y,

(7.2.7)

w h a t li is the unit vector in the dimtion of the wavenumber vector I(.

Thus. for any p a r t i ~ l a wave r mode a,= a(%), the phase velocity depends on the wavenumbers k,.k,. and k,. In other words, different waves propagate with different phase velocities, and such waves are called dispersive provided the

7.2 tinear D!sprrsrve Wows

265

determinant la'O/dk,ak,l# 0. On the other hand, waves me called mndispersivr if the phase velocily does not depend on the wavenumber. For one-dimensional dispersive waves, the determinant reduces to O " ( k ) # 0 . However. in general. the governing dispersion relation for one-dimensional waves is w = O ( k ) that may give a complex w for a real k. In such a case, the phasc velocity depends not only on the wavenumber. but also on the amplitude of the waves. This means that the amplitude will grow or decay in time according as Im(w) > 0 or < 0. So, the former case leads to instability The group velocity vector for the three-dimensional wave motion is defined by the result (7.2.8) dk

21 dm

For one-dimensional waves, the gmup veloeity C ( k ) = ( d o / & ) .

It is noted that the gmup velocity plays a fundamental mk in the theory of dispersive waves. and IS much more imponant than the phasc velocity. The following an examples of linear dispersive waves: the governing dispersion relation for surface W B W S on water of constant depth h w2 =gktanh I .

(7.2.9)

the K0rtewcg.de Vries (KdV)equation for long water waves: @,+co@,+a@- = O

(ci =gh).

w=cok-akl.

(7.2.1Oab)

the Boussincsq equation for long water waves: @s-ci@s-&m =0,

w'=e:k'(l+~'k'r'.

(7.2.11ab)

ule Benjamin. Bona and Mshony (BBM) equation: @, -co@, -a@, = 0,

w = -c,k(l+ a k ' r ' .

(7.2.IZab)

the Klein-Gordonequation.

@" -e'V'$+d'@

= 0,

)"'.

w =f ( c * d +d*

(7.2.13ab)

the internal waves in B stratified owan an governed by the equation

(7.2.14ab) whne ~ ( x , z , r )npnsenu the s t n a m function and

7.3 lnitiol-Value Problem and Aqmpfofic Solvtionr

167

The phase and the p u p velocities associated with these examples can readily he ealeulaud. For dispersive waves, the former are different from the latter. The general ~oluuonfor any drspersive wave system can then be obtained by the Fourier superposition of the plane wave solution (1.2.3) for different wavenumbers (k.1.m) with the corresponding frequencies dispmion relation (7.2.6).end is represenled by

#(r,r)

=I-II

0 to

F ( r ) ~ ~ p [ i { r . ~ - - R ( r ) f&, }]

satisfy the

(7.2.21)

where the spectrum function F ( r ) is determined fiom appropriate initial or bmndmy conditions. Although (7.2.21) is the exact solution, the principal features of the dispersive waves cannot be described wvlthout an exact or approximate evaluauon of the integral in (7.2.21). We shall discuss this p i n t in the next Section which is concerned with initial value mblcms. I t is relevant to mention here that, in general, the governing equations for dispersive waves arc inherently nonlinear, and the corresponding dispersion relation is also nonlinear in the sense that the frequency a, is not only a function of but also of other parameters. such as amplitude and local propcRic~of the medium. The theory of water waves provides an excellent example of nonlinear dispersion. In the sNdy of linear dispersive systems. the preceding parameters ere assumed to be small, and the governing equations for such Systems are obtamed by linearizing the original nonlinear equations The classical theory of water waves satisfymg Laplace's equation and the lineaned free surface conditions can again be cited as an excellent example of the process of lincariration. However, in his pioneering work based upon averaged variational principles. whitham (1914) gave a new dsscnption of nonlinear water waves and a completely different approach to nonlinear dispersive waves in general. I t has become clear from Whitham's theory that nonlinear dispersion is much more imponant than the corresponding linearized concept and plays a agnificaa role in the general theory of dispersive wave systems.

I.

7.3

Initial-Value Problems and Asymptotic Solutions

We consider the initial-value problem of wave propagation m which the disturbance $(x,t) is given by f(x) at time 1 = 0. me general solution for any particular made of propagation with the known dispersion relauon a,= a(=)is given by the three-dimensional integral

168

Nonlinear Dirpmivc Wows and Whirhorn's &maom

-

9 ( x , t ) = IF(K)exp[i(K.x-R(K)t}]

dK,

(7.3.1)

where j d u nprerentr a line, area. or volume integral, depending on the number

ofdimenrims n=1,2. o r3 . At t=O,mult(7.3.1)givcs

f(=)= 9(r,O)= ~ F ( r ) e x p [ i ( r - x )&, ]

(7.3.2)

w h m F ( K )is the generalized Fourier transform ofthe initial disturbance f ( x ) , EO that

(7.3.3)

Thus. the specrmm function F(K)can be determined from the given initial da b Whatever the particular mitial-value problem may be. the general solution can always be represented by the integral (7.3.1). The numkr ofwave modes of type (7.3.1) depends on the panicular problem. So, the complete solution will be the sum of terms such as (7.3.1) with one integral for each of the solutions 0 = R(K).

Note that the physical interpretation of the integral solution (7.3.1) is not at all obvious from its present form. On thc other hand, the exact evaluation of (7.3.1) is also a difficult task. To investigate the principal features of dispersive waves, it is important to consider the asymptotic evaluation of (7.3.1) for large x and t. It is. then, necessary to resort to arymptotlc methods. We next turn our attention to asymptotic evaluation of (7.3 I ) by using the method of stationary phase (Jones, 1966). According to this method, the main contribution to the integral (7.3.1) for large r 1s fmm the terms associated with the stationary poinu, if any, when the phase function r . x - r R ( r ) is stationary. that is, Wherc

(7.3.4)

It EBD be shown that the dominant eonmibution to the integral (7.3.1) will come from those values of k, = k , ( x , , t ) that satisfy (7.3.4). Thus, the asymptotic solution from one such p i n t statlonary k, is obtained in the standard form

7.3 Inmol-Value Problem and Arympiotic Solutionr

269

where the summation convention is used. The complete asymptotic solution for )@,I) will be the sum of terms like (7.3.5) with one term for each of the stationary wints k, of equation (7.3.4).

The asymptotic solution (7.3.5) can be expressed in the form of the elementary plane W B W solution O(X. I)

- ~e[.(x,.

I) erp

[i ~ ( x ,I)}]. .

(7.3.6)

where Re denotes the real part and the complex amplitude A(., ,I) and the phase function b'(x,.r) are given by

(7.3.8)

O(x,,r)=K-x-ta(K).

with the fact that k,(x,,r) is the solution of equation (7.3.4).

The asymptotic solution (7.3.6) has the same form as the elementary plane wave ~ I u U o n and . dso represents an oscillatory wavetrain. But. in contrast to the elementary solution, (7.3.6) is not a uniform wavetrain because k, IS a function of .r, and I. Moreover. the quantities A and 0 ) = R ( k , ) involved in (7.3.6) are no longer constants. but they are functions of x, and I. It follows from (7 3.8) combined with stationary phase equation (7.3.4) that *=k,+ Jr,

[

1-1

- z = 1 2 ( k r ) - [ ,x

dk,

}

%=k,(x,,I), Jx4

,I

aiI -Jk, -r -O(k,)=w(z,,I). Jk

JI

(7.3.9)

(7.3.10)

This means that JOldx, and -dO/dr still have the significance of a wavenumber and B frequency. They are no longer constants. but functions of x,

and I. Thus. k, = Je/Jz, and 0 ) = - JO/a represent the local wovenumber and the loealfiegucncy and are Still governed by the dispersion nlation (7.2.6). The most remarkable difference between the asymptotic solution (7.3.6) and the elementary plane wave solution (7.2.3) is that the former represents an oscillatory nonuniform wavetrain, but the parameters A , k,. and w arc no longer constants. Indeed,they me slowly varying functions of I, and I in the

271

7.4 Nonhneor Dispersive Wwcr Md Whithm'r Equations

"+.(E)-O. m

(7.4.7)

ax

The solution ofthis equation determines the geometry of the wave pattern. We climinate 0 from (7.4.6ab)to obtain the equation

?++LO.

(7.4.8)

m ax

This is known rn the Whilhnm rguorion for the conservationof waves, where k l~pre~en the s denriry of waves and (u is the flux of waves. Using t k dispsrsian relation (7.4.3). equauon (7.4.8) gives dX+C(k)l=O.

a

(7.4.9)

ax

w hen C(k)= W'(k) is the group velocity. This represents the simplest nonlinear wave (hypsrbolic) equation for the propagation of wavenumberk with group velocity C(k). Since equation (7.4.9) is similar to (5,2,Il), we can use the analysis of Seeuon 5.2 in Chapter 5 to find the gens& IolUtion of (7.4.9) with the initial condition k = f ( x ) at t = 0.In this case, the solution takes the form k(x.t)=

I(<), x = F+tF(<).

(7.4.10ab)

when F('(5)= C(j('(5)).This funher confirms the propagation of wavenumberk with group velaeiry C. Some physicd interpretation ofthis kind of solution has already been diSfwSed in Section 5.2. Equations (7.4.9) and (7.4.3) reveal that 0) also satisfies the first-order. nonlinear wave (hyperbolic)equation

d"+W*(k)do = 0. m

(7.4.11)

ax

It follows from equations (7.4.9) and (7.4.11) that both k and constant on UK characteristic curves defined by = w q k ) = C(k)

0)

remain (7.4.12)

t

in the (x,t)-plane. Since k or (u is constant on each characteristic, the characteristics are suaight lines with slop C(k). The solution for k(x.1) is given by (7.4.lOa.b). Finally. it follows fmm the preceding analysis that any constant vduc of the phase 0 propagates according 10 e(x,t)=constant, and hence.

e,

+($) ex=o,

(7.4.13)

which gives, by (7.4.6ab).

dx=_l=”=, dt

0.

(7.4.14)

k

Thus, the phase of the waves propagates with the phase speed e. On the other hand, equation (7.4 9) ensures that the wavenumber k propagates with group velocity C ( k ) = ( d o / & ) = W’(k). We next investigate how wave energy propagates in B dispersive medium. We consider the following integral involving the square of the w w e amplitude (energy) between any two points x = x, and x = x, (0c x, < 1%): Q(f)

=]I

All dr = 7AA.k

(7.4.15)

(7.4.16)

which is, due to a change of variable x = f W’(k),

= 2 x JF(k)F’(k)dk,

(7.4.17)

what x , = l W ’ ( k , ) . r = 1 , 2 .

When k, is kept fixed as f varies. Q(f) remains constant so that

In the limit, as x2 - x , - 0 , this result reduces to the first-order, panial Bfferentialequation

(7.4.19)

This represents the equation for the conservation of wave energy where IA

IA

12 and

W’(k) are the energy dcnriry and enargyflur. respectively. It also follows

that the energy propagates with -up velnity W’(k). It has been shown that the wavenumber k also propagates with the gmup velocity. Thus, the group velacity plays a double mle.

7.5 Whirhnmk Theoory oJNonlincar Dispersive Wows

213

The preceding analysis reveals another important fact that (7.4.3). (7.4.8). and (7.4.19) constitutea closed set of equations far the three functions k, a,and A. Indeed. these me the fundamental equations for nonlinear dispersive waves and are known as Whirhnm's equntiom

7.5

Whitham's Theory of Nonlinear Dispersive Waves

The theory of linear dispersive waves is essentially based on the Fourier superporitian principle. However, in nonlinear problems, the Superposition pnnciple IS no longer applicable to fonsmct a more general sol~tionfar a wavetrain It follows from the asymptotic solution (7.3.6) that the form of the elementary solution $=Aerp(i8). 8 = r . r - o t , can still beusedtodercribe nonlinear WaveIrains provided that A. a,and I( me no longer consIants but slowly varying functions of x and 1corresponding to fixslow mdulation of the waveuains. For slowly varying waveuains, there exists a phase function e(x.1) so that the local frequency m(r.1) and the wavenumber vector K ( x , ~ ) =k, are defined in terms of 8 by o(r.1)=-8,, k , = 8 , .

Inview of the slow variation of kr and linear dispersion relation in the form

(7.5.lab)

0 . it

seems reasonable to assume the

a=R(k,).

(7.5.2)

We note that, for nonlinear waveuains. R may also depend on amplitude and other parameten. Elimination of 8 fmm (7.5.lab) immediately gives

4+da)=o.

ax

dr

4-3=,, (7.5.3ab) ax, ax,

where the former ISa nlauonship between the wavenumben and frequency. This can be regarded BS an equation of continuity for the phase. Substitution of (7.5.2) into ( 7 . 5 3 ) and use of (7.5.3b) imply dk.+C,(k)

at

w

?%=-dR, ax, ax,

(7.5.4)

k C, = C, ( t )IS called the three-dimensionalgmup vebci@,defined by

(7.5.5)

174

Nonliwar Dkpemve Waves and Whirhorn’s Equofionr

This represents the propagation velocity for the wavenumber k,, An observer traveling with the local group veloeity C , ( k ) of the wavetrain

moves along a path in (x,,r)-space known as a my. From the existence of the phase function +,.I) and the dispersion relation (7.52). it turns out that changes along rays me given by

-dx._-an * dt ax,

&=” dt

h

(7.5.6ab) ’

Equation (7.5.4) can also be wtinen in the characteristic form

These are identical with Hamilton’s equations of motion for a panicle with position x, and generalized momentum k,, where R plays the role of the Hamiltonian. If R IS independent of time, that is. if m = I l ( k , . x , ) , thcn, equation (7.5.6b) imply that the frequency a, remains constant for the wave packet. This is identical with the important fact that, for B dynamieal system with a time-independent Hamiltmian. every motmn of the system carries a constant value of the Hamiltonian equal to the total energy. Thus. the frequency behaves like energy and the wavenumber like momentum. This remarkable analogy between classical (partide) mechanics and quantum (wave) mechanics leads to the well-known duality exploited in quantum phyncs. 10 a uniform medium. aR/ax, = 0,equation (7.5.6a) or (7.5.74 implies that the wavenumber k, is constant on each characteristic, and the charaetcristicr are straight lines in (x,,f)-space. Also, each k, propagates along the characteristics with the constant group velocity C,. Funher, if the medium is independent of time I, dRlh = 0.and thcn,thc frequency 0 is constant on each characteristic. In a nonunifmedium. equstionr (7.5.7ab) show significant diffcnnecs. FinL k, is no longer constanl, but varies st the rate -aR/ax,. Second, it propagates along the charactensuer with the variable group velocity C , ( k , , x , , t ) , end third. the eharaetenbcs are no longer straight lines. It is interesting to observe that the local dispersion relation (7.5.2) with n.5.lab)give~thc~ialdiffenntialequationforphssee(x,.r) in thcform (7.5.8) Thlha IS the well-known Ifmilton-Jocobi equation.me solution of this equation determines the geometry (or the kinematics) of the wavetrain.

275

7.5 Whithem’s Theory of Nonlinear Dispersive W o v e

The preceding Ltnematic theory provides no indication of changes in the amplitude or energy of a wavevain or wavepacket. whitham (1974) initiated further study of the amplitude function (7.3.7) and the related energy density to derive B differentialequation for the amplitude,and U, determine the second role of the group velocity. For the one-dimensional CBSC, the energy density described in (7.3.7) is propodonal to IA so that

12,

(7.5.9) where F ’ ( k ) is the complex conjugat, of F(k).

The amount of energy Q(t) between two poinu Q(t)=

x , and x, is given by

Ix(k)A’dx.

(7.5.10)

whac p(k) is an arbitrary propodonality factor associatedwith thc quare of the

amplitude and cnergy. I n a new coordinate system moving with the group velocity, that is, along ulc rays x = C(k)t ,equation (7.5.10) reduces to the form

Q(r) = Zx I g ( k ) F(k)F‘(k)dk,

(7.5.11)

when n”(k)>o and kl and k, arcdefined by x , =C(k,)f and x2 =C(k,)r. nspsetively. Using the principle of consawtion of energy, that is. the energy between the points x , and x I traveling with the p u p velocities C(k,) and C(k2)remains invariant, it turns out from (7.5.10) that

(7.5.12)

which is, in the limit as x2 -1,-0,

This may be treated 8s the energy equation. The quantities g(k)A’ and g(k)C(k)A‘ represent the energy density and the cncrgyflw, so that they me

276

Nonlrmar Dirprrsivr Wove, ond Whihm's Equalions

proportional to I A I) and C(k)lA I),nspectively. The flux of energy propagates with the group velocity C(k). Hence. the group velocity has the double role, and it is the propagation velocity for the wavenumber k and for the energy

dk)l A r . Wus, (7.5.3a) and (7.5.13) are known as Whithorn's conservotion quotiom for nonlinear dispersive waves. The former represents the conservation of wavenumber k, and the latter is the conservation of energy (or more generally, the conservation of wave action). It IS important to observe that, even in a uniform medium (JD~J., = 0 ) . equation (7.5.3a) or its equivalent form given by (7.5.4) is the most fundamental hyperbolic equation. even if the original equation (7.2.1) for 4 is linear or nonhyperbolic. The con~ervationequations can more rigorously be derived h m a general and extremely powerful approach that IS now known as Whitbrn'r meraged voktiomiprinciple.

7.6

Whitham's Averaged Variational Principle

In all dynamical problems where the governing equations admit uniform, periodic. wavewain solutions,it is generally uuc that the system can be described by the Hamilton variauonal principle

SIIL(u,,u,, .u) &df

=0

t7.6.1)

(7.6.2)

Equation (7.6.2) IS a sefond-order. panid differential equation for u ( x , , ~ ) We . assume that this equation has periodic wavevain solutions in the form U=a(@), @=K.X-Wt, (7.6.4ab) where K and (u are constants and represent the wavenumber vector and frequency, respectively. Since (7.6.2) ir a second-order equation, its solution depends on two arbivary constants of mtegrahon. One IS the amplitude n and the other is the phase shift. Omitting the latter constant, it turns out that solution of

7.6 Whilhom's Awraged Vonolionol P t i m i p k

277

(7.6.2) exists provided that the Uure parameters w . L. and (I are connected by a dispersionrelam D(oJ.Y,~)=o. (7.6.5)

In linear problems with

B

waveuain solution in the form u = O(8)= m",

the dispersion relation (7.6.5) does not involve the amplime a.

For slowly varying dispersive wavetrains, the solution maintains the elementary form u = O[E.a),but 0 . Y , and n me no longer constants, so that B is not B linear function of x, and f. The local wavenumber and local frequency are dekincd by k = - ,JE (7.6.6ab) Jl ' Jx, The quanritier w , k,, and n are slowly varying functions of x, and t corresponding to the slow modulatron of the wavemi". The Whithorn ovcroged Lagrangion over the phase of the integral of the Lapngiao L is defined by

and is calculated by assuming the uniform periodic solution u = O(0.a) in L Whitham first formulated the ovcragcd vorl?rionolp"nciple in the form

SjjX&dt=o to dmve the equations for a,.

(7.6.8)

and a. It IS noted that the dependence of 3 on x and I reflects possible nonuniformity of the medium supporting the wave motion. In a uniform medium, P is independent of x and I, so that the Whitham function 3 ~ . Y ( m , ~ . a However. ). in a nonuniform medium, some additional variables also appear only through their derivatives. They reprerent potentials whose derivatives are important physical quantities. The Euler equations resulting from the independent variations of & and SB in (7.6.8) with X=3(o,r.o) are &:

Y

Po(m,r,o)=0,

(7.6.9) (7.6.10)

The 0-eliminant of (7.6.6ab) gives the consistency equations

Nonlinror Dispersive Wows and W h r t h ' r Equorions

278

4+*=o, dk.-3=o. Jt

Jx,

(7.6.1lab)

J x ~ Jx,

Thw.(7.6.9)-(7.6.11ab) represent the Whirham equations for describing the slowly varying wavetrain in a nonuniform medium and constitute B closed set from which the triad (u , I, and a can be deUrmmed. In linear problems, the Lagrangian L, in general. is a quadratic in u and its derivatives. Henee. if O(@)=oeos@is substituted in (7.6.7), S must always take the form S(o,u.a)= D((u.u)a',

(7.6.12)

so that the dispersion relation (3-= 0 ) must L9Lc the form

D((~,K)=O.

(7.6.13)

We note that the stationary value of Y: is. in fact. zero for linear problems. In the simple ease. L equals to the difference between kinetic and potential energy. This proves the well-known principle of equipartition of energy,stating that average potential and kinetic energies are equal.

7.7

The Whitham Instability Analysis and Its Applications to Water Waves

As already discussed. Section 7.6 dealt with Whitham's new remarkable variational approach to the theory of slowly varying. nonlinear. dispersive WBYCS. Based upon Whitham's ideas and. especially, Whitham's fundamental dispersion equation (7.6.10). Lighthill (1965, 1967) developed an elegant and remarkably simple result determining whether very gradual - not necessarily small - variations in the properties of a wavetrain are governed by hyperbolic or elliptic panial differential equations. A general account of Lighthill's work with special reference to the instability of wavetrains an deep water was described by Debnath (1994). ?his section IS devoted to the Whitham inrtabiltty theory with applications to water WBYOS. Accordmg to whitham's nonlinear theory, Sa= 0 gives B dispersion relation that depends on wave amplitude LI and has the form o=@J(u,a), (7.7.1)

where equations for u and LI an no longer uncoupled and constitute a system of panial differential equations. The first important questmn is whether these equations are hyperbolic or elliptic. This can be answered by a standard and simple method combined with Whitham's conservation equations (7.5.3a) and (7.5.13). For moderately mall amplitudes, we use the Stokes expansion of 0 in terms of k and (I' in the form

219

7.7 The Whrthom lnrrobility Analysis a,,( k)o‘+...

a,= a,&)+

.

(7.7.2)

We substitute this result in (7.5.34 and (7.5.13). replace m’(k) by its linear value a,;(k), and retlin the terms of order a’ to obtain the equations for k and a’

in the form

m;(t).’]

dd + mi(.&)dax.’+

dk+ uJ,(k) d .’= 0 . ax ax

(7.7.3)

= 0.

(7.7.4)

a,;(k)ald”

ar

Neglecting the term O(a‘), these equations can be nwrilten BS

d”+ a,; L+ a,*.hl= 0, a ax ax

(7.7.5)

d”’+ a,; .hl+ zax = 0 , ax

(7.7.6)

at

Thesz describe the modulations of a linear dispersive wuevchsin and reprerent a coupled system due to ths nonlinear disprsion relation (7.7.2) exhibiting the depdencc of (u on both k and a. In mamx form, these equations read

Hence. the eigcnvalues A me the -1s

of the determinant equation

when a,, and b,, are the coefficient mavices of (7.7.7). This determinant equation gives the characterirue vclocitics

A

=(*)=

C(k)= C,(k)*a[a,2(k)a,;(k)]”’ +O(o*),

(7.7.9ab)

where C,(k)= a,;(k) is the linear group velocity, and. in general. o$’(k)+ 0 for dispersive waves. Thc equations arc hyperbolic or elliptic depending on whether m2(k)a,;(k)>0 or c 0 .

Nonlinear Dispcrswr Wows and Whithorn's Epuotionr

280

In !he hyperbolic case,!he charmastics an real, and tht dwhle charactenstic velocity splits into two separate velwitiss and provides a generalization of the gmup velocity of nonlinear dispersive waves. In fact, the chaaftenrte velocities (7.7.9ah) a n used to define the nonlinear group velocities. The Splitting of the double charactenstic velocity into two different velo~iliesis one of the most significant result^ of the Whitham theory. This means that any initial disturbance of finite extent will eventually split into two separate disturbances. This prediction is radically different from that of the lineanzed theory, when an initial disturbance may suffer from a distortion due to dependence of the linear group velocity C , ( k ) = w i ( k ) on the wavenumber k but would never split into two. Another significant consequence of nonlinearity in the hyperbolic casc is that compressive modulations will suffer fmm gradual distortion and steepening in the typical hyperbolic manner discussed in Chapter 6. This leads to the multiple-valued sdulions and hence. eventually, breaking of waves. I n tht elliptic e s e ( w 2 m ~ < 0 )the . characteristics me imaginary. This leads of nonlinear wave propagation. Any small sinusoidal disturbances in r? and k may be represented by solutions of the form e x p [ i a ( x - C ( k ) r } ] . where C ( k ) is given by (7.7.9ah) for the unperturbed

to ill-posed problems in the theory

values of a and k. Thus, when C ( k ) is complex. the above disturbances will gmw exponentially in time, and henee the +odic wavetrains become definitely unSkdJ1.2.

An application of the above analysis to the Stokss waves on deep water reveals that the associated dispersion equation is elliplle in this case. For waves on dccD water. the dawnion relation is

This is compared with the Stokes expansion (7.7.2) to give m,(k) = @ end

o1( k ) = l k '

a.

Hence, m,"w, c 0 . the velocities (7.7.9ab) am complex, and

the Stokes waves on deep water are definitely unsroblc. The instability of deep water waves came a B real surprise to researchers in the field in view of the very long history of the subject. The question of instability went unrecognized for a long period of time, even though speclal efforts have been made to prove the erisknce of a permanent shape for periodic water waves for all amplitudes less than the critical value at which the waves assume a shapererted form. However, Lighthill's (1965) theoretical investigation into the elliptic casc and Benjamin and Feir's (1967) theoretical and expenmenlsl findings have provided conslurive evidencc ofthe inslsbility of Stokes waves on deep water.

For more details of nonlinear dispersive wave phenomena, the reader is n f e d to Debnath (1994).

7.8 Exercises

7.8

181

Exercises

(a) For the plane wave solutions $(x,r)=Aexp{t(h-w)} pmml differenualequation

of the real

$=O.

?")+ -.pi-( I

where P is an odd functron, show that the dispersion IS m = P(k). (b) For a wave packet described by $(..r)=R~[A(~,I)w(=,I)], where A(z,I) represents a slowly varying complex amplitude function, and ~ ( x , I=) erp[i(k,x-m,I)]

mo = P(ko)

denotes the basic carrier wave of frequency

for a given wavenumber ko # 0 , show thal the amplitude

A(x.r) satisfies h e equation

"4 + P'(k,)A,]+lP"(h)A_

= 0.

where P(k) has a Taylor snes expansion about k = ko.

The dispersion relation (Whitham, 1967a) for water waves in an inviseid fluid of arbimry depth & is m=m0(k)+I2,(k) g + O ( E ' ) .

P CQ where m,(k)=(gktanhk&)i. E = i g p o ' a n d

co=ru,o,

=.=*

and where T,=tanh(k&), k, dko charactenstic velociues given by (7.7.9ab) M given by

(a) Explain the significance of the solution for I2,(k)

Show that the

> 0 or < 0

282

Nodinmar D!rprrme W m a m d W h i t h ' s Equarionr

(b) If the critical value for instability is determined by the numerical value of (4)for which i l , ( k ) = O . show that equations are hyperbolic or elliptic according to whether k% < 1.363 or > 1.363.

If the whithamaveraged variational principle is given by

sJJe(-e,,e,)dtm=o, whae

0 = -9,

and k = ex,show that the Euler equation is

dre,-dz

-0

'-

Jx ' Hence. or otherwise. show that the phase function 8 satisfies the secondJf

order,quasi-linearquatian 3 , en-2 e, e, + e, e_ = 0. me Whitham equations for the slow modulation of the wave amplitude L? and the wavenumkr k in the ease of twdmtensional deep water are and :+$=O.

$$]+$(:.)=O w h a t mo = @ and C =

is the p u p velocity. Using Chu and Mci's

2% (1971) disrzrsion relation

derive the following equations for the phase function #(..I), used Chu and Mei's notation W = -24,:

where we have

Show that the second equation can bc integrated with respect to x to obtlin

4 +i4~-lo'-!=L=0.

'

Using 'f'=ocxp(4i4), s c w n g e r equation

2

8

320

show that

Y(x,t) saitsfies the nonlinear

i ~ + ~ y ~ + ~ y l y ~ = o 8

2

Chapter 8

Nonlinear Diffusion-Reaction Phenomena, Burgers' and Fisher's Equations 8.1

Introduction

Many phywal phenomenaBIC dcwnhed hy the interactionof conve~uonand diffusion and also hy Ihc mlcracuon of diffu,ion and rcacliun From a p h y w a l p i n t of n e w the eonweson-diflunon pmccrr and !he ddunon-reacson pracsr are quite fundamental to dcscrik a wide varicly of problems i n physical chemical biological and engincenng SCIC~CCI Some nonlinear. pariial diffcmnual equauun, that m d c l there ~ ~ ( T C P Y Sprovide many ncu maghu into the question of inlcrsetion of nonlmcanly and diffusion It IS well known that the Burgem equation IT the wnplc-1 nonlinear model equation reprcrenling phcwmenn d e w i t 4 by a balanrr k l w c c n C U ~ V C L U U ~and diffurion On the other hand the Firher cquauon )I also anahcr rimplcrt nonlmca mudel cquation which a n w in a vide tanel) of prnhlms involving diffuwn and r e a ~ t m n

To undcniand the above physical processes this chapter IS de\olcd to the qNd) of h t h Bug-' and Fisher 5 equauons and their differens kinds of S O ~ U I ~ O ~ with physical qnificance Special ~IIcnlion I* given to diffuvrc U B Y ~ soIulions and mucling wave soIulions of the Burgcrv and the Fnher cquauonr In adJiuon lo the standard msthcmrtieal mcthd, urul f a ~ d ~ i the n g Burgen and Rrhcr equation, rimilanty methods arc dcvcloped to And the wnilartty soIulions of both linear and nonlinear diffusion equation3 with example, of appllcatlonr

8.2 We d

Burgers' Equation and the Plane Wave Solution l thc diffcmnud furm o f h e nonlinear m n ~ l v a u o ncquaoon

3?+d4=o dt m

(8 2 I)

284

Nonlinear Dzfiion-Reoctron

Phenomena, Burgerr' and Pishcr'r Equotionr

To investigate the nature of the discontinuous solution or shock waves, we assume a functional relation q = Q(p) and allow jump discontinuity for p and q. In many physical problems of interest. it would k a better approximation to assume that q is a function of the density gradient p, as well as p . A simple model is to taLe

4=a P ) - %.

(8.2.2)

where v is a p i l i v e comtant. Substituting (8.2.2) into (8.2.1), we obtain lhe nonlinear d i f i i o n equnrion

P,+.(P)P, = %.

(8.2.3)

where e(p) = Q ( p ) . We multiply (8.2.3) by c'(p) to obtain c,

+ cc, = v.'(p)p_ = V{CY -c"(p)p,q.

(8.2.4)

If Q(p) is aquadralic function in p , then, c(p) is linear in p . and c"(p) = O . Consequently, (8.2.4) kcomcs cc+cc'="c".

(8.2.5)

As a simple model of Nrbulcnce, c is replaced by the fluid velocity field "(1, f ) to obtain the well-known Burgers "puntion as

u,+ug=vu,,.

(8.2.6)

when v is the kinematic viscosity. Thus the Burgers equation is B balance between time evolution. nonlinearity, and diffusion. This is the simplest nonlinear model equation for diffusive waves in fluid dynamics. Burgers (1948) first developed this equation ptimady to throw lighl on turbulence described by the interaction of two opposite effects of convection and diffusion. However, turbulence is more complex in the sense that it is both three-dimensional and statistically random in nature. Equation (8.2.6) arises in many physical problems including one-dimensional turbulence (where this equalion had ill ongin). sound waves in a viscous medium, shock waves in B viscous medium, waves in fluidfilled viscous elatie tubes. and magnetohydrodynamic waves in a medium with finite elu'hieal conductivity. We note that (8.2.6) is pmbolic, whereas (8.2.6) with v = O is hyperbolic. More imponantly, the pmpenier of the solution of the parabolic equation are significantly different from those of the hyperbolic equation

8.2 Burgers' Equntzon m d the Plmi Wove Solution

285

We first solve (8.2.6)for two simple cases: (i) a linearized Burgers equation and (ii) an equation with the linearized convective term, cu, where c is a C0"Sta"t.

In the fmt case, the lincarizcd Burgers equation

y=vu,.

--<x<-.

(8.2.7)

,>O,

with the initial conditions at t = 0 (8.2.8ab) can readily be solved by applying the Fourier transform with respect to final ~olutionof (8.2.7) with (8.2.8ah) for t > 0 IS

X.

The

~ ( ~ , f ) = ~ ( " , + ~ ~ ) - ~ ( " , - ~ ) ~ ~

This shows that the presence of the diffusion term, vu, is

to smooth out the

miUal distribution like (mi;.The solution (8.2.9) tends to constant values u,. as x + and uI.as x + +.a. The absence of the diffusion term in (8.2.6) leads to gradual nonlinear steepening. and eventual breaking. Indeed, equation (8.2.6) combines two opposite effects of nonlinearity and diffusion. In the absence of the diffusion term in (8.2.6). the resulting equation reduces to the

--.

first-order, nonlinear wave equation which admill a progressively distoned wave profile as a solution. Eventually this solution develops a discontinuity as B

shock wave. In the second case. equation (8.2.6)reduces to the linear parabolic equation u, +

e g = Y"".

(8.2.10)

We seek B plane wave solution of (8.2.10) in the form " ( X , 1) = aexp{i(kr

whne ImO) =-vk'

< 0,since

Y>

-m)].

(8.2.1 1)

0

Thus, the solution (82.11) becomes u ( x , 1) = a i * ' , crp[ik(x -ct)].

(8.2.13)

This represents a diffusive wave with wavenumberr k and phase velocity c The amplitude of the wave decays exponentially with time f and the decay time to =(vk')-' which kcomes smaller as k inereese~with fired v . Thus the waves of smaller wavelengths decay faster than the waves of longer wavelengths.

On the other hand, for a fixed wavenumber k . the decay time decreases as Y increases so that waves of a given wavelength attenuate faster in B medium with

286

Nonlinear Dimion-Reaction P k n o m e ~Burgers' . and Fisher's Eguotionr

a larger v . This quantity v may be regarded s a mnsurc ofdtfision. Finally, after a sufficiently long time (r >> ro), only disturbances of long wavelength

will survive, whereas d l waves of short wavelength will decay very rapidly. F a v
8.3

Traveling Wave Solutions and ShockWave Structure

To investigate the effects of nonlinear steepening and diffusion, we seek a mveling wave solution of the Burgers squation (8.2.6)in the form u(x, r) = ~

( 5 ) . 5 = x - ur,

(8.3.lab)

where tho wave speed U is to be determined and "(5) represents the wave form with the property that it tends asymptotically to constant values u,.as 5 + and y. as 5 -,+-.Weassume that u, > u 2 . Substituting (8.3.lab) into the Burgers q u ~ t i o n(8.2.6) gives the ordinary diffcnnual equation

--

-uu'(g+uu'(+

vu"(5)=0.

Integrating lhis squation yields -Uu(<)+'u'-

vu'(e)=A,

where A is a constant of intcgrauan. Or. equivalently,

"'(5) = -?-(u' - 2Uu - 2 A ) . (8.3.2) 2" Clearly, this suggests that u, and y am the mots of the quadraticequation u2 -2Uu - 2 A = 0 . (8.3.3) Hence, U and A are determined from the sum and the product of the mots u, and u1 of (8.3.3). and therefore.

U = ~ ( U , + " ~ )A, = - : " , % .

(8.3.4ab)

Ihus, the wave speed is the nvrmge of the two speeds at symptotic states at infinity. Equation (8.3.2) can now k written as 2v~=u'-2Uu-ZA=(u-u,)(u-u2). d5

As u, > 4.the wave profile u ( c ) dccreascs monotonically with 5 h m the constant value u , , as + to thc constant value u2,as 5 ++- as shown m

--

Figure 8.1. At 5 = 0 . u = i ( u , + + ) . The shape of the waveform (8.3.8) is significantly affected by the diffusion coefficient v . This means that the presence of the diffusion processes pnvenu the gradual distonion of the wave profile. and so. it does not break. On the other hand. If the diffusion term is absent ( v = O ) in (8.2.6). the wave profile will suffer from gradual distonion and steepening. and hence, it would definitely break with development of a shock. Shock waves are formed as B result of B balance between the steepening effect of the convective (nonlinear)term and the smoothing effect of the linear in the equation of motion. The tendency to steepening has bsen diffusive -1 demonmated for plane w a v e by many authors including Riemann (1858) who first introduced the moSt general approach.

I 0 Figure 8.1 Traveling Wave Solotion rz(E), C = x - U t .

The upshot of the above analysis ISthat t k convection and diffusion terns in the Burgen equation exhibits oppasite effects. The former introduces sharp discontinuity in the solution profile, whereas the latter tends to diffuse (spread out) the discontinuity into a smooth profile. In view of this property, the solution IS called the diffusive wave. In the context of fluid flow. Y mpr e sno the kinematic viscosity which measures the viscous dissipation. Multiplying exp[-&(u,

both

numerator and denominator

of

(8.3.6)

by

-4)]. we write the solution (8.3.6) in the form

The exponential factor in this solution indicates the existence of a thin tmnsition lnyerofthichess 6 of the order v/(q - u , ) . This thickness 6 canbe referred to as the shock thickness, which tends 10 zero as v+ 0 for fixed u, and",. Also. 6 inoreases as u, + u, for a fixed v . If 6 is m a l l compared with other typical length scales of the problem. the rapid shock uanrition can satisfactorily be approximated by a direontinuity. Thus. in the limit as v - 0 , we might expect that solutions of (8.2.6) tend to solutions of the nonlinear equation ", +U", =o (8.3.10)

together with discontinuous shock waves satisfying the jump condition (8.3.11)

8.4 The Cole-Hop1 Transfornoam

289

when U is the shock speed given hy

U =I(", +.*).

(8.3.12)

m i s IS in complete agreement with the above analysis.

8.4

The Cole-Hopf Transformation and the Exact Solution of the Burgers Equation

We solve the initial-value problem for the Burgers equation y+uu,=vu_, x s R . f>O.

(8.4.1)

u(x. 0) = F(x).

(8.4.2)

x E R.

Special attention will be given to small values of v or to large values of Reynolds numbers. Hopf (1950) and Cole (1951) independently discovered a transformation that reduces the Burgers equat~onto a linear diffusion equation.First, we write (8.4.1) in a form similar to a conservation law

u,+(~u'-vu.)'=o.

(8.4.3)

This can be regarded as the compatibility condition for a function qI to exist. such that and v u , - L v ' = ~ , .

U=V,

(8.4.4ab)

We substitute the value of u from (8.4.44 in (8.4.4h) to obtain V

V"

-$ v: = V,.

(8.4.5)

Next. wcintmduee v = - Z V l o g Q so that u = I y =-2v-.

'

4

(8.4.6)

This is called the Cole-Hop/ nonsfomtion which, by differentiating. gives

Consequently. (8.4.5) reduces to the linear diffusion equation

2 90

Nonlineoor D5qirrion-Reoction Phcnomnn. Burgers’ ond Fishrr’r E q u f m s

0,= V

(8.4.7)

h .

Many solutions of this equation are well known in the literatun. We substitute the given solution for 0 10 find solutions of the Burgers equation.

We now solve quallon (8.4.7) subject to the initial condition @(x.O)=b(x).

ZER.

(8.4.8a)

Tbir can bc written in terms of the initial value u(x, 0) = F ( x ) by using (8.4.6)

(8.4.8b) Integrating this re~ultgives

}.

o(x.0) = b ( x ) =exp{-LrF(ct)dct 2vo

(8.4.9)

The Fourier transform method or the joint Fourier-Laplace transform technique can bc used to solve the linear initial value problem (8.4.7>(8.4.8a). and hence, thc standard solution of this oroblem is

to where a(<)is given by (8.4.9). We then substitute the value of S(<) rewrite (8.4.10) in the convenient farm

Thus, k ( X * 1 )=

j(y)

--

exp(-&)dC.

(8.4.13)

Therefore, the exact solution of the Burgers initial-value problem is obtlined fmm (8.4.6) in the form

It(% f )

=

i(+)w( -

-5)

-j e v ( - & ) d <

dC (8.4.14) ’

291

8.4 7'hhr Cole-Hopf Tronsformotion

This is clearly single-valued and continuous for all values of 1 . The physical interpretationof this exact solution C M hardly be given unless B suitable simple form of F ( x ) is specified. Even in many problems. an exact evaluation of the integrals involved in (8.4.14) is almost a formidable task. It is then necessary to reson to asymptotic methods. We next consider the following example to investigate the formauon of discontinuities or shock waves. Eromplr 8.4.1 Find the solution of the Burgers initial-value problem with physical significsncc for lhe disconunuow data F ( x ) = A6(x)H(x), (8.4.15)

where A is a constant, 6 ( x ) is the Dirac delta function, end H ( x ) is the Hcaviside unit step function. To find lhc SOIYUOII,first we calculate

(")

(1- <)/2-& = a and writing =R zv which can be intcrprctcd BS the Reynolds number (see Whilhem, 1974). Clearly, by small Y , we mean large R(R>> 1). and the large Y corresponds to small

which is obtained by substituting

R ( R c 1).

Similarly. the intcgral in the denominator of (8.4.14) gives

292

Nonlinaor [email protected] Phcnomcm Burgers' and Fisher's Eguorions

So the final ~olutionu ( x , I) is obmned from (8.4.14) in the form

Thus, this solution takes the similarity form

+ =Rz(v, I)

-

R),

(8.4.17)

where q = ( x / Z f i ) is thesimiloriry variable. Two limiting cases (i) R + O and (id R --f

are of interest. and they are considered below.

Case (i) In the limit as R + O ( v + - ) . the effect of diffusion would k more significant than that of nonlinearity. So, in this case, we can approximate c"

- 1+ R and ecfc(&)

+ 0 in (8.4.16) +.I)

-&exP(-4M).

to obtain

the final solution

2

(8.4.18)

This is thefundamenrol (or source) rolution of the linear diffusion equauan. At my time I , the velocity field ISGaussian. The pcBk height of u(x. I) decrcass inversely with

&.

&,whereas the width of the p e a

(x

- &)

increases with

These are remarkable features of diffusion phenomena.

Case (ii) I n this ease. R + - ( v + O ) . So. the nonlinear effect would dominate over diffusion. and hence, dircontinwty BS a shock is expected to develop. We inuoduce the similarity variable 11 =z/= so that d 6 = q f i , and then, rewrite the ~olution(8.4.16) as

which has the similanty form

8.4 The Cole-Hopf Trmrfomtoaon

293

(8.4.20)

-0

as R + -

fa f c 0 and f > l

when R + -and 0 < f < 1. we use !he asymptotic result erfc(qv%)-+exp(-f'R) f * to obtain the solution from (8.4.21) as

(8.4.23) It tums out that the final asymptoue solution, as R +-, is (8.4.24)

This result represem a shock wave at x = a , and the shock speed IS

and hence, the shack condition is satisfied. whitham (1974) investigated the stmcture of the shock wave solution for large "dues of R. He found two wansition layen-nc at f = I, and the other (weaker) layer at q = 0. For large but finite R. (8.4 22) indicates a rapid tansition from exponentially small values in q > I to u !he trsnsition layer f

-

-

11 in 7 < 1. In

1. solution (8.4.22) can k approximated by

294

Phmomew Burgen' ond Fisher's .!?qmtiom

Nonlinear Di@sion-Remlion u(x, 1 )

-

[I+2&

exp{ZR(g-I)}r'.

(8.4.25)

This shows that this uansition layer has thickness O(R-'). On the other hand, the second layer at g=O does smooth out the

-

discontinuity m the derivativebetween u 0 in 9 < 0 to u

- 5 in 0 < g < 1. It

follows from (8.4.21) that this weaker layer ~ecumfor g = O(

)'fi .

and henee,

(8.4.22) assumesthe appmximate form

This solution for large R is drawn against g =

j . This represents zw

10 Figure

8.2. where

2v

a triongular wave solution

u

is plotted

of the Burgers

quation. As R + c-, the shock layer at x = 0 hecomes a discontinuityof u,.

Fiyre 8.2

8.5

Trimplar Wave Solution of the Burgers Equation.

The Asymptotic Behavior of the Exact Solution of the Burgers Equation

We use the smtiionsry phase appmximstion to examine the asymptotic nature of the Burgers solution (8.4.14). We consider any typical integral

8 5 The Asymprotie Behavior of th h " ( x , I)

t Solution of the Burgem EquofWn

=I

be approximatedasymptotically as, I+

w

(8.5.1)

F(k)cxp{itB(k)} 1 ,

- (9

where F(k) is a given function and e(k)=

k-w(k).

29 5

Integral (8.5.1) cm

for fixed x , by

h k, is a stationary point determined by the solution of the equation B'(k)= T - r ' ( k ) = U .

a c k, < b

(8.5.3)

and e"(k,)#U According to this result, the significant contribution to intcgralr involved in (8.4.14) comes from statlonary pornts for fixed x and 1 , that is, from the mots of the equation

-a_f

<

a<

-F(q+.

(8.5.4)

= {(x. I) is a solution of (8.5.4) representing a stationary point. Invoking the stationary phase formula (8.52). integrals in (8.4.14) have the asymptoue npnsentation as v +0,in the form,

Suppose thal

-jeXp(-;)d5- [GIexp{-$} lherefore. the final asymptotic solution is u ( x , I).*, w

h C(x,r) satisfies the equation (8.5.4).

In other words. the solution assumes the asymptoticform

(8.5.5)

196

Nonlinear D5qusion-ReacrionPhmwnma. Burgers' ond Fisher's Equnrions

1.

u = F(<),

(8 5.6ab)

5 = x -IF ( < )

This is identical with the solution of the Burgers equation without the diffusion cornsponds to the characteristic term ( v = O ) . Here. the stationary point variable in the context of the first-order, quasi-linear equation. As discussed in Chapter 5 , the W B V profile ~ dexrihed by (8.5.6ab) suffers from gradual distmion and stapemng. then leads to multiple-valued solution after a sufficiently long time, and, eventually, it would break with the development of discontinuity as a shock wave. When this State is reached, mathematically, there will be two stationary pints of (8.5.4). and then. some modification is required to complete the asymptotic analysis. This leads to WhiIhrn'J grorncrricol mle of equal area which is equivalent to the shock condition. We shall not pursue this analysis further. and refer to Whitham (1974) and Burgers (1974) for B detailed discussion.

<

8.6

The N-Wave Solution

To find BD N-wave solution of the Burgers equation, we begin with the source solution of the linear diffusion equation (8.4.7) of the form (8.6.1) when

r is a constant.

Substituting (8.6.1) into (8.4.6) gives the solution of the Burgers equation as

(8.6.2) 4 "I

:[ (41]] I.

= - 1+

Lexp2-

.

(8.6.3)

For any time t > 0. this solution is shown in Figure 8.3 and has the N-shaped form. Because of this particular shape of the wave profile, it is known as the Nwave solurion of the Burgers equation. At I= > O, the profile may he taken as the initial profile. Such N waves M observed in many physical situations including problems governed by the cylindrical or spherical Burgers eqoation.

297

Figure 8.3 N-WPW Solution of the Burgers Equation.

ThC area under the negative phase of the wave profile is ule same as that under the positive p h s e of the profile. So, the area under the positive phase of the wave profile is given by f u ( x . I)* = -2V[logQ(x, I)]= ;Zvlog[l

+[I.

We denote the nght-hand side of (8.6.4) mule initial time I= intmduce a Reynolds number R by

A

R = 2” = log

sothat f=[e‘-l),

When R >> 1, eR- I

[‘+El,

(8.6.4)

I,by A and. then. (8.6.5)

and solution(8.6.3)redueesto

-

u(x.1)

e’, so that (8.6.6) reduces to the farm

-

(:]

p+[exp{-R(-&]]].

(8.6.7)

298

Nonitnear D~@mon-RrocrionPhmomno, Burgers' and Fiskr's

-

Eqwrrionr

In the limit, as R + with fixed I, (8.6.7) gives the shock wave solution of UX B u r ~ mmuation in the form

In the limit as I +

-. for fixed v and - (;)(;)'

?,

solution (8.6.2) takes the form (8.6.9)

+&).

U(& I)

Thn corresponds to the dipole solution of the linear diffusion equation.

--

Finally. in the limit as R + 0, en I R , and rerull(8.6.6) gives u(x, I)

- ( ~ ) ~ e x p ( - *2AI)

is identical with (8.6.9) as crpcctcd becaue R

8.7

(8.6.10)

-,0 cornspondsto v -+ - .

Burgers' Initial- and Boundary-Value Problem

We solvc Burgers' equation (8.4.1) in 0 < x c I, I > 0 with the following initial and boundary conditions

+,0) =%sin(?).

0sx
"(0,r) = u ( l , r ) = O , I > 0.

(8.7.1) (8.7.2)

It follows fmm the Cole-Hopf urnsformation, that IS. from (8.4.9) that

so that boundary conditions (8.7.2) BX satisfied.

The standard solution of h e linear diffusion equation (8.4.7) is given by

8.7 Burgers' Initial- md Boundary-Volw Probkm

-

$(x,t)=~+&~exp

299

[-q+%(y!),

(8.7.4)

1-1

ulhen om=

f

lexp[-(2)(L ...9)] -

hr

=exp[-3L),p!!L)

(8.7.5)

zm

ZITV

sd a" =

SjCXP[-( -c0s7)] 7) &)(I

2nv

hr

p).

-%)

= Zcxp[

m(

(8.7.6)

2m

when the above integrals are evaluated by using standard integrals fmm Abramowm and Stcgun (1964). lo(x)andIm(x) me the modifted Besscl functionsof the first kind. 'Ibus. the solution of ths Burgm initial-bmndsry problem is given by

At 1=0, thc value of the denominator in this square bracket IS ex{(-$)cw(7)].

We use this result combined with (8.7.1) and the value

of a ( x . 0 ) from (8.7.3) and (8.7.4) to verify that the solution (8.7.7) sstisfici

the initial condition (8.7.1). We note here that the quantity

- R npmena

(TI-

Ihc Reynolds number. me solution of the linear diffusion equation (8.4.7). subject to the initial and boun~data(8.7.lt(8.7.2),isgivenby O(x.r)=u,cip(-~)sin(~).

(8.7.8)

300

Nonlmmor Difuiorion-Reacrion Phonommo, Burgers' and Fisher's Equntions

The following condusions are in order. First, the nonlinear solution (8.7.7) contains an infinite set of higher harmonics with decreasing amplitudes, whereas the linear solution (8.7.8)contains only the fundamental harmonic. Second, the former solution depends OD the Reynolds number R rather than the initial amplitude u0. We next use the asymptotic expansion of the mdified Bersel function

to find the asymprotic

rolulion for u ( x , I), BS R +

-.

It is interesting to point out that this asymptotic solution is identical with the exact solution. This series S O ~ U U OIS~e r a m e l y rapidly convergent for all values

of

($),and hence, it is easy to calculate the waveform. When I'

term with

>> 1, the

n = 1 in (8.7.9) dominates the series and hence. we obmn u(x. I )

- ( y exp( ) -?) sin( 7).

(8.7.10)

This represents a sinusoidal waveform with an exponentially decaying attenuation. mu,the ultimate decay ofthe solution is dominated by diffusion. Another form of h e solution (8.7.9)can be derived by using the Iacobi theta

funcuondefined by J,(X. T)= I+ZCexp(-nn'T)

cos(2nX).

.=I

where X = E a n d T = Z sothat 21 I' '

[ [F)]

d l o g J , ( X . T)= 2 t ( - l ) " sinh dX "=I

sin (ZnX).

In view of these resulb. solution (8.7.9)becomes "(1,I ) =

(y); { F]l)JI 7). (-I)*'

s.i('

sin(

(8.7.1I)

8.8 Rshrr's Equation and D!q.Twion-Remrion Process

301

This is another form of the solution. As suggested by Cole (1951), this solution is approximately equal to u(z, f)

-f [ t a n h ( e ) - ( F ) ] .

(8.7.12)

This doer not depend on the initial amplitude. Physically, the initial sinuroidal profile for large R suffers from a nonlinear steepening effect near x = 1 because of higher harmonics generated by convection. However, this waveform steepening may be prevented by the effects of diffusion. As time f becomes large, this effect spreads in the entire wave profile, leading to exponential decay of the harmonics according lo (8.7.1I). Finally, for a detailed discussion of the solution of the spatially periodic initial-value problem with general initial conditions, we refer to a paper by Walsh (1969).

8.8

Fisher's Equation and Diffusion-Reaction Process

u,

- vu,

= ku( I-f).

(8.8.1)

where v ( > O ) is a diffusion constant. k ( > O ) is the linear growth m e , and

( -3

r(> 0) is the carqing capacity of the environment. The term f ( u ) = ku I

represents a nonlinear growth rale which is proportional to u for small u. but deneases as u increases. and vanishes when u = c . It corresponds to the growth of a population u when there is a limit k on the size of the population that the habitat can support: if u > f , then f ( u ) < O . so the population decreases whenever u is greater than the limiting value k. This interpretation suggests that Ule habitat can support a cerlain maximum population so that OSu(x,O)IK

far x e R .

(8.8.2)

In recent y c m , the Fisher equation (8.8.1) has bcen used as a basis for a wide va&y of models for the spatial spread of gene in population and for chemical wave propagation. It is pertin~ntto mention recent work on gcnc-culture waws of advance by Aoki (1987). on propagation of chemical waves by Arnold et of.

(1987). and on the spread of early farming in Europe by Ammerman and CsvaliSfona(l971, 1983). The Fisher equation (8.8.1) is a panicular case of a general model equation, called the nonlinear reaction-diffusionequation, which can be obtained, by i n d u c i n g the net growth rate f(x, t, u). so that it takes the form u , - Y u , = ~ ( x , ~ , u ) ,X S R , t > O .

(8.8.3)

The term f is also refemd to a soume or reaction term and it represents the birth-death process in an ecological context. This equation arises in many physical. biological, and chemical problems involving diffusion and nonlinear growth. For example, if a chemically reacting substance is diffusing through B medium. &en. i s concentration "(1.1) satisfies (8.8.3) where f represents the rate of increase of the wbsrance due to the chemical reaction.The temperature distribution u ( x , I ) satisfies (8.8.3) when chemical reaction generates heat at a rate depending on the temperature. other problem described by (8.8.3) include the spread of animal or plant populations and the evolution of neutron populations in a nuclear reactor, where f represents the net p w t h rate. We study the Fisher equation BI a nonlinear model for a physical system involving linear diffusion and nonlinear growth. It is convenient to introdurn nondimensional quantities x ' , 1'. U' &fined by

.:4

where

k-' and K represot the length scale. time sale and population

scale respectively. Using (8.8.4), and dropping the asterisks, equation (8.8.1) !&es the non-dimensional fnm, u,-u,=u(l-u), x s R . t>O. (8.8.5)

In the spatially homogeneous problem, the stationary states arc u = 0 and u = I which represent unstable and stable solutions, respectively. It is then appropriate to look for traveling wave solutions of (8.8.5) for which 0 5 " 5 1. We seek all solutions of (8.8.5) subject to (8.8.2) with c = l such that all derivatives of u vanish as 1x1 +-, and

-

lim u ( x , t ) = l . and limu(x.r)=O, 1 2 0 .

,+

.*-

(8.8.6)

Physically, the first condition implies that the population has its maximum value as x+--, and the second condition represents zero population BI x+ It was pmved by Kolmogorov ct 01. (1937) that. for all initial data of +OD.

303

8.9 Travrlhg Wove Solutions and Stnbilim Anolyds

the type, O < _ u ( x , O ) S l , rR. ~ the solution of (8.8.5) is also bounded for all

xand 1 , that is 0s

U(X,I)

Sl, x s R . f > O .

They also showed that, for the two sets of discontinuous initial data (i)

+.o)={;,

(ii)

"(=,a)=

.)",;:

1.: 1 f(x).

o<x
,

when f(x) is an arbitrary function. the solution of (8.8.5) evolves. as I+-,

into a shocWike traveling wave satisfying the condition (8.8.6) and popagates to the right with minimum chaacteristic velocity cm,"= 2 . Finally. it is imponant to note that equation (8.8.5) is invariant under the + --x.So, it ISsufficient to eonrider waves propapdog to the

h;nnsformaUon x

right only.

8.9

Traveling Wave Solutions and Stability Analysis

The Fisher equatron (8.8.5) is one of thc pmtotyps equations which admits uaveling wave solutions. We reek a traveling wave solution of (8.8.5) in the form u(x, t) = u ( 5 ) ,

5 = x -ct,

(8.9.1)

w h e n the wave speed e is to bc determined and Ihe waveform u ( 5 ) satisfies the

<+

boundary data(8.8.6). as T - . We substitute (8.9.1) into (8.8.5) to obtain lhe nonlmcar,ordinary diffePs"tial equation

""(<)+ ."'(<)+ "(5)- "'(<) = 0 . Writing

!

d5

(8.9.2)

= u. equation (8.9.2) gives two first-orderequations

Or, equivalently.

du = 0." + u,

du = U(" - I)-cu.

d5

d5

(8.9.3e.b)

du -=

u(u-l)-cU (8.9.4) U du This system (8.9.3ab) has a simple interpretation in the P o i n c d phase plane The singulrr points of this system an the solutions of the equations u =o, u(u-l)-eu=O. (8.9.5ab)

Thus there are two singular points ( u , u) = (0,O)and (I. 0 ) which represent the steady states. W Efollow the standard phase plane analysis to examine the nature of the nonlinear autonomous system given by -du =p(.,v). fi=q(u.u).

4

d5

~ h matrix c associated with this system at the critlcd p i n t (u.,

u0) is

In the prescnt problem, the matrix A at (0.0) is

he matrix at (1,O) is

me eigenvalues 1 of the matrix A at (0.0) an the -IS

of the equation

This equation giver the ergenvalues

a=-+

K].

(8.9.6)

The eigenvalues are real. distinct. and of the same sign if the discriminant D = c ’ - 4 > 0 . According to the theory of dynamical systems. the origin is a stable node for e 2 em,, = 2 , and. when e = cm,”= 2, it is a degenerate node. The phase plane trajectories of the equation (8.9.2) for the traveling wavefront ~olulionfor c > 2 M shown in Figure 8.4. ’ k r e is a unique separatnxpning the stable node ( 0 , O ) with the saddle point (LO). If c2 < 4 , elgenvalues an

8.9 Traveling Wave Solurions and Smbiliq Analysu

305

complex with a negative nal part, and hence. the curve is a stable spiral, that is, u oscillates m the vicinity of the ongin.

On the other hand. the eigenvalues Iof the m a h i A at

(1,O) ara the mots

Of the equatlon

I

0-a I

This equation giver eigenvalues

a=

1

+hl=o

"-.

fj e'].

(8.9.7)

These eigenvalues am n a l and of opposite sign, and hence, (1.0) is a saddle P0,"t.

Figure 8.4

Phase plane trajectories lor c > 2 .

Finally, the traveling wave solution "(6) of the Fisher equation for e > 2 , satisfying the boundary conditions (8.8.6).as <+1-. IS a monotonically decrearing function of 6 when its first derivative vanishes, as + &-. Clearly. this corresponds to the srporotrir joining the singular p i n t s (I, 0) and

<

(0.0). as shown in Figure 8.4. A typical traveling wave ~olulion" ( 6 ) for

306

Nonlbror Difmron-ReoctL-n P h m o m u Burgers' and Fisher's Eguofions

e Z Z is shown in Figure 8.5. On the other hand, if c < 2 . there are traveling wave solutions, but they are physically unrealistic since u ( < ) < 0 for some 5 , hause, in this case, u spirsls around the origin. Funher, u +0 at the leading edge with diminishing oscillations about u = 0.

A, Figure 8.5

Traveling Wave Solotion.

We cxmme the stability of the traveling wave solution of (8.8.5) in the form " ( 4I)= U(<).

5 = I - ct ,

whne e is the wave speed.

(8.9.8)

-

Such solution is said to be osymproricolly stable if a small perturbation imposed on the system at time t = 0 decays to zero, as I --f . We inmduee a ewrdinatehamemovingwiththc wavespeed C . that is. < = x - c r . r = r . so that (8.8.5) nduces to the form ",- C"< =I&<< + u(l - "). (8.9.9) We lmk for B perturbed solution of (8.9.9) in the form

"(5, I)= U(5)+E u(C. 1).

(8.9.10)

whae E(O c E << I) ISa small p m e t e r and the second tm representsa small

u( 5. I) is assumed to hsvc compact support in the moving frame, that is. u(<. t ) = O for

permhation fmm the given bade solution (I(<). The pmurbed state

some finite n > O . We mbstitute (8.9.10) into (8.9.9) to find the for the pemrbcdquantity,to order B , u,-c"<~=u<<+(l-2u)". (8.9.11)

1<1tl1 for €qUBtiO"

8.10 Pcnurboiion Solutions of thr Fisher BowdoryVolue Problem

301

We next seek solutionsof this quation in the form u(<,t)=V(<)eip(-k).

(8.9.12)

Substituting this solution into (8.9.1 I) gives the following linear eigcnvalue problem (8.9.13) v"(<)+N(<)+(d+I-2u)v=o, where the growth factor 1 is considered an eigenvaluc. Since the original perturbed state has a compact suppon, we may take the boundary condition on V(<) as V(fo)=O. According to the general theory of eigenvslue problems. the perturbation will gmw in time if the cigcnvalues d m negative. and hence the system will become unstable. On lhe other hand, if the eigenvducs L are positive. then, the perturbation will decay to zero, as I - + - . and hence, the system is asymptotically stable. To reduce thc problem in the standard form, we introduce a new uansformatio"

u(<)=-(<)

+$<)

(8.9.14)

so that the above eigenvalue problem rrdumr to the form

(89.15ab)

q ( e ) = C - ( I - Z U ) 2 2U(C)>O for e 2 2 .

(8.9.16)

This is a standard cigenvalue problem. end its all eigcnvalucr are real and positive provided c 2 2. This means that all small perturbations of kintte extent decay expanenually to zero. as t -. We conclude that the waveling wave solution of the Fisher equation is asymptotically stable. However, the present perturbation analysis is not completely general. The general problem has been shldied by several authors including Hoppensleadl (1975) and Lamon (1978).

-,

8.10 Perturbation Solutions of the Fisher Boundary-Value Problem I h e existence of the traveling wave solution of the Fisher equation was established by using a geometric argument. However. it has not been possible to

308

Nonhnior Diffurion-RcacrionPhmommd. Burgers' and Frsher's 4 w t i o n r

determine the exact or approximate representalion of the solution. We now employ the standard penurbaliunmethod to find the asymptotic solurion of h e boundmy-valueproblem for ~ ( 5 )which . sausfier the differential system

""+ c"'+"(l-")

-- 5

< < ",

= 0.

.(--)=I,

(8.10.1)

(8.10.2)

u(+w)=O.

what e 2 2 . Sibce equauon (8.10.1) IS aulonomo~s,the solaion IS translation-invariant. we cnoose the value " ( 5 )at 5 = 0 to bc any number in the range of u and hence, we tahe u ( O ) = i , which is needed to solve the problem. We introduce a new independentvanable L = f, = &g,

when f = c-' is B small parameter. to

transform(8.10.1) and the boundary conditions info the form

=o,

EU"(Z)+i(z)+"-"*

--
u(o)=l,

"(--)=I,

We seek a pemxbation series crpnsion for u(2. E ) = "&)+ &",(Z)+

"(+") U(L)

in powers of

u 2 ( z ) + . . .,

€2

(8.10.3)

=o.

(8.10.4) E

as (8.10.5)

where u&). u , ( L ) , and u 2 ( z ) are to be determined. We substitute (8.10.5) into (8.10.3) and set the coefficients of various powem of f to be zero to obtain u; +%

-*:

= 0,

wo(o)=I..

",(-")=I,

(8.10.6)

&(+-)

=o.

(8.10.7)

an3

":+",(I

- 2%)+ = ";

u,(-")=u,(o)=U,(+c.?)

0.

(8.10.8)

= 0.

(8.10.9)

The general solulim of (8.106) is given by u&)

=(I-A@)-',

when A = -I.smee u @ ( O ) = i .

Thus the S O I UUO~reduces to the form %(Z)=(l+c')-'.

(8.10.10)

4

+(

-)u,

-&

(8.10.11)

=0

This can be solved directly by elemenmy methods and the solution is gwen by u , ( r ) = 4 1 0 g (I+C.)

{ ]

(8.10.12)

7

Thus, the asymptotic solution of the original problem is given by

whm

z =(:)=f(x-cr).

This represents an asymptotic traveling wave solution far c 2 2. The present problem IS not a singular perturbation problem because u and

tend to finite

dz values in the limit,

as e - 0 , equation (8.10.3) has

a uniform limit and the

regular perturbation method gives an accurate solution in the whole domain.

8.1 1 Similarity Methods and Similarity Solutions of Diffusion Equations Birkhoff (1950) first recognized that Boltzmann's method of solving the diffusion equation with a concentration-dependent diffusion coefficient ISbased on the algebraic symmetry of the equation. and special solutions of this equation can k obtained by solving a related ordinary differential equation. Such rolutionr ace called sirnilnn'fy solurions because they are geomevically similar. He also suggested that the algebraic symmetry of the pMial differential equations can be used to find similarity ~olufmn~ of other panid differential equations by solving associated ordinary differential equations. Thus. the method of similarity solutionshar become a very successfuldealing with the determination of a group of transfornations under which a given partial differential equation is invariant. The simplifying feature of this method is that a similarity transformation of the form w(.~,t)=Pv(q), q = x i ' can be found which can. then, be used effectively to reduce the panid differential equation to an ordinary differential equation wilh 7 as the independem variable. The resuhing ordinary differential

3 10

Nonlinear DiBwion-ReociimPhmomma. Burgers' and Fi&r'r Eguoeons

equation is relatively easy to solve. In practice, this method is simple and useful in finding SOIUUOOS of both linear and nonlinear p d a l differential equations. We illustrate the method of similarity solutions by examples of applications.

Example 8.11.1(Sirnilwiry SoluIions of Linear Digusion Equntions). We consider the classical linear diffusion equation with con~mntdiffusion coefficient K in the form U,'KU,.

OSx<-,

I>O,

(8.11.1)

subject to the following boundmy and initial condrtions

u(O,I)=l,

u(x,t)+O

as x + -

for I > O ,

" ( x . o ) = o for o s x < - .

(8.11.2ab)

(8.11.3)

We innoducs a one-paramcter set of stretching transformations in the (x,~.u)-space&hcdby i=a"x.

i=oPt,

i=n'u.

(8.11.4)

under which equation (8.11.1) is invariant, where (I is real parameter which belongs to an open interval I containing LI = I, a,B. and y BE fixed, real con~mns.Usually the set of transformations in the (x, I. u)-space is denoted by T.,andwswritethcsetexplicitlyas T . : R ' + R ' f a e a c h o ~ l . Thesetofall such mnsformations IT.} form a Lie group OD R3 with an identity element T I .This can be seen BS follows:

The set {To) obeys the composition (multiplication) law T-T,x=T,x,

forall o , b s l .

(8.11.5)

This law is wmmumtive because Td ,= Tb. The assmiatwe law is also satisfied since T.(T,T.) = T,(T,) = T , = (T,)T, = (T.T,)T,.

In view of the fact that T,Ts = T,. = T-, = To f a all n + 0 , T, repscnts the identity transformation.

(8.11.6)

(8.11.7)

Finally. we obtain T.Ta, = T, = T..,T-. This shows that the inverse of To is Ti,.

Clearly,

(8.11.6)

8.11 Lmiiariq Methods mul Similoriry Solul!ons of Dzfiim Equations

ii = and hence,

-

P u , .

iE=

r u , ,

-

u.--Tu.. I Y =07-5",-kn'-~*Uy=a~-~(",-x",).

pwded

/?

E

3 11

(8.1 1.9)

=o'-l=uy'

(8.11 10)

Z a . Hence. equation (8 I 1 1) is invariantunder the vansformatm

i=a'"r.

i=a"x.

for any choice of

OL

i-n'u,

(8.11.11)

and y . The quantities

v =In"(,.

t),

q =X t P ,

(8.11.12ab)

Toprovidcdp=-(y//?)and q = ( a / / ? ) .Thus. the invarianls of the msformauans are given by arc invariant under

v ( q )= ur-"' = U

q = xi-*" = x i

~ P ~ ' ~ ,

. .

'

(8.11.13ab)

Substituting (8.11.13ab) into the original equation (8.11.1)-(8.11.3) gives an ordinary differential equationof the form

The msfonned data am then given by u(0, r) =t"n'u(0) = I and u(-) = O .

(8.11.15ab)

To make (8.11.15a) independentof 1. we require that y = O . Consequently. (8.11.14) (8.11.15ab) become

u(O)=I and u(-)=O. (8.11.17ab) Thus equation (8.11.1) admits the set of transformations (8.11.13ab). which reduces the panid differential equation (8.11.1) to the ordinary differential equation (8.1 1.14). Result (8.1 1.13ab) is called B similarity ironsfomorion. and the new independent variable q is called a stmrloriry variable. Integrating(8.1 1.16) yields the general solution

where Aand B are integrating conitants to be determined by using (8.11.17ab).

It turns out that A = 1 and B = -

f . Thus, the 001ution takes the form ITK

8.11 Similarily Mathods and Similorily Solutionr of DIJwion Equrriona

3 13

pmvided p = ( p - Z a ) / p and q = a / p . Therefore. equation (8.11.20) admits the similarity Vansfonnation given by 2a-8 .(8.11.27ab) u=tPv(q) = t u(q), q = x t

’

’.

Todetermine aandp,weuse(8.11.23)toobtain

which is independent of

t

provided

(alp)=’.

Thus. the similarity

h.anrformations(8.1 1.27ab)assume the form .“(Z,t)=t

..

’u(q),

q=xr

’.

(8.1 1.29ab)

Putting these results into the given equation (8.11.20) gives an ordinary differential equation for u(q) 3(UU’) + q v , + v = o .

(8.11.30)

Integratingthis equation once with respect to q gives 3uu‘+qv=A.

(8.11.31)

where A is a c o n ~ t m tof integration to k determined from the fact that the solution is symmetric about q = O . that is, u.(O,t)=O, and hence. u’(0)=twu~(O,f)=O. Thus the constant A must be zero and hence the resulting equation (8.1 1.31) can be integrated with the boundary condition (8.lI.Zl). that is. u(+-) =0,to obtain the solution

(8.11.32) where c is B constant of integration which can be determined from the condition (8.11.28), and hence,

giving c = ( t y .

(8.11.33) -<

9

3 14

Nonlinear Dij%sion-Reoerion Phenomma, Burgers' and Fisher's Equnrionr

The imponant point about the solution u(q) is that u'( q) doer not tend to zero continuously. as q + -, as was the case for the linear diffusion problem. The solution ~ ( q represents ) a parabola, which intersects. the q axis at q=ic,andioverterisat ( O , ~ c ' ] . Thus the final solution for U(I,I) k a m e s

(8.11.34)

This represents shock-wavelike behavior with propagating wave front at x = x, = cri. The wave front propagates in the medium with the speed

( ~ ) = ~ c ~ ' w h i c h d e c r e a s ewith i time I. This means that the wave slows down as I increases. The solution (8.11.34) shows that u is zem ahcad of the wave, and its derivative has a jump discontinuity at the front. Recent studies of various nonlinear diffusion models discussed. by Murray (1993) and Okubo (1980) for insect and animal dispersal, show that grasshoppers exhibit a dispersal behavior similar to that of the above model.

Thus, the major conclusion of this analysis is that the solation of the nonlinear diffusion problem is significantly different from the smooth fundament1 solution (8.11.35)

of the linear diffusion problem with a point source at x = 0 and I = 0.

E r m p l e 8.11.3 Use !he Boltmano transformation )"('=q

2

6

torcduee

the Donlinear diffusion quation Ju= _ _

&

Jx{

r(u)&,

Jx}

O
f>O,

(8.11.36)

with the boundary and initial dam u(O,t)=%.

I > % u(x,O)=u,,

O<x<-,

(8.11.37)

8.11 Similarity MethDdr and Imilority Solutions o/Di#uion Equottonr

31 5

to an ordinary differential equation with the independent variable q . Hence.find the solution ofthe ordinary differential equauan. Since q='

x L i , we obtain

Equation (8.11.36) beeomes (8.11.38)

The given boundary conditions reduce to and "=a,. as q + - .

"=urn. q=O.

(8.11.39)

To find the solution, it is convenient to write (8.11.38) with i=y,mdi=I

on q=Ointhcform

(8.1 1.40)

We solve this equation subject to the condition i = 0, as q --t -. Integrating equation (8.1 1.40) twice, we ohm"

where A is a constant of inlegration to be determined, so that A

in order that i

e x p [ - I ( T ) d $ ] d $ = 1,

=%, as 11 + -,

(8.1 1.42)

is satisfied and where K = ~ ( isi ) a known

function. The solution (8.1 1.41) sausfies the condition 6 = 1 on q = 0. Considerable attention has heto given to the solution of equation (8.1 1.36) in both semi-infinite (0 < x c -)and infinite (-- < x < -) medm for various kinds

3 16

Nonlinear Dz,9wim.React;on Phmomcw Burgers' and Fixher's Eqwrions

ofconcenoauon-dcpendentdiffusioncoefficients.The reader is ref7 of Crank's book (1975).

to Chapter

~ ~ ~ m p n p8.11.4 lc (Sirnilmiry Solution of o More Generol Nonlinear Diffusion Model). We considerthe following nonlinear diffusion equation

m= whm

u"

J!l

,

(8.1 1.43)

ax[ Z]

i s a constant.

C,

We follow the analysis of Municr el 01. (1981) to obtain the similarity solution of (8.11.43). We tint inmoduce Ule Kirchhoff transformation "*I

I

u=lr"dr=L,

(8.11.44)

n+1#0,

"+I to express (8.11.43) in terms of u as

-= au

~J(n+l)"]'q=

ax

'UK

d'v ax' '

(8.11.45)

where r = ~ ~ ( n + l ) ' * ' . We next introduce a onc-p~~meter set of vansformations i=oPr,

z=n=x,

i=o'u,

(8.1 1.46)

to obtain

=L".xa 6; - C 6-l 6* = ar-'U, - C ."*I

5

U"' U ,

= .7-@( uc- Cu'

(8.1 1.47)

provided ulat y-B

= Ey+ - Za, that is, 1=($)(:-I]. B

n+l

(8.11.48)

The invariants of the transformationsare given by q=~j-@=xt-miP=q,

and ~ i . = 6 i - 7 i u = u r - 7 ~ u = w (8.11.49) .

Thus, equation (8.1 1.43) admits the similarity mansformationin the form U ( ~ , ~ ) = ~ ~ ~ ~ W ( X I ~ " ) .

(8.1 1.50)

3 17

8 I 1 Simrlori~Methods and Similon'ty Solutions of Difwrllon Equottons

Substituting (8.11.50) into (8.11.45) gives

qw'(q)=~$'w"(q)

(+)(F-l)w-(;)

This equation can be integrated by replacing

;-

E = l p "+2

(8.11.51)

and hence.

- _ -"+'sotha1(8.11.5l)ducesto "+2

.-

w " ( q ) = - ~ ~ I'w ( w + - ! - . q w ' ) . "+I whne

'ca

(8.11.52)

= ( n + l ) / x ( n + Z ) .lntegrating(8.11.52)oneewithrsspeetto q gives

q w"*'.

w'(q)=A-Ko

(8.11.53)

where A is an integrating constant. The only case in which equation (8.11.53) admits an explicit analytic solution for w corresponds to A = 0. So, in this case, we integrate (8 11 53) to obtain the general solution wn:l(q) - =B-( (8.11.54) "+I 2

5)f.

when B is a constant of integration. Hence. the solution for v(x.f) follows from (8.1 1.50) and (8.1 1.54) as

Finally, resull(8.11.44) gives u ( x , 1) in the form I

" ( X , f)

= [(n

+I)V]h

=( n + l ) z

n

$1.

b i z -i2(5 "+I )

(8.1 1.56)

Invoking translation and sealmg in f , we obtain the final solution

where band p are arbitrary constants. It is noted that the Ilmllarity ~olutionin

terms of q = z L i is possible only when

a0 = 12. with an arbiwary n.

3 18

Nonlinear D!fiiomReaction Phenomena Burgers' and Fisher's Equlltiom

Erompir 8.11.5 (He01 Transfer in o Superfluid Helium). At a very low temperature, helium reduces to a liquid phase (called superfluid helium or He-I0 with rather unusual properties. One of the most remarkable propenies is that heat tranqan in st~tionaryHe-II 1%dcarrikdwt by the l i w Fw"cr law, but by the nonlinear Goner-Mdlink law

.=+)

mi

(8.11.58)

where q is the heal flux. K IS a kind of thermal conductivity and T is the

absolute temperature. Combined with the heat balance squatwn

S(s)+($)=O

end in suitable units, (8.11.58) giver the nonlinear heat

diffusion equation (8. I1 3 9 )

5 = I and S is the heat capacity per unit volume.

whne

This equation IS invariant under a one-parameter set of transformations (8.11.4) with a=lprovidedZy-3P=-4. The invariant solutions for the tcmperaturc described by the similanty transformationsare given by .-

~ ( ~ . ~ ) = t " f l ~ ( q q) ,= x t D , W

h

(8.1 1.60)

"( q ) satisfies an ordinary differentia1equation y-

d",

[zy

- +q%9v=o.

(8.11.61)

dq

'Ibis equation is also invanant under ulc arsoeisted wnsfmation group f i = p q . and G=p-'v,

O
(8.11.62)

The clamped-flux case is of special interest befause it has k e n investigated experimentally in connection with the stability of superconducting magnets cooled with supemuid helium. The initial and boundary conditions for this special problem are given by T,=-(:Iatx=O,

and T ( x . t ) + O as T(x.O)sO

I+-.

for x > O .

forr>O, (8.11.63ab) (8.1 1.64)

8.12 Nodincar Reocnon-Dsffrsion EquIionr

319

We require that y = I and 0 = 2,so that these conditions become

d(q)=-(:r

u(q)+O, as 9'".

a t q = O and

(8.11.65)

In this CBSC. the ordinary differential equation is given by (8.11.61)with y = I and 0 = 2.

8.12 Nonlinear Reaction-Diffusion Equations Tbe general conservation equation m hree space dimensions is *+v.

F = / ( x , I. "),

(8.12.1)

where F is a generalflux i r m p n due to diffusion or some other praeesser and / ( x , u , I ) IS the source or reaction term. For the ease of general diffusion pmsser,wsean tahe F = - r V u , sothatequation(8.12.1)redueer tothe form *=v~(Kv!&)+/(X,I,U),

where K = K ( X , u ) is a function of

iand

u

(8.12.2)

.

For the ease of several chemicals or interacting species, the vector ",(..I). I = 1.2. ....n , represents concanmations or densities each diffusing with its own diffusion ccefkiicicnt K, and interacting according to the veclm source term f(r,u,r).Ihcn.cquation(8.12.2)baomcs

d" =v.(KvV)+f(X.

1,U).

(8.12.3)

where K represents a diffusivity matrix with no cross-diffusion among the speeles,so that K simply becomes a diagonal matrix. In fact. Va IS a tensor. so that V . r V u becomes a vector. Equation (8.12.3) represents a system of nonlinear di@ieion-rroclion (or inrerocringppulmion-difisim) equations. Turin (1952) used it as an imponant model for the chemical basis of morphogenenr. Such systems have been widely investigated in the 1970s. One simple model equation as a special case of (8.12.3)is *=v.(KvU)+f(U).

where K IS a diagonal mauir.

(8.12.4)

320

Nonlincor Dij%rion-Reoction Phenomena. Burgrrd and Frsher's Equanonr

In particular. when K is independent a f i and u,equation (8.12.4) reduces to the form

m

= KV'U

(8.12.5)

+f(U).

Physically, the components of u r e p s e n t the concentrations of certain species, which are reacting with each other at each point and also diffusing through the medium. Funher, if f ( u ) = 0 , (8.12.5) becomes a system of diffusion equations

IS rcgardcd BS a mathematical model of Ihc LISLI)UI kinelics of iysicms of r a c l i o n For more m i o m a i m . Ihc reader I S refcned 10 Kopcll and Howard (1Y73, who studied plane wave sol~tionsofrracuon diffwon equauonr

In the one-dimensional case with f as a funelion of Y onl) ( X 12 2,

u c can wnlc

Bs

(8.12.7)

where. usually. ~ ( u=)kn', t and r are positive constants.'Ibis equation with f c 0 has been SNdied much more widely than that of the nonzero f . However, it is of interest to examine equation (8.12.7) with functions f(u) with two simple zeros. one at u = 0 and the Mher at u = 1.

0,

We write another version of (8.12.7) with f(u)=au"(l-u") m, and n are positive constlnt~in the form

a,, = k ?(u'*)+an'(l dr

Jx

Jx

-u").

where

(8.12.8)

Pattle (1959) solved this equation without the reaction tern ( a = 0).On the other hand. Newman (1980) obtained a number of traveling wave solulions ~ ( 5 ) .S = r - n , f o r v a n o u s v a I u e s o f ra n d m with n = l . His study shows that, when I > 0,"(5) = 0 at B prescribed positive value of 5 . Funhermore, when I = 0 and rn = 2. the solutm represents an ordinary logistic distribution. According to Newman, this model equation can be applied to problems with growth phenomena. and also to population genetics and combustion pmcesscs. Monmll and West (1973) obtained B number of solutions for various problems involving diffusion and growth

321

8.12 N o d m a r Reocnon-Diffusion Equations

It is convenient to use dimensionless variables x' = rewrite (8.12.6) in nondimenrional form, dropping the asterisks, _=_ Jt

Jx(

u

Jx)

+*-(I-"").

(8.12.9)

With u ( x , t ) = ~ ( < ) , < = x - c l , we obtain the traveling wave solution of the equation

(8.12.10) Several authors including Kaliappan (1984) and Murray (1993) obtained an exact traveling wave s ~ l ~ t mofn(8.12.10) for the case where I = 0 and m = I

with the boundary conditions and u(+-)=O.

"(--)=I

(8.12.11)

Making reference to Murray's (1993) detailed analysis with r = 0 and m = 1, we seek an exact ~olutionof (8.12.10) in the form

" ( c ) = L'

(8.12.12)

where n. b. and s arc positive e~nrtantsto be determined. This solution automatically satisfies the boundary conditions (8.12.11) at 5 = f -. We substitute (8.12.12) in (8.12.10) to obtain the following results s=-

', "

u(O)=u,,

b

=

e

2("+2)

,

.=-

2(n+2)

o="o'-l.

(8.12.13)

(8.12.14)

The second derivative of ~dution(8.12.12) gives the point of inflexion

The above analysis provides a more general traveling wave solution of the power law logistic equation with diffusmn. In panicular, when r = m = 0 and n = 1, (8.12.10) reducer to the Fisher equation which was salved earlier. If we chaose uo =

f , then a = 6 - 1 ,

and hence all quantities b, C.

calculated, and sol~tlon(8.12.12) is known exactly.

5, and u, can be

322

Nodineor Difiiion-Reaction Phenomena. Burgers' ond Fisher's Equarionr

Ifweset n = 2 in (8.12.13)-(8.12.15), then, s = l , b =

5, =O.

and u,

=A.

l,o=l, J?'e = 6

In this case, the solution (8.12.12) reduces to that of the

ordinary logistic equation with diffusion.

When r=Oandnr=n+l,~uation(8.12.9)bccomes

;

_ = _2+u"*l(l-u"). (8.12.16)

Following Murray (1993). we obtain the traveling wave solution (8.12.16) in the form

" ( 5 ) with

5 = ~ - - dofequation

(8.12.17) whens=i, b

Jn+l. = L and c

= L a'

When n = l . equation (8.12.16) is further simplified, and then. s = l ,

b= c=&,

and the resulting solution obtained from (8.12.17) is in a p m e n t

with Newman's (1980) solution in population genetics. This solution also describes the situation in which the reaction term is modified by the coalition growth instead of the logistic growth. Another interesting and useful exact solution was obtained by Murray (1993) for the case I = rn = n = I . This case describes density-dependent diffusion with logistic population growth, which is represented by the equation

a!,= _ _ dr

"*

+u(l-u).

We seek a traveling wave solution U ( with u(-..)=landu(+..)=O.Thus.cquetion ("U')

(8.12.18)

ax( ax)

X . ~ ) = U ( ~ ) , ~ = X - Cof ~

(8.12.18)

(8.12.18)reducesto

+cu'+u(l-u)=O,

when the p rim denotes ths derivative with re~pectto

(8.12.19)

5.

lbmus. the associated phase plane system is given by

8.11 Nonlinear Reocrion-Diffixon Equorionr

323

In (8.12.20b). u = 0 is a singularity which can be removed by definmg a new vanable 5 by

d = u d . so that equations (8.12.2Oab) become d5

du

d5

-=NU,

d5

du and - = - [ c u + v * dS

+"(I-u)].

(8.12.21ab)

Clearly. the second equation is not angular. The singular points in the ( u , u)phaseplanean ( u , ~)=(0.0).(1.0)(O.-e). Aphase-plane analysis shows that (1.0) and (0,- c ) are saddle poinm, whereas (0.0) is a stable nonlmear node because of the nonlinear term uu in (8.12.21a).Murray (1993) described the phase trajectories of (8.12.21ab) for different values of c with graphical repreJentations. Physically, the model equation (8.12.18) indicates that the population disperses to regions of lower density more rapidly as the population increases. The traveling wave solution "(5) with boundary conditions

= I, u(+-) = 0,and c = 1is given by

u(--)

w

d

h

& is the wave h n t and

H ( x ) is the Heaviside unit step function

Several authors including Aronson (1980). Crank (1975). Fife (1979). Ghez (1988), Gurney and Nisbet (1975). GuRm and MacCamy (1977). Newman (1980). and Shigesada (1980) have studied various problems of density-dependent diffusion and their extensions. The above analysis. showing the existence of the traveling wave solutions, can be generalized to more general, nonlinear, diffusion phenomena in which the diffusion caefficient is u', for I # I, or even more general K ( U ) in (8.12.4)with ccnain conditions imposed on ~ ( u )In. general, nonlinear reaction-diffusion interactions radically change the mathematical end physical features of the solutions of many biological and chemical problems. Funher. we write out the diffusion term in (8.12.4)in full to obtain u, = r ( u ) + K ' ( U ) U : + K ( U ) U , .

(8.12.23)

This shows that the nonlinear diffusion term can be thought of as an equivalent convection with velocity h'(u) = -K'(N) a,. If the convection arises as a natural extension of a ~ o n ~ e r ~ a tlaw. m n we can derive an equation with the eonvcetian velocity h'(u) in the form

(8.1 1.24)

Such equations mse in a wide variety of problems in physical, chemical, and biological sciences. It IS imponant to point out that the nonlinear convection ~n reaction-diffuaon equations can have a significant effect on the solutions. This is to be expected since there is another major transport process. namely, linear or nonlinear convection. This process may or may not dominate over diffusion effects. However. If the effect of diffusion is negligible compared with that of convection, convection obviously tends to make pans of the waveform grow steeper and steeper, and solutions develop discontinuities in the form of shock waves. Indeed, the development of shock waves and the bnakmg of waveform are typical nonlinear convection phenomena.

8.13 A Brief Summary of Recent Work with References To understand various physical phenomena described by the interaction of diffusion. convection or relaration, Lighthill (1956) gave a fully self-contained account of the basic physical feaNres of shock-wave development, propagation and decay, the internal structure of shocks, end their confluence with the derivation of a nonlinear coupled system intermediate between the Navier-Stokes equations and Ule Burgen equation. He then discussed approximate solutions of several physreal problems by wing graphical construction of characteristics and the method of steepest descent. He also showed that the Burgers equation is the appropriate one for the investigation of weak planar-wavc phenomena. Subsequently, many authors including Blackstock (1964), Wakh (1969), Benton and Platzman (1972). Crighton and Scott (1979), Parker (1980). Rodin (1970). Lmon (1978). and Lardner (1986) studied initial-boundary-value problems for the Burgers equation and investigated the physical significance of their sohtion~. It is pertinent to mention that Benton and Platzman (1972) have also compiled an exhaustive list of the possible solutions of the Burgers equation and have illustrated the physically interesting ones by isochronal graphs. At the same time, considerable attention has been given to the determination of self-similar ~ o l ~ t i oasn rntermediatc ~ asymptoties for nonlinear diffusion equations (see Bannblan and 2el'dovich (1972), Zel'dovich and R a m (1966.1968). Barenblalt (1979) and Newman (1983)). Many generalizationsof the Burgers equation have been made by several authors including Case and Chi" (1969), Murray (1970ab, 1973). Penel and Brauner (1974). and Crighton (1979). Kricss and Lorcnz (1989) discussed the existence and uniqueness of solutions of boundary value problems associated with Ihe Burgers equmon. Recent developments in the reactiondiffusion equation in general and the Fisherequation in partreular are available in

8.14 Exercises

325

papcn by Johnson (1970). McKean (1975). Barenblatt(1979). Manoranjan and Mitchell (1983). Hagstrom and Keller (1986). Gazdag and Canosa (1914), Mumy (1993). Smoller (1983). Bntton (1986), G r i n d (1991). Dunbar (1983). Loganand Dunbar(1992). andFife 0979). TheFidxrcqumonalsareprerentsa nonlinear model equation for the evolution of B neutron population in a nuclear reactor. Canosa (1969, 1973) showed that the Fisher equation describer a balance between linear diffusion and nonlihear local multiplication. and 11 admits ShoCktype solutions. Gacdag and Canosa (1971) used the pseudo Spectral method accurately to discretize spatial derivative for numerically solving the Fisher equation. Larson (1918) studied a more general Fisher equation and determined lower and uppep bounds of 11s solutions. Ablowtlz and Zcppctella (1979) have investigated the exact IoIUtiOn and PainEve tanscendcnu. They have also shown that equatim (8.9.2) can be transformed into a simpler nonlinear equatmn which admits 10Iution in terms of the Weicntrass p function. For Ule generalized Firber equation with f(u)io placeof " ( 1 - u ) . Kametaka (1976) proved the existence theorem for the traveling wave solution when certain conditions on f ( u ) hold. and he investigated the stability of traveling wave and transient solutions. Finally. we refer to the work of Abdelkader (1982) who extended the resulfS of Ablowitz and Zeppctella (1979) for the generalized Fisher equation of the fam

""+ CU'+"

-u"

= 0.

1 < n < -,

(8.13.1)

-.

2
8.14

Exercises

Determine a set of stretching transformations (8.11.4) under which the following equations are invariant (a)

U" =c'uy,

@) u u , + u : = o ,

(C)

",+ ""' = 0,

(4 ", =

(C)

uu,=uy,

<

(r>O. r>o).

(0

.(+ ;.). U"

e'u.+u,=O,

(.e).

Use = r-er to reduce the reaction-diffusion equation (loner and Sleernan (1983))

Solve this ordinary differential equation with Ule naetlon term

and !he boundary conditions

"(--)=I

and u(+-)=O.

Dixusr the solution for cases (i) c t 2 and (ii) c = 2. Show that the similarity transformation of the Boltemann nonlinear quaUan (8.1 1.20) with boundary and initial conditions (""')_=-.=CO"Smt,

u(x.t)+0

as x + - ,

r>o,

u ( x . 0 ) = 0 for r > O ,

. .

u ( x , I ) = I ' u(q),

XI

',

where u( q) satisfiesthe nonlinear ordinary differential equation

3(uu') +277u'- u = 0. Show !hat the Boltrmann equation (8.11.20) is invariant under the oneparameterset oftransformations i = x c x p ( ~ ) , i = r + s i = u e x p ( a r ) ,

0 c r < -.

-

We boundary condition (Bannhlatt and Zel'dwich (1972)) "(0.I) =Uoexp(I), - - < t

<

also invariant under the above set of transformations when a = 1. Show !hat the most general invariant soluuon has the form 1s

8.11 Exercirrr

327

{ (-L

.

When a = l , u(x.r)=exp(t) v xexp

If it is a solution of

211

(8.11.20). then iU image under the above set of transformations must be a solution, that is. emu(x.r)=

):

u(q)exp(r+r). q = x exp(

e+?)

is a solution. 5 . Show that the similarity solution of Burgers' equation (8.2.6) with v = I is

u ( w ) = (zr +n)-'f(q),

q = (zr+a)i

1.

where /( q) satisfies the nonlinm ordinary differential equation j"-(/-q)f'+/=0.

6 . Show that the similarity solution of the Burgers equation (8.2.6) with v = I is given by u(x. t) = l / ( q ) - I, q = (x +r)

Ji

whne

/"+ f'( 7.

P,

/ satisfies the squation

5- /)+$

= 0.

me Prandtl-Blarius boundary layer problem for a flat plate can be desmbed by equations +uu* = "Yn

U",

u,

+ vy=o

with the baundary conditions ( u . u ) = ( 0 , 0 ) at y = o . x > o .

u=u u=u

as y - - t - , x > o . at x = o ,

y>o,

where U is the incident velocity. Show that the above equations are invariant under the set of transformations i=n'x,

i=ay.

i = n P u , and

U=o-'u,

where a-B = 2 . Show also that the boundary eondltions are all invariant under the above transformations when a = 2 and 0 = 0. Show that the first

Nmlrnnrar Di,9kion.Reocrzon Phenomena, Burgen' m d Fisher's Qwlionr

32 8

boundary conditions are also invariam even if boundary conditions become P U.

two

=I'

0f

O and the last two

8. In terns of the stream function v defined by (u. v)=(v,,- v.) with v = I and U = 1, the Randtl-Blasius equation in Exercise 7 is given by

v,

= v,v,

- v-v,.

Show that the similarity transformation of this equation is . .

q=YX

v(x.y)=x'f(rl).

2

where f ( q ) satisfies the Blasius equation

Zf-

+ f f" = o

with boundary conditions f(O)=O.

and f'(-)=I.

f'(O)=O,

9. The Sparrow ernl. (1970) problem of B flat plate with uniform suction or injection IS obtained by changing the variables

(e,q ) = ( vufi.

YE)

in the Randtl-Blasius equation tn Exercise 8 where vo 15 the eonstant suction velocity. Show that 5 and r) are invariant under the set of transformations introduced in Exercise 7. Show that the simrlarity transformation of this problem with v = l and U = I is given by

v=x'f({,r)), whcnfsatisfiestheequation

Zf,,, +ff.. = e ( f , f,, -f,

fd

with boundary conditions

f,(q=O)=O,

and f + Z e = - S f ,

f,(q+-)=I.

a1

q=O

.

10. In Exercise I(c), show that

__ v(q)=rfl

"(Z.t),

q=.r'1-.

arc invariant undcr(8.11.4). Using the data u ( x , O ) = O , x > O , u ( x , t ) + O .

as x +

-.

I > 0.

and g ( 0 . I ) = -4, r > 0,where q is a constant. show that

p = I and a = 1.Hence, show that 3u"- v( v- qv')-O,

v( q ) satisfies the problem

o < q <-*

329

8.14 Ezercires

and u(q)+O

v'(O)=-q,

as

q+-.

11. InExereise l(0,showthatthequantities

.u ( q ) = t ' u and ~ = X I - "

are invariant under (8.11.4). Hence prove that v ( q ) satxsfies the ordinary differentialequation U""

+.I (;I

+ ;-p)..,

?aU"+;(I

Derive the equation for

5= = P

+(;

- I)

;)3

[

= 0.

-x

2

P

12. Show that equations

(a) u , + u u , + ~ u , = O , a n d ( b ) u , + u ~ u , + ~ u ~ = O ,

are invariant under the wnrformatians i=o'x.

providedthat a = - i and

P = -2for equation (b).

i=a'r.

P=-!

and G = o u for equation (a), and

a=-E

and

Chapter 9

Solitons and the Inverse Scattering Transform 9.1

Introduction

Dispersion and nonlinearity play a fundamental role in wave motions in nafure. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind wiul the development of a venical profile. I n particular. the dispersive term in the Koneweg.de Vties equation daer prevent this from ever happening tn its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the questions of how nonlinearity affects dispersive wave motions. Another interesting feature IS the instability and subsequent modulation of an initially uniform wave profile. To understand these features. this chapter is devoted to the Boussinesq and Koneweg.de Vdes equations and solitons with emphasis on the methods and solutions of these equations that onginated from water waves. Special attention is given to the inverse scattering transform, e~nservationlaws and nonlinear uansformations. BIcklund translurmatiuns and the nonlinear superposition principle, and the Lax formulation and IU KdV hierarchy, Z S (Zakharov and Shabat) Scheme. and AKNS (Ablowitr, Kaup. Newell, and Segur) methd.

9.2

The History of the Soliton and Soliton Interactions

Historically, John Scott Russell first experimentally observed the solitary wave, a long water wave without change in shape. on the Edinburg-Glasgow Canal in 1834. He called it the "grcor wow of franslarion" and, then. rcponed his observations at the British Association m his 1844 paper "Repon on Waves." Thus, the solitary wave rrpresents. not a periodic wave, but the propagatton of a

single isolated symmettical hump of unchanged form. His discovery of this remarkable phenomenon inspired him further to conduct a series of extensive laboratory expenments on the generation and propagation of such waves. Based on his experimental fmdingr. Russell discovered. empirically. one of the most

332

Solitons ond the Inverse Scarrering Trmsfom

imponant relation^ between the speed U of the solitary wave and its marimum amplitude a above the free surfaceof liquid of finite depth h in the form (9.2.1)

U'=g(h+n),

wh en g is the acceleration due to gravity. His description o f solitary waves contradicted the theories of water waves due to G.B. Airy and G.G. Stokes;they raised questions on the existence of Russell's solitary waves and conjecturedthat such waves cannot propagate in a liquid medium without change of form. It was not until the 1870s that Rurrell's prediction WBS tinally and independently confirmed by both 1. Boussinesq (1871) and Lord Rayleigh (1876). From the equations of motion for an inviuid incompressible liquid. they derived formula (9.2.1) I n fact, they also showed that the solimy wave pmfilc (see Figure 9.1) z = q ( w ) is given by

-

q(1.r) = a sech' [ p ( x ut)], where

8'

(9.2.2)

= 30 +{4h2 (h+ a ) ) for any a > 0

q(x Figure 9.1

Although th-e

A solitary

WWL.

authan found the sech' solution, which is valid only if

a c< h , they did not wnte any equation for q that admits (9.2.2) BS a solution. However,Boussinesq did a lot more and discovered several new ideas including a nodincarevolution equation for such long water waves in the form

9.2 The History of the Soliton and Soliton Inlerocrionr where c = @ is the speed of the shallow water waves. This is known

333

as the

Bourrinesq (bidirectbm0 rqution, which admits the soluuon q ( x , t ) = o sech' [(3n/h')"'(x+U1)].

(9.2.4)

This represents solitary waves Waveling I" both positive end negative xdireeuons. More than 60 years later, in 1895, two Dutchmen, D.J. Koneweg and G. de Vties. formulated a mathematical model equation to pmvide an explanation of the phenomenon observed by Scott Russell. They' derived the now-famous equatlon for the propagation of w w e i in one direcum on the surface of water of density p m the form

where X is a ewrdinatc chosen to be moving (almost) with the wave, c = @, E IS a small parmeter,

and

(9.2.6) when the surface tension T

(<
is negligible. Equation (9.2.5) is

known as the Konrwcg-de Vder (Kdv) oqlultion. This is one of the simplest and most useful nonlinear model equations for solitary waves. and it represents the

longtime evolution of wave phenomena in which the steepening effect of the nonlinear term IS counterbalanced by dispersion. It is convenient to inuoduce the change of variables q = q ( X ' . t ) and X' = X + ( c / h ) c t which, dropping the asterisks, allows us to remite equation (9.2.5) in the form,

$4 1

'I,=-'I?+ a 7 m .

(9.2.7)

Modern developments m the theory and applications of the KdV solitary waves began with the seminal work published as a Los Alamos Scientific Laboratory Repon in 1955 by Fermi, Pasta. and Znam on a numetical model of B discrete nonlinearmawspring system. In 1914. Debye suggested that the finite thermal conductivity of an anharmonic lattice IS due to the nonlinear forces in the springs. This suggestion led Fermi, Pasta, and Ulam to believe that a smooth initial state would eventually relax to an equipanition of energy among all modes because of nonlinearity. But their study led to the striking conclusion that there is no quipartstion of energy among the modes. Although all the

Solirons Md the Inverse Scorlenng Tronrfom

331

energy was initially in the lowest modes,after flowing back and fOah among various low-ordermodes, it eventually returns to h e lowest mode, and the end stace is a series of recurring states. mlts remarkable fact has become known as the Fern-ParrO-Ukm (FPW recurren~ephanornsnon. This curious result of the FPU experiment inspired Manin Kruskal and Noman Zabusb to formulate a contilluummodel far the nonlinear mass-spring system to understand why recurrence occumd. In fact. they considered the mitialvalue problem for the KdV equation, u, +U"" +6u, = 0 , (9.2.8) when

.=(!I,

I is a typical horizontal length scale, with the initial

condition u(x.O)=cosnx.

(9.2.9)

OSXS2,

andthepenodic boundary conditions withpcriod2, sothat u ( x , r ) = u ( x + Z . I )

for all I . Their numerical study with fi=O.O22 interesting results, which are shown 10Figure 9.2

produced

B

lot of new

4

2 1

0 -1

Figure 9.2 Development of solitary waves: (a) initial profile at t = O , (b) profile at I = $ , and (r) wave profile at t=(3.6)=" (From Zabusky and Krusksl (1965)).

They observed that. mibally, the wave steepened m regions where it had a negative dope. a conrequcnce of the dommant effccs of nonlinearity OYU the

9.2 The Hislory of the Soliton ond Soliton lnlemclions

335

dispersive term, S u m . As the wave steepens, the dispersive effect. then, becomes significant end balances the nonlinearity. At later times. the solution develops a sene$of cighr well-defined waves. each like sceh' functions with the taller (faster) waves ever catching up and overtaking the shorter (slower) waves. These waves undergo nonlinear interaction according to the KdV equation and, then, emerge from the interaction without change of form and amplitude, but with only a small change in their phases. So the most remarkable feature IS that these waves retain their identities after the nonlinear interaction. And another surprising fact IS that the initial profile reappears, that IS, very close to the FPU recurrence phenomenon. In view of their preservation of shape and their resemblance to the particlelike character of these waves, Krurkal and Zabusky called these solitary waves solirons (as photon. proton. ctc.). Hmmeally, the famous 1965 paper of Zabusky and Kruskal marked the birth of the new ~onceptof the soliton, B name intended to signify paniddike quantities. Subrequently. Zabusky (1967) confirmed, numerically, the actual physical interaction of two solitons, and Lax (1968) gave a rigorous analytical proof that the identities of two distinct solitons are preserved through the nonlinear interactmn governed by the KdV equation. Physically. when two ~olitonsof different ampliNdeS(and hence. of different speeds) are placed far span on the real line, the 1a11er (faster) wave to the left of the shorter (slower). the taller one eventually catches up to the shorter one and. then, overtakes it. When this happens, they undergo a nonlinear interaction according to the KdV equation and emerge from the interaction completely preserved in form and speed with only a phase shift. Thus, these two remarkable features, (i) steady progressive pulselike solutions and (ii) the p t t ~ e r v a t i ~ofntheir shapes and speeds,confirmed the paniclelike property of the waves and, hence. the definition of the Soliton. Subsequently. Gardner et ol. (1967, 1974) and Hirota (1971, 1973a.b) eonsrmctcd nnelyticol solutionr ofthe KdV equation that provide the description of the interaction among N SOlitonS for any positive integral N. Experimental confirmation of 10liton~and their interactions has been provided succes~fullyby Zabusky and Calvin (1971). Hammack and Segur (1974), and Weidman and Marwonhy (1978). Thus, these discoveries have led, in turn. to extensive theoretical. experimental. and computational studies over the last 30 years. Many nonlinear model equations have now been found that Q O S ~ S Ssimilar pmpenies, and diverse branches of pure and applied mathematics have been required to explain many ofthe navel featurn !hat have appeared Finally, it seems pcnincnt to mention the followmg. It IS not easy to give a precis definition of a soliton. However. the term can readily be associated with any solution of a nonlinear partial differential equation (or system) that (i) represents a wave of permanent farm. (ii) IS localized, so that it decays or approaches a con~tantvalue at infinity. and (iii) can undergo a $song interaction with other solitons and retain its identity. I n the context ofthe KdV equation and other similar equations. the single-soliton solution IS usually referred to as the solrm)y wave, but when more than one of them appear tn a solution. they are called solmnr. In other words. a soliton is a solimy wave when it is infinitely separated fmm any other soliton. Also, for equations other than the KdV

Solitons a d the lnvcrsr Scartwining Tmmfom,

336

equation, the solitary-wave solution may not be a sech' function, but a rech or

tan-'(en') profile. In fact, some nonlinear equations have solitary-wave

solutions but not solitons. whenas others (like the KdV equation) have solitary waves that are sol~ton~. Indeed. the soliton has formed a new paradigm in mathematical physics. In recent y e m , the concept of the soliton has been used in a rather very broad sense. For example, the nonlinear Schriidingcr equation (NLS), describing waves in plasmas, in superconductors. in nonlinear optics. etc., yields the envelop soliton solution which has the form of scch modulating a monochromatic c m e r wave. Unlike the KdV soliton, a NLS solmn does not depend on the amplitude. Physically. the former is called a low-frequency soliton, whereas the latter IS called a high-frequency soliton (Makhankov, 1978). Another important model equation, the so-called sine-Gordon (SG)equation, IS used to describe nmlmear wave motions m the unified theory of elementary pmicles, magnetic flows. and dislocations in crystals. This equation admits SG Soliton solutions that are called either kinks or ontikinks. whose velocities do not depend on the wave amphtude.

9.3

The Boussinesq and Korteweg-de Vries Equations

We consider an inviscid liquid of constant mean depth h and constant density p without surface tension. We assume that the ( x . y)-plane is the undisturbed h e surface with the z-axispositive upward. The fne surface elevation above the undisturbed mean depth h is given by z = q ( x . y , I ) , so that the free surface is at z = H = h+ q and z = 0 is the horizontal rigid bottom (see Figure 9.3).

I

.

x

*

oE rl(sY, t )

F l y r e 9.3

A shallow water wave model.

337

9.3 The Bowsmeq and Korleurg-dr Vrks Equotionr

It has already heen mognmxi that the parameters E = a/h and K = d ,where n IS the surface wave amplitude and k is the wavenumber.must bo!h be Small for the linearized theory of surface waves to he valid. To develop the nonlinear shallow water theory. ~tis convenient to i n v d u c e the following nondimensional flow variables h a r d on different length scaler. a typical horizontal length scale I (which could be wavelength), and a typical venieal length scale h (which could k fluid depth):

(x..y.)=l(x,y),

<=:.

<=

h

(4)

, q . =a~ .$.=($)$.(9.3.1)

where c = @ is typical horizontal velocity (or shallow water wave speed).

We next invoduce two fundamental parameters to characterize the nonlinear shallow water waves as E=- a

and

6=c fa

(9.3.2ah)

'

In terms of the preceding nondimenrional variables and the parameten, the basic equations for water waves (see Dchnath (1994). equations (1.6.5)-(1.6 8)) can be written in the nondimenrional form. dropping the asterisks:

a($- +en)+

= 0,

*+f($:+$;)+l):+,=o at

(9.3.3) on Z = l + E q ,

26

2

+

6 [q,++.q,

=o #,=O

on

2=1+Eq,

on z = o .

(9.3.4) (9.3.5) (9.3.6)

It is noted that the parameter K = & docs not enter explicitly in equations (9.3.3)-(9.3.6). hut an equivalent parameter y = n/f is associated with E and 6 h u g h y =(o/h)-(h/f)=E&

.

If E is small, the terms involving E in (9.3.4)-(9.3.5) can be neglected to mover the linearized free surfaceconditions. However, the assumption hat 6 is small might be interpreted as the characteristic feature of the shallow water theory. So. we expand $ in terms of S without any assumption about 8 . and write

$ = $m+s$,+6'$1+..-,

(9.3.7)

and. hen.substitute in (9.3.3)-(9.3.5). The lowest order term 8" (9.3.3) is

=o.

(9.3.8)

Solifom and the hvcrsr Scoffwing Transfoform

338

which, with (9.3.6). yields SO, for all I . or $ o = & ( x , y . r ) . which indicates that the honmntal velocity components are independent of the vertical coordinate z in lowest order. Consequently, we use the notation and

&=u(x,y,f)

@o,=v(x,y,f).

(9.3.9ab)

' h e f n t - and second-order terms in (9 3.3) are given by

oo= + h,,+ o,, 6.. + o,,

+

= 0.

(9.3.10)

h. = 0.

(9.3.11)

Integrating (9.3.10)with respect to z and using (9.3.9sb) giver

o,, = -"=,

+ u,)+ C ( X . Y , f ) .

(9.3.12)

where the arbitrary functton C ( x , y , f )becomes zero because of the bottom boundary eondmon (9.3.6). Integrating the resulting equation (93.12). again, with respect to z and omitting the arbitrary constant. we obtain I

= -; L u . +(% ) .

so that = 0 at z = 0 and u and at the bottom boundary.

(9.3.13)

u are then the horizontal velocity components

We next substitute (9.3.13) in (9.3.10)-(9.3.l1), and. then. integrate with condition (9.3.6)to determine the arbivary function. Consequently. $, =~z'[(Vzu),+(Viu)y],

& =Lz'[(V2#), 24

+(V'v),].

(9.3.14ab)

where 8' IS the two-dimensional Laplacian. We next consider the free surface boundary condiuons retaining all terms up toorder 6, E in (9.3.4), and S', E ' , and 6~ in (9.3.5). It turns outthat o~-~(",+u,)+rl+~E("'+u')=o, 6

[Iv,

+E

(uq. + vq,)}+(I+eq)(y + u,)]

=c

[(V'u),

(9.3.15)

+(V'~)~].(9.3.16)

Differentiating (9.3.15) first with respect to x and, then, with rwpect to y gives two equations: u, + E ( U U . +w,)+qz -16(um +vw)=O,

(9.3.17)

u, + E (uu, + uu, ) + 'I, - 16(u,.,

(9.3.18)

+ v,)

= 0.

9.3 The Bowrincq and Korreweg.de Vnrs Equrfionr

339

A simple simplification of (93.16) yields

q, +["(I

+ E q ) ] ,+[ u( I + E q)],= ~ [ ( V ' U ) , +(V'v),].

(9.3.19)

Equations (9.3.17)-(9.3.19) represent the nondimcnsional shallow w a u r eqUor,onr.

Using the fact that 4, is irrotational, that is. uT = u . and neglecting terms

O ( 6 ) in (9.3.17)-(9.3.19), we obtain the fundamental shallow water equations

",+ E("U, + uu,)+

9, = 0.

(9.3.20)

", ++g

q, = o .

(9.3.21)

+U"J+

rl, +[U(l+Erl)]. +[U(l+Eq)],

=o.

(9.3.22)

This system of three. coupled, nonlinear equations is closed and admits some interesting and useful solution^ for u, v. and q . It is equivalent to the boundary layer equations in fluid mechanics. Finally, it can be linearized. when, E < I lo obtain the following dimensional equations: u,+gq.=O,

u,+gq,=O,

q , + h ( u , + u p ) = O . (9.3.23ak)

Eliminating u and u h m there equations gives 'I,=c'(11- + gn).

(9.3.24)

This 1s a well-known two-dimensional wave equation. It cornsponds to the nondispcrsivc shallow water waves that pmpagate with constant velocity c =@. This velocity is simply the linearized version of .ip(h+s). The wave equation has the simple dAlcmbcn solurion representing plane progressive WBYF.

tl(x.y,I)=/(k,x+l,y-K,~I)+g(k,x+l,y-K,cI),

wherefand g are arbivary functions and K: = (k:

(9.3.25)

+ 5). r = 1,2.

We considcr the onc-dimcnsional cme retaining both E and 6 order terns in (9 3.17)-(9.3.19) so that these equations reduce to the so-called Boussinesq equnfionr(1871a.b. 1872)

", + E X " " + 1 '.

=0 ,

q , + [ " ( l + E q ) ] , - ~ 6 " ,=o.

(9.3.26) (9.3.27)

340

Solitons and the Inverse Scatmining Transform

On the other hand. equations (9.320)-(9.3.22). expressed in dimensional form, read

+ vu) +gHr =0,

(9.3.28)

+ vv, + g H ,

=0.

(9.3.29)

H , + ( ~ H ) , + ( v H )=o. ,

(9.3.30)

u, + u g 0, + u

v,

when H = (h + v) is the total depth and H. = I ) , , since deplh h IS constant

In panicular, the one-dimensional version of the shallow water qualions follows from (9.3.28)-(9.3.30) and is given by u,+uu,+gH,=O, (9.3.31) H , +(uH),= O .

(9.3 32)

This system of approximate shallow water equations is analogous to the exact governing equations of gas dynamics for the case of B compressible flow involving only one space variable (Riabouehinsky, 1932). It IS convenient to rewrite these equations in terms of the wave speed c=m by using dc=(g/Ze)&, so that lheybecome

", + "U' + 2CC'

= 0.

(9.3.33)

2e, + CrA, +2uc, = 0 .

(9.3.34)

?he standard method ofcharacteristies can easily be used to solve (9.3.33) and (9.3.34). Adding and subtracting these equaltons allow us to rewrite them in the

characteristicform [$+(c+u)$]("+2c)=0,

(9.3.35)

I:

(9 3.36)

E

+ ( e - ")-

(u

~

2 c ) = 0.

Equations (9.3.3W9.3.36) show that u + 2 e propagates in the positive x direction with velocity c + u , and u -2c V w e k in the negative x direction with velocity c - u . that IS, both u + Z c and u - 2 c propagate in their respective directions with velocity e relative to the water. In other words, *=u+c u + 2 c = c o n r t a n t o n e u ~ ~ Conwhish ,

* dr

u -2c = constant on C U N ~ S C.

on which

=u -c

1

(9.3.37ab)

f

when C, and C. are char(~cle~sislic curves of the system of pamal differential equations (9.3.31)-(9.3.32). Disturbance propagates along these characteristic

9.3 The Bourrinrrg ond Korrrwrg-h V m s Equorionr

341

curves at speed e relative to the flow speed. The quantities u f 2 c arc called the Rirmam invariants of the system, and B simple wave is propagating to the dght into water of depth h. that IS, u - Zc = c, Then, the solution is given by

=a.

.=j(<). x = < +

c (

whm

U(X,I)=~(X)

0

3

+ "

I,

(9.3.38)

at t=O.Howcver,wenotethat

giving

(9.3.39) Thus, if I

+-2/(3

f({)is

anywhere less than zero, u, tends to infinity as

j').In terms of the free surface elevation, solution (9.3.38) implies

rhat the wave profile progressively distotts itself. and, in fact. any forwadfacing ponion of such a wave continually steepens. or the higher p a of the wave tend to catch up with lower parts in front of them. Thus. all of these waves. carrying an increase of elevation. invariably break. Breaking of water waves on heaches is perhaps the most common and the most stnking phenomenon m nature. An alternative system equivalent to the nonlinear evolution equations (9.3.26)-(9.3.27) can be derived from the nonlinear shallow water theory. retaining both E and 6 order terms with 6 < 1 . This system is also known as the Bowsinesq equruions, which, in dimensional variables. an given by

q, + [ ( h+ 7) u], = 0. u,

+ uu, + g q.

= 1h ' y

(9.3.40) ,

(9.3.41)

They descnhe the evolution of long water waves that move in both positive and negative x-directions. Eliminating q and neglecting terms smaller than O(e.6) give B single Boursmesq rquorion for u(x.t) in the form u,,-c*u_+i(u') = i h ' (9.3.42) 2 " 3 u " . The linearized Boussincsq equation for u and q follows from (9 3.40) and (9.3.41) as

342

Sohronr ond the lmerse Scattering Transform

This is in perfect agreement with the infinitesimal wave theory result expanded for small kh. Thus, the third derivative term in (9.3.41) may be identified with the frequency dispersion.

Another equivalent version of the Boussinesq equation is given by

(9.3 44)

There are several features of this equauon. It is a nonlinear panid differential equation that incorporates the basic idea of nonlinearity and dispersion. Bouinesq obtained three invariant physical quantities, Q. E. and M. defined by

I I

provided that ? + I 0 as x + -. Evidently. Q and E represent the volume and the energy of the solitary wave. The third quantity M is called the momenf of instability. and the variational problem. S M = O with E fired, leads to the unique solilary wave solution. Boussincsq also derived the results for the amplitude and volume of a solitary wave of given energy in the form

The former result shows that the amplitude of a solitary wave in a channel vanes inversely as the channel depth h. The Boussinesq equation can then be written in the normalized farm u,, -uy

- A("2) 2

This panicular form

--u-

=O.

(9.3.47)

=

because it admits inverse scattering formalism. Fquatim (9.3.47) has steady progressive wave E O I U I in ~ ~the S form IS of special interest

u(x.f)=4k1f(X),

X= h - m f ,

(9.3.48)

where the equation for f ( X ) can be integrated to obtain

f, = 6A + (4 - 66)f- 6f',

(9.3.49)

where A is a constant of integration and

w1 = k l + k'(4-6B).

(9.3.50)

For the special ease A = 6 = 0 , a single solitary wave solution is given by f(X) = %chi (X - X.),

(9.3.51)

343

9.3 The Bousrincsp m d Korreweg-de Veer Equofionr

where X, is a constant of integration. This result can be used to construct B solutionfor a series of solitary waves. spaced 2 0 apart. in the form (9.3.52)

f(X)= &ech'(X-Zno).

".._

Th~lllsis B 2 0 periodic function that satisfies (9.3.49) for certain values ofA and B.

We next m u m c that & and S BIT comparable, so that all terms O(&.S) in (9.3.17)-(9.3.19) can be retained. For the care of the two-dimensional wave motion (u I 0 and a/+ I 0).these equations become

u,+q,+&U",-Lb~=O.

q,+["(l+aq)],-$s"&"

(9.3.53)

=o

(9.3.54)

We now seek steady progressive wave Solutions traveling to the positive xdinctiononly,sothat u = u ( r - U f ) and q = q ( x - U r ) . Withthetermsofzero order in & and S and (I= 1. we assume a solution of the f m

u=q+~P+sQ.

(9.3.55)

where P and Q BIT unknown functions to k detemmed. Consequently, equations (9.3.53)-(9.3.54) become

( q + E P + 6 Q ) , + q, + &qq, S'q,+[(I

+ & q)(q + & P + S Q ) ] . -S'

q= = 0 ,

(9.3.56)

= 0.

(9.3.57)

q,

These equauonr must be consistent so that we stipulatefor the zero order q,=-qx, P = - i q ' , Q='q =-is (9.3.58) 3 *" 3 = We use these r e s ~ l t sto rewrite both (9.3.56) and (9.3.57) with the assumption that & and S BIT of equal order and small enough for their products and squares to be ignored, so that the ratio (&IS)= (al*/h') is of the order one. Consequently, we obmn a single equation for q ( x , r ) m the form

This is now universally known as the Kortrweg and de Vries cquorion as they discovered it in their 1895 seminal work. It is imponant to point out that

344

Solitons and the Inverse Scottoring TrmsJom

(a/S) = o f / h ' IS one of the fundomental parameten in the theory of nonlinear shallow water waves. Recently, lnfeld (1980) considered three-dimensional generalizations ofthe Bourrinerq and Kornweg.de Vries equations.

Exontple 9.3.1 (Ion-Acoustic Waves and the KdV Equation). The high temperature plasma is a fully ionized gas consistingof elecvom and ions that are governed by the equations of continuity and momentum combined with the classical Maxwell equabons. Using the subscripts e and I for quantiuen related to electrons and ions and neglecting dissipation due to collisions. we write the following equations of mouon for plasma. The equation of eonunuity IS ?L+V.(",",)=O,

(9.3.60)

when n,(j = c, i) is the density and u, is the flow velcat'lfy.

The equation of motion IS m,n,[%+(u,

V)n,]=-Vp,+n,q,[E+f(u,xB)].

(9.3.61)

The Maxwell equations are given by V-E=4n(q,n,+q,nc). V B=O, ?+c(Vx

E)=O,

- s + c ( V X B ) = 4X(q,n,u, + q p , u < ) .

(9.3.62ab) (9.3.63ab)

t

The equation of state IS given by P, =',TI.

(9.3.64)

In the above equations, E is the electric field, B IS the magnetic field, T is the pmduct of the Bolamann constant and the temperature. q and m am charge and mass.nspcctively. and c is the speed of light. For an electrostatic wave, that is. for a one-dimensional longitudinal wave ( V x E - 0 , B=O), we set q, =-qe i e to obtain

(9.3.65)

dr+,dy=*LE-ld JE -=4ne(n,-q), JX

aE

-=4ne(np-n,u,). t

(9.3.67ab)

345

9.3 Tho Boussindsq ond Konnulg-L Vnes Equorionr

The motion of the cold ions in a hot electron gas with the classical Boltzmann distribution is governed by the simplified form of the above equations (9.3.60)-(9.3.64) In this case. with me - 0 and T, =constant wlth T. >> T , , it follows from equations (9.3.65)-(9.3.67ab)that (9.3.68) (9.3.69)

&+", ?!!.E=-4 Ld(n,T,), dt

ax

m,

m,n,

(9.3.70)

Jx

%=4,.(., -nJ. (9.3.71) ax "hex arethe governingequations forquantities n,,u,, E, and a, if T, IS a given

constant or, by the adiabatic law, so that (n,T,) 1s proportional to m,'. Introducing the elecuostatic potential so that E=-@.,, equation (9.3.69) gives the Roltzmann dismbution for electrons n. =noerp

?.!

(TI

, where no i s

the number density ofelesuanr (or ions) in the unpenurbed state. To examine the motion of the cold ions in a hot electron gas with the Boltzmann distribution, we use equations (9.3.68)and (9.3.69).It is convenient to nondimenrianalire the physical quantities BJ follows:

where

On.=(4nnO/me)'c

is

the

electron

,IDc = w;: ( T e / m c p i s the electron Dcbye length,

plasma

frequency,

on, =m,~(me/m,)~ is. the

ion plasma frequency, n, = n o = ionstant and e, =Ao, .wn,= ( T , / ~ L , ) ~ . Consequently, omitting the mericks. equations (9.3 69)-(9.3.71) reduce to nondimensionalform Jn, -+a

d (",+O, ax

(9.3.72)

&+,4=-4 at Jx Jx' Substituting

n, = 1+n,' crp[i(k.-ot)]

and (u,.@)=(u:.

(9.3.73)

@') erp[i(k.-or)]

in Ule above equationsgives the linear dispersionrelauon

m2 =k'(l+k'r'.

(9.3.75)

For long waves ( k ' e I). equation (9.3.75) reducer to (9.3.76) Under the assumption of a quasi-neutral state, n, =n. =exp(@) so that

@ = l o g ( n , ) ,the equationofmotion(9.3.73).becomes (9.3.77) This i s identical to the equation of motion for a gas with unit sound ~elocity (c, =I). Physically. this means that CICCVO~S eren a pressure on ions through the electric field E = - @ s , and hence, waves describedby the above dispersion relation me called the ion-acoustic WL~VCS.As Ulc waves steepens by the inenia

-n,) cannot be neglected. Consequently,a soliton i s generated by the dispersion involved with the deviation from the quasi-neutral state (the second term of the dispersionrelation). term, @-(=n,

To obtain l h e soliton solution of the above system, we Iwk for a stattonary wave solution with = x - c I and the boundary conditions

<

0-0,

nr+L u,+o, It turns out that

%, J5

as

1{1+-.

(9.3.78)

r(<.I) satisfiesthe nonlinearequation

[*I

z a<

=erp(@)+c(c'-z&

Thts gives Ule value of c2 when

-(c'+1).

1= O u. @ =em,so that a5

(9.3.79)

We assume that

ern<1,

so

that c - 1 = 6 1 4 1. Consequently, equation

(9.3.79) reduces to the form (9.3.81) Integrating this equation gives B 6oliton rolution of the form O(x,r)=3(Se) s e c h ‘ [ E ( x - c l ) ] .

(9.3.82)

This represen6 a KdV soliton when the amplitude IS finite but small. As the amplitude of the potential tends to Om =’

c’.

that is. eem=I m,c’. the ions

cannot pars the potential and so will be reflected. As a nsult, ion orbits will intersect and hence. the cold ion approximation ceases to k valid. Accordingly,

the maximum amplitude of the soliton

-

15

given by #- = 1c2, and, from

-

equation (9.3.80). we obtain qm 1.3 and, hence, c 1 6. Thzs reveals Ulat the maximum Mach number of the soliton is 1.6 ar predicted by Sagdecv (1966).

9.4

Solutions of the KdV Equation, Solitons and Cnoidal Waves

To find ~ o l u t i o nof~ the KdV equation. it is convenient to rewrite it in terms of dimensional variables as

when c=@,andthetotaldepth H = h + q . The fimttwoterms ( q , + c q , ) describe wave evolution at the shallow water speed e. the third term with coefficient (3c/2h) represents a nanhnear wave steepening, and the last term with coefficient (ch‘/6) describes dispersion Thus the KdV equation is a balance between time evolution, nonlinearity, and dispersion. The dimensional velocity u is obtained from (9.3.55) with (9.3.58) in the form

(9.4.2)

We seek

frame X so that Substitwingthis solution

a naveling wave solution of (9.4.1) in the

IxI+-.

q = q ( X ) a n d X=x-U1 with q'0,as in (9.4.1) giver

where q' = dI)/dX lntegraung this equation twice wilh respect to X yields (9.4.4) 6

4k

where A is an integrating consmt.

We multiply this equation by 21)' and integrate again to obtain

where B IS also B constant of integration. We now consider a special case when q and its derivatives tend to x r o at infinity and A = B = 0, so that (9.4.5) gives (9.4.6)

LI = 2h(

f - ]).

(9.4.7)

Ihe right-hand side of (9.4.6) vanishes 81 q = O and q = a , end the exact solution of (9.4.6) is given by

q ( X ) = a sech' (bX).

b

=(

%r.

(9 4.8ab)

4h

Thus, q(x.1) = a sech'

where the velocity of the wave IS

[(

$r(x-

UI)].

(9.4.9)

9.4 Soluoons of the KdV Equnlion. Solitons and Cnoidol Wows

U=

e(l) ;+

349

(9.4.10)

This is an exact solution of the KdV equation for 811 o / h : however. the equation n is called a is derived with the approximation (o/h)c< 1. The ~ o l ~ t i o(9.4.9) (or solitary wove) describing a single hump of height (I above the undisturbed depth h and tending rapidly to zero away from X = 0.The solitary wave propagates to the right with velocity U ( > c ) . which is directly

soliton

proponionalto!he mplitudenandhas width b-‘ = (30/4h’~’‘’,

that is. b-‘

IS

inversely proponlonal to the squaw root of the amplitude 0 . Another significant feature of the soliton solution IS that it travels rn !he medium without change of shape. which is hardly possible without retaining 6-order terms in the governing equation. A solitary wave profile has already been shown in Figure 91 In the general ease, when both A and B are nonzero, (9.4.5) can be rewritten a3

when F( q) is a cubic with simple zeros.

We seek a real bounded solutionfor q ( X ) , which has a minimum value zero and a maximum value a end oscillates between the two values For hounded solutions. all three zeros 9,. q , . q , must be real. Without loss of generality, we set q ,= O and q 2 = a _ Hence, the third zero must he negative so that

q, = -(b - 0 ) with b > LI > 0. With !huechoices. F ( q ) = q ( n - q ) ( q- n + b ) and equation (9.4.1 I) assumes the f m

h’3 (2)’ = q(a- q)(q-a+ b), d y

(9 4.12)

(9.4.13)

which is obtained by companng !he cocfficienls of q‘ in (9.4.1 1) and (9.4.12). Writing

L?

- q = p a , 11 follows from equation (9.4.12) !hat

Solitonsand the Inverre Scamring T r m g m

350

Substituting p = & q in (9.4.14) gives the standard elliptic integral of the first kind (see h t t a and Debnath, 1965)

and then, function q can k expressed in t e r n of the Iacobian snfunnion

where m is the modulus of

m(z. m)

Finally.

when en(& m) is also the Jacobian elliptic function with a period Z K ( m ) , when K ( m ) is the complete elliptic integral of the first kind defined by “I‘

K ( m )=

I

(1

- m’sin’

e)-“’de,

(9.4.18)

and cnyz)+s”*(z)=l It is important to note that cn L is periodic, and hence q ( X ) representsa train of periodic waves in shallow water. So these waves are called cmidnfwaves with wavelength (9.4.19) We upshot of this analysis is that solution (9.4.17) represents a nonlinear wave whose shape and wavelength (or penod) all depend on the amplitude of the wave. A typical enoidal wave is shown in Figure 9.4. Sometimes, the cnoidal

waves with slowly varying amplitude are observed in rivers. More often, wavetrains behind a weak bore (the so-called undufar bore) can be regarded as cnoidal waves. Two limiting eases are of special physical interest: (i) rn + 0 and(ii) m + l .

In the first case, snz+sinz. c n z + c o s z as m + O (a+O). This corresponds to small-amplitude waves where the linearized KdV equation is appropriate. So. in this limiting case, the solution (9.4.17) becomes

9.4 Solutions ofrhr KdV Equorbn. Solirons and Cnoidal Waves

tl(~,t)=l,[l+,,,(~-"t)],

k-61"'.

351

(9.4.20)

where the correspondingdispersion relation is 0 = Ok

=ckb-:k2h2).

(9.4.21)

This corresponds to the first two terms of the series expansion of (gk tanh kh)". Thus. these results arc in perfect agreement with the linearized IheOry.

In the second limiting case, m + 1 ( a+b). m z + seehz. mus, the cnoidal wave solution tends to the classical KdV solitary wave solution where the wavelength L , given by (9.4.19). tends to infinity because K ( o ) = - and K ( 0 ) = n/2.The solution identically reduces to (9.4.9) with (9.4.10)

We next nport the numencd cmpu1BUon of the KdV equation (9.4.1) due to Bemiin and Karpman (1%6). In t n m s of new variables defined by x =x--ct,

t =t.

.

3c

11 = - 7 ,

(9.4.22)

2h

omitting the askricks, equation (9.4.1) bccomcs

when

B

=(:)

11, + 1111, + B 1,

=O.

(9.4.23)

ch'.

We examine the numerical solution of (9.4.23) with the mind condition (9.4.24)

(e)

what qo 1s constant and f it B nondimcnsionalfunction charactcnzing the initial wave pmfils. It IS convenient to i n d u c e the dimensionless variables

352

Solironr and the inverse Scomnng Transform

<=q,

r=x* .=A

(9.4 25)

'lo

so that equauon (9.4.23) and (9.4.24) reduce to UI

+

""(

+ u-'

"(((

(9.4.26)

= 0,

.(<.O)=f(<). where the dimensionless paameta u is defined by

(9.4.27)

[T

u =I - .

Berezin and Karpman (1966) obtamed the numerical solution of(9.4.26) with the Gaussian initial pulse of the form u(<,O)=f(<)=exp(-<') and values of the pameter u=1.9 and cr=i6.5. Their numerical Solutions are shown in Figure 9.5.

Figure 9.5 The solutiow of the KQV equation u ( x , t ) for large values of I with the values of the similarity parameter 0 : (a) mz1.9, (b) 0=16.5. (From Berezin and Karpmsn (1966))

As shown in Figure 9 5 for case (a), the pemrrbation splits m10 a ~olitonand a wave packet. I n case (b), there are six solitons It is readily seen that peaks of the ~ ~ l i t o nlies nearly on a swaight line. This is due to the fact that the veloclty

of the soliton is proporlronal to i s amplitude, SO that the dtrtanees vaversed by them would be proportional to their amplitudes Zabusky's (1967) numerical investigation of the interaction of two solitons reveals that the taller soliton. initially behind, catcher up the $honer one, they undergo B nonlinear rnteractionand, then. emerge from the interaction without

9.4 Solutionr oftha KdVEqualzon. Solrlonr and Cnoidol Waver

353

any change in shape and amplitude. The end result is that the taller soliton reappears m front and the shorter one behind. This is essentially strong computational evidence of the stability of solitons. Using the transformation x. = E p ( x - c t ) ,

f. = g’t. q’ = ( . E l ) ’

q

with a,9(=3c/Zh)=6and p3(=ch2/6)=I,wewritetheKdVequation(9.4.1)

m the normalizedform. dropping the asterisks.

II, + 6111, + rl, = 0 . We then seek a steady progressive solution of (9.4.28) in the form q=zk‘f(x).

(9.4.28) (9.4.29)

X=lu-a,t.

Then. the equation for f ( X )can be integrated once to obtain f, = 6A + ( 4 - 6B)f- 6f’.

<9.4.30)

where A is a con~tantof integration and the frequency a, is given by a, =t’(4-6B).

‘

(9.4.31)

A single soliton solution corresponds to the special case A = B = 0 and is given by

f(X)=scch’ ( X - X o ) , where

(9.4.32)

X, is a constant.

Thus. for a series of solitons spaced Zo apm, we write f ( X ) = x s e e h ’ (X- Zno). (9.4.33) ”=.This is a 2a periodic function that satisfies (9.4.30) f a some A and B. The general elliptic function solution of (9.4.28) can be obtained from the i n t e p l of (9.4.30) that EBD k written as

f: =-4C+lZAf+(4-6B)f’

-4f’.

(9.4.34)

where C i s a constant of integration. Various asymptotic and numerical results lead to the relations (see Whitham, 1984)

c-

I Mu)

1 d’B(4

2 do 4 and the cubic in (9 4.34) can be factorized as - C + 3Af +(I -+B)f’

do’

(9.4.35ab) ’

- f’ = (f,- f)(f -f2)(f- f,),

(9.4.36)

Solirons end the Inverse Seofiwing Transform

354

j,(u)(r=1,2,3)are&teninedfmm A(u), B(u)).and C(u). I f w e s e t j , > j , > j , and, then. the periodic ~olutmnoscillate between f, at X = 0 and j , at X = u,one panieular form of the solution IS given by

w

h

f ( x )= f, +(f,- j 2 )cn*

(mx).

(9.4.37)

where the modulus m of m ( z . rn) is given by

(9.4.38) Thus, it follows from (9.4.33)and (9.4.37)that the following identity holds: X ) = Tsech’(X-Znu).

j 2 + ( j , - f , ) m’(=

(9.4.39)

“=-

which EM be verified by companng the penods and poles of the two ader. It is interesting to point out that there IS also a theta function representation of the penodic ~olutionof lhe KdV equauon. The solution may be written 8“ the farm

f ( X ) = z l o g Q(X).

2x1

(9.4.40)

where B(X) is a slight modification of the theta function B,(r.q) in Dutta and Debnath‘s (1965)notauon, the imaginary period c in q = exp(im) has been set equal to (Ziu/z) so that q = erp(-Zu), and the independent variable z is changedto(iX+z/Z)and Q(X) = xexp(-ZnY-

Zn’u).

(9.4.41)

I-..

A simple direct proof of (9.4.40)is not obvious. However, 11 IS equivalent to

(9.4.33). which can be verified by using lacobi’s infinite product formula for the theta functions (see Dutta and Debnath, 1965,p. 108). With the appropriate modification m (9.4.29).the produetrepresentation of Q(X) IS

Thus, the series representations (9.4.41)follows fmm the preceding analysis by replacingxby X + u . There is B similar situation for the modified KdV equation in the normalized f m

9.4 Solumm of rhc KdV Equation, Solitons and Cnoidnl Waves

"_=a.

",+3U'",+

355

(9.4.43)

If we seek a steady progressive solution in the form v=&kf(X),

X=kx-wf,

(9.4.44)

then, f ( X ) satisfies the equauan Fm =2A+(28+1) f - z f ' ,

(9.4.45)

where A is a constant of integration and

28 =

(s

- I).

(9.4.46)

A single soliton corresponding to the special ease A = B = 0 is given by

f ( X ) =sech(X-X,). and hence, a S

~ C E of

(9.4.47)

solitons spaced 2a apart is given by f ( X ) = &ch(X-2na).

(9.4.48)

".._

This is, indeed, an exact solution.The required identity between the series ofthe sech function IS found by substituting (9.4.48) into (9.4.45). it fallowr from whitham's argument (1984) that (9.4.49)

Integrating (9.4.45) once gives

f: = - C + ( 2 8 + I ) f' - f'

E

(f: - f2)( f' - f:),

(9.4.50)

where C i s a constant of integration and f,Z f Z f,. This equation admits the penodic solution f ( x ) = f , dn(f,X. m).

(9.4.51)

where the modulus rn of the elliptic function is m' = (f:

- f:)/f:.

(9.4.52)

and (9.4.48) gives an identity for the dn function m terms of a sum of the sech functions Therelation between a and m is given by af,(a)= K ( m ) . Finally, the higher order mcdified KdV quauon

", + ( p + 1) d v , + "admils single-soliton solutions in the form

= 0,

p

> 2,

(9.4.53)

356

Sofitom ond the Inverse Scarrering Tlonsform

v(x,r)=n sechZ"(kx-ot).

(9.4.54)

However. in view of the fractional pawen of the seeh function, it seems. perhaps. unlikely that there will be any simple superposition formula.

The Lie Group Method and Similarity and Rational Solutions of the KdV Equation

9.5

The KdV equation describes the generation and propagation of maderately small-amplitude shallow water waves and many other important phenomena. where a small nonlinearity is combined with a cubic dispersion relation. In many physical situations. its similarity solutions becomes important. A simple transformation X+CX. f+bt, and I + u + ( I u (9.5.1) can be used to wnsform the KdV equation

",+ u, + u g +","

=0

(9.5.2)

into h e general form

" , + ( ~ ) "2u , + =o,b . , "

(9.5.3)

when 0 , b, c are nonzero real constants. With suitable choices of the con~tants. this equation reduces to the standard form (9.4.28) which is invariant under the vansfonnation 7. definedby i=a,

i=&,

i=i'"

(9.5.4)

for a nonzero real a. The set of all such transformations {To)form an infinite Lie group with parameter Y with the composition (multiplicative) law Tar, = Td . This law is commutative. Also, the associative law is satisfied

since

T,(TJ.)=TJ& =T& = T J

=(T~T,)T<.

In view of the fact that TIT, = TI. =To, =To for d l

(1 #

0, T,represents the identity mnsformation.

Finally, we find that

T J . , =T,= T e , T a .

357

9.5 The Lie Croup Method

This confirms that T#.>is both a left-hand and fight-hand inverse of To

The fact that the KdV equation (9.4.28) with 6 replaced by -6 is invariant under the continuous (or Lie) group of transformations T- suggests that we can seek invariant properties of the solutions. In fact, the quantities u ( x , r ) = t P f(<),

5 = xrs

are invariant under the transformation To provided p =also easy to check that, if u(x.r)=-(3r)-"f(<)

(9.5.5ab)

1 and q =-i.It is

where

5 =x(3r)-"',

then.

j ( 5 ) satisfies the ordinary differential equation

f'" +(6f - <)j' - 2f = 0.

(9.5.6)

Another substitution of j = A ( d V / d < ) - V ' in (9.5.61, where V = V ( C ) and A is a eonntant to be determined. yields a P a i n l e i cquolion ojrhe second kind

v"-~v-2v'=o.

(9.5.7)

This equation gives a solution describing a wave profile that decays. as ++-, and oscillates as 5 + --. The presence of a Painled equation, for which each movable singularity is B pole, is not accidental. It is believed that then is a direct correspondence betwen the appearance of a Painlev6 equation for a given pmial differential equation and the existence of an inverse scattering transform (and hence. soliton solutions) for that equation. Zakharov and Shabat (1974) proved an exact reduction of the Boussinesq equation

<

."-."-I(..) -I=o (9.5.8) 2 y 4uto an ordinary differential equation by looking for a uweling wave solution u(i.r)= j ( x - U t ) = j ( X ) ,

(9.5.9)

when j ( X ) satisfies

(,-UZ) j 2 J + j"" = I 0. (9.5.10) 2 4 This can be integrated twice to obtain the following Painlev6 equations, depending on the cun~tantsof integration:

j"+I(

f"+2f' + A = 0.

f"+2f' + X = O .

(9.5.1 lab)

In general, cenmn partial differential equations admit similarity solutions in the form u(x.r)=r-"f(xiB),

(9.5.12)

Solitons and the h v e n e Scaiienng Trmsfom

358

where a and p are constmts and f satisfies an ordinary differenual equation. The equation is invariant under the transformation T.: i=opx,

i=ar,

i=i"u,

(9.5.13)

where d IS a scalar. The set of all such transformations T. also forms a conunuous group. It a n easily be verified that the normalired KdV equation u, +6uu, + u, = 0. (9.5.14) has the similarity sduuon u(x,r)=(3t)-'p f(I;),

I;=+r)-'".

(9.5.15ab)

Substituting (9.5.15ab) nn (9.5.14) gives the following ordinary differential equation for f ( 5 ) in the form

fm+(6f-C)f'-Zf =O.

In the limit

BS

(9.5.16)

< +-, ~ o l ~ t i oofnequauon ~ (9.5.16) lend to solutions of the

lineariredequation

fW-Zf -
(9.5.17)

This equation admits a solution f ( I ; ) = A i (I;). where Ai (I;) is the Airy function. The other two solutions Bi (I;), which is exponentially large. and

G/ (I;) - - ( z z ' ~ ' do not decay exponentially (see Abramowilz and Slegun (1964). pp. 446-450). Thus we set the boundary condition f(<)-aAi'(I;),

as I;+-,

(9.5.18)

where n is an amplitude parameter. The numencal eomputauon of Berezm and Karpman (1964) suggests that. when n is mall enough. f ( 5 ) is meillatory. as + --, but otherwise, f ( 5 ) may develop singularitier. Rarsler (1978) showed that there exim a critical value a, of LI which separates the oscillatory nature from the singular solutions.

<

For 101=a,, f ( I ; ) - i <

as

<+--.

For

Ial
oscillatory. as I;+--. On the other hand, when = > a , . f ( I ; ) develops a singularity. Some special nonlinear evolution equations admit solutions that are rational functions of theindependent variables. For example, u ( x , r ) = ( x / t ) is a rational solution of the eauatian

9 6 Conre~arionLaws and Nonlinear Tlonsformorronr

359

(9.5.19)

u,+uu,=O.

It turns out hat the KdV equation (9.4.28) with the coefficient6 replaced by 4 also has a simple rational ~ o l ~ l i o Assuming n. u = u ( x ) and u,u‘.u”+ 0. as I + c-, the resulting KdV equation u’” - 6uu’= 0 can be integrated twice 10

1 I

obtlin u” = Zu’, which admits the rauonal SO~UUOO

(9.5.20)

u(z,1)= zx-‘.

This is singular at x = 0. It can also be shown that equation (9.4.28) with the coefficient 6 replaced by - 6 has a rational solution in the farm 6x(x’ -241) ”(&I)=-

(9.5.21)

(x’+1212)

This is also singular on ~ ’ + 1 2 =1O~. It is conjectured that all rational solutions of the KdV equation M singular ! Zabusky (1967) showed that the modified KdV equauon u, + 6u2u, + uyr

=0

(9.5.22)

-6A’f)I +I],

(9.5.23)

has the rational solution u(x.r) = A- ( 4 A ) / p ( x

where A is B real constant.

9.6

Conservation Laws and Nonlinear Transformations

~n equation of the form

Z+aX=O

(9.6.1)

J f d x is called a consenafion low, where T and X arc known as the dcnairy and the flux.. respectively, and neither T nor X involve derivatives with respect to I. If both T and X are integrable on (-c-,-), so that X tends lo a constant. as

I x I+-. then, equation (9.6 1) can be integrated to obtain

1

-Tdr =O

dt[J

or

:

ITdr=constant.

(9.6.2ab)

The integral (9.6.2b) is usually called a eon~mnfof mofion. provided interpreted as a urnelike variable.

I

is

Solitonr and lk lnvrnr Sconering Tronsfom

360

It is convenient to write the KdV equation in the standard form ulal IS much used for the developmen? of thc lhwry of solimns

Ku = N, - 6 u g

+u,

=O.

(9.6.3)

This IS already in conservation form with T-u Hence.

if

and

X=uy-3u'.

(9.6.4ab)

Tand X are integrable. and u satisfies (9.6.3). then,

-

Judr=conslant,

(9.6.5)

which applies to all ~ o l ~ t i oprovided n~ that u and its gradients tend to zero as Thus, the first constant of motion is simply the spatial integral of U.

I XI+=-.

The second conservation law for (9.6.3) can be obtained by multiplying it by ",SO

that

Thw,

-

(9.6.7)

for all SoIutionS, u of the KdV equation, which vanish fast enough a1 Infinity. Thus, both u and u2 are conserved densities for the motion associated with the KdV equation. We know that the KdV equation describes a certain class of nonlinear water waves. I n fact. u is associated with the free surface elevation function which, m turn, is also proportional to the velocity in the r-direction. Thus, equation (9.6.5) describes the conservation of mass, and (9.6.7) the conservation of horizontal momentum for water waves within the scope of a shallow water approximation theory. n i s immediately suggests that there should be a comsponding conserved density that could be associatedwith the energy of water waves. Indeed, this is true sad can be confirmed by adding 3"' x (9.6.3) to us x { ( J / J x ) ( 9 . 6 . 3 ) )so that the resulting equation can be nwntten as

This gives athird constant of the motion

J (u'+$u:)dr=conrra"t. -

(9.6.9)

361

9.6 Conservorion Lnws and Nonlrneor Trmsfomatiom

Then. by using very laborious methods, Miura, Gardner and Kruskal (1968) obtained s i g h t more conservation laws for the KdV equation. They then conjectured that there exist an infinite number of pdynomial conservat~onlaws for the KdV equation. At the same time. a question was raised whether the existence of soliton solutiom of the KdV equation is clorely related to the existence of an mlinite number of conservation laws! Miura e l al. (1968) developed an ingenious method of determining an infinite number of conservation laws by i n d u c i n g the so-called Mium rmmfomtion u = "2

+ v,.

(9.6.10)

This IS similar to the Cole-Hopf transformation used in the Burgen equation which E M be reduced to the linear diffusion equation. A direct substitution of (9.6.10) in (9.6.3) gives

Zvv, + v,, -6(v'

+ 9)(2uv, + v,)+6v,y

+Zvv-

+ v,

=O.

which E M then be put into the form

where the operator M ISdefined by M u = v, -6v'v,

+ v,

(96.11)

=O.

This is called the rnod$ed KdV (mKdV) equation. Thus, if M v = O . then u satisfies the KdV equation (9.6.3). In contmsl to the Cole-Hopf transformation, the Miura transformation reduces one nonlinear equation to another nonlinear equation, neither of which can easily be solved. However. the Miura transformation establishes a connection between the KdV equation and the Sturm-Liouville problem, and hence. it leads to the inverre scattering method for the exact solution of the initial-value problem for the KdV equation. Another remarkable fact about the Miura transformation ISthat there exists an infinite number of conservation laws. It is now convenienl to work with w , rather than v , where v = LE-'+ E W and

& IS M

arbitrary real parameter.

Consequently. the Miura vansformation becomes u=

1E - l +

W+&W,

+E

V .

(9.6.12)

Since any arbitrary constant can be incorporated in the solution for u, we just write

" = w + c w , + &V.

(9.6.13)

This is known as the Gordner transfornoton. It is important to note that. since v exists. so does W . Substitution of (9 6 13) m (9 6.3) gives

362

Solitom and rhr lnvrrrc Seamring Tmnsfom

+ u= w, +&w,, + Z&2ww, - ~ ( w + E w , +&')(w,

0 = u, - 6 u g

+ w,,,

=(I+ *+

2&'w){w, - 6 ( w

&

+EW,

+ZE*WW,)

+2&*(ww')"

+&W,

+ &'W')W, + w,,,}.

JX

Th"S. w satisfies the Cardncr C*UariO" w, - 6 ( w

+ &*)

w,

+ w,

(9.6.14)

=0

for all &. W e note that, when & = O . this equation becomes !he KdV equation with w = u. Another interesting fact is that equation (9.6.14)is already in the consewation form

2 + "(.

~

ax

~t

3w' -2&%J) = 0 ,

(9.6.15)

with w BE the conserved density. that is,

-

J w d x= c o m m t .

(9.6.16)

To obtain m infinite number of conservation laws for the KdV equation, we "$6 the arbitmy parameter E . Since w + u as E + 0,we w i t e an asymptotic expansionof w ( x , t , & ) in E : w-x&"w"(x.t)

as

&+O.

(9.6.17)

"4

Treating the constant in (9.6.16). similarly. as a power series in &, and using (9.6.17). we obtain an infinite number of constants of motion. Finally, we substitute (9.6.17) in the Gardner vansfomation (9.6.13) and. then. equate coeklicients of 8".for each n = 0.1.2,. ... It is then easy to see that wo = u,

-

w1 =-war

w1 - -w2"

and so on

w> = -

=--U,.

- z w p , = -( U"

"2).

Wlj

- w;

+ zu"x,

=u"

- "1,

9 7 The I n v m c Scararing rrmsfom (ISTJMethod

363

In particular, the integrals which BIT generated hy wo,w,. and w, h a m e our fmst three integrals (9.6.5). (9.6.7). and (9.6.9). respectively. On the other hand. wI and v , BIT exact differentials in x. and hence, the comsponding integrals

jw"*=eonrta"t

(9.6.18)

will not give us any useful result. In other words, if n is even, then (9.6.18) generates an infinite set of conservation IBWS, and. if n is odd, wm IS an exact differential and, hence, does not lead to any conservation law. Much of the significant work on the theory of KdV equation was initialed by the publication ofthe several papem ofcardneret al. (1967, 1974).

9.7

The Inverse Scattering Transform (IST) Method

Historically, Gardner et al. (1974) formulated an ingenious method for finding the exact solution of the KdV equation. This novel method has been generalized to solve several other nonlinear equations. I n this section, we describe the Inverse Scattering Transform (IST) for the KdV equation. The exact solution of the KdV equation is obtained by associating its ~olutonwith the potential of a time-independent Schrodingcr equation. The next step IS the ~ o l ~ t i oofn the quantum mechanical problem with the initial value for the KdV equation taken as the potential. This involve^ calculations of the discrete (bound) eigenfunclions. their normRlizmon conmnti and eigenvaluer, and the reflection and transmission coefficients of the continuous (unbounded) states. These resulu are collectively known as the rcaffenng dafo of the Schriidinger equation. The next step is to determine the evolution of the scattering data for any potential that ~VOIYOS, according to the KdV equation. fmm a prescribed initial function. Both discrete and continuous eigenvalues are found to be invariant under such changes, and the normalizalton ~onilantsand the reflection and mansmission coefficients evolve according to simple exponential laws. The final step of the method deals with the determination of the potential for any time f from the inversion ofthe scattering data The main prohlem is to find the exact solution of the general initial-value pmhlem for the KdV equation y-6uux+u-=0, xsR, f > O . (9.7.1) with the initial condition ~ ( x . O ) = u ~ ( x ) , reR,

(9.7.2)

when ",(I) satisfies cemin fairly weak conditions so that the ~olutionu(x.1) emts for all x and 1.

364

Solitons and the lnvcrse Scottiring Tmnrfom

We begin with the Miura (1968) transformationdefined by

" = u1 + p.

(9.7.3)

where u = u ( x . I ) . Substituting (9.7.3) in (9.7.1) gives

Zuu,+ u,, -6(u'+ v~)(Zuu,+ u , ) + 6 u , y

+2uu_

+ v,

=O.

This C M be rearranged to obtain equation

( m)( i+ + u, -6U'U,

u",) = 0.

(9.7.4)

Thus, if v is a solution of the modified KdV equation. then, (9.7.3) defines a solution of the KdV equation. We note here that, if u is given, equation (9.7.3) representsthe Riccati cqwrrion for u.$0 that the transformation

"=%

V'

(9.7.5)

where V(X.I) is B n o n m differentiablefunction, reduces (9.7.3) u) the form V"

-u v =0 .

(9.7.6)

The function u involved in (9.7.6) is a solution of the KdV quation (9.7.1). We further note that KdV equation (9.7.1) is Galilean invariant. that is, it is invariant under the hansformation i = x + 6 L I. i = u + 1,where L E R. so that we can replace u by u L to obtain a second-der linear equation Lv=lv, (9.7.7) where the operatorL is given by

-

8

L=--+u.

ax1

(9.7.8)

T%isis thcSehr6iingcrequation for v with the potential u ( x . t ) andeigenvalue 1 = 1 (f) which depends on I because of the parametric dependence on I. However. this Schraingcr equation for is different from the one in quantum

v

mechanics because the potential U ( X . I ) is nor known and it is the solution of the KdV equation. So, the method of solution of (9.7.7) is not so simple. However, it is well known that equation (9.7.7) admils two distinct kinds of solutions recognized as bound stores, where v +0 exponentially as II I --t -, andsenneringsmres. where v oscillates with x at infinity. Since b t h u and L depend on I. the eigenfunctions v(x,I) also depend on I. The quantity I involved in (9.7.7) connor k regarded BS time. but simply as a parameter,so that u = u(x,t) represents B family of potentials. At I = O . the initial condition is

9.7 The Inverse Scattering Tmmnrfom VSTI Method

365

given by u ( x , O ) = u , ( x ) . and hence, the problem is first to solve the onedimensional Schredinger equation for the known potential "&)

-

[-$-

"4

v(x.1)= A (1) v(x.1).

(9.7.9)

This will determine how the eigenfunctions and eigenvalues evolve as potential changes from the given data according to the KdV equation (9.7.1). The potential ~ ( 1 . 1 for ) m y 1 > 0 can then be determined from the scattering data at time I by the inverse scalering uanrformation.

To determine the eigenvalues A ( I ) , we fin1 differentiate (9.7.7) with respect to ito obtain

v-

-a.

v - (. - ' I )v, = 0

and, then, with respect to f to denve

v " , - ( % - A , ) v - ( ~ - Av,=o. )

(9.7.10)

The quantity u, can he eliminated from (9.7.10) by means of (9.7.1). so that (9.7.9) becomes

where the term (u-

(-L+A) v,+(u--6uur) v+A,v=O, (9.711) v ) can be obtained from the result ( ~ . v ) " = ~ ~ v + ~ , v " + ~ ~ " v '(9.7.12) .

We solve this equation for (u- y) so that the resulting expression can be expressed hy the original S c M i n g e r equation in the form u-

v=(u, v)=-u, v"-Zu_ v, =(-L+A)(usv)-Zu,v,.

(9.7.13)

Substituting this result in (9.7 11) yields

(-L+A)( V, + ur V ) - 2(3uu, v+u_ v,)+A,v = 0 .

(9.7.14)

To replace the quantity (3"". B+U_ v,), we rewrite the term, u_ first using the identity

(.

V,)" = u',

v, + 2 5 V"

+ u v-.

v,

by

(9.7.15)

The quantity vI, in the second term on the nght-hand ride of (9.7.15) can be eliminated by using the SchrMinger equauon, and v- in the third term can he eliminated by the x-derivative of the SchrMinger equation, $0 that (9.7.15)

homer

366

Sohronr m d rk Inverse Scarrrnng Transform

bvJs = u _ ~ , + z u , ( u - A ) v + u [ u . ~ + ( u - A ) ~ , ] v+3uu,

=unv,-2Au,

We next solve for ( 3 u y v+us

v.)

v+(~-A)uv,.

(9.7.16)

by means of the Schrodinger equation and

its x-derivative. so that

3"".

v +"v, ,= 3UUr

v + ( u v,)-

+ 2 1 u,

=(-L+1)(uv,)+2Aux

v-3uu.

v- (Id - A)" y,

v

=(-L+A)(uv.)+~~[v~-(u-A)~,]

= ( - L + A ) ( ~ v ~ + 2v.). 1

(9.7.17)

Substituting (9.7.17) in (9.7.14) gives ( - L + A ) [ ~ , + u ~ ~ - ~ ( u + ~ A ) ~ ~ ] + A , ~(9.7.18) =O.

We next i n d u c e a new quantity Y defined by Y = v,+u= v-2(u+2A)vZ, which is, by replacing (u,

(9.7.19)

v) with the derivative of (9.7.7) with respect to x. = V, + V- - 3 ( u + A ) ~ . .

(9.7.20)

I n view of (9.7.19). quation (9.7.18) BSsurneS the form

(-L+A)Y=-A,

v.

(9.7.21)

Equations (9.7.19)-(9.7.21) arc the main results for determining the behavior of all quantities involved in the scattering problem as the potential evolves according to the KdV equauon. As stated earlier. the solution of the inverse scattenng problem requires a priori knowledge of the scattering data. This includes the discrete sigenvalues, the normalized coefficients for the eigenfunctions, and the reflection and franSrnissmn coefficients. We fist consider the discrete ~ p c c m mof eigenvolues

(bouedsrares).

In view of the fact that u ( x , r ) decays rapidly BS I x I + for all I, the Sehrtidmger equation (9.7.7) admits a finite number of discrete cigcnstates with negativeenergy 1"=-K:. n=1,2. ...,N andaconunuauEEpecrmmofpositive energy ( A = k' > 0). The corresponding discrete eigcnfunctions

belong to

the Hilben space L'(--,-). To determine the evolution of the eigenvalues, equation (9.7.21) is multiplied by and use (9.7.7) to slmphfy the result to obmn

vm

lntegmting this equation with respect to x over (--,-)

gives

Normalizing the cigenfunctions vn(x) of the bound Elates by (9.7.24) and using the fact that v.(z) and its spatial derivatives vanish as x concludethat the right-hand side of (9.7.23) vanishes 30 that

+f-, we (9.7.25)

for a potential "(x.1) that evolves with I according to the KdV equation. the cigcnvalucs of the bound states are invariant with respect to t : An(1)=AR(0) for "=I,%. Theinvarianceof thcdisereteeigenvalues represented by (9.7.25) has several imponant consequences. First, equation (9.7.21) with A n c ( f ) = O implies that Y nrcpre~enlsthe discrete eigenfunctions of the Schdingcr equation (9.7.21) with the cigenvalues A =Am. Second. it follows from (9.7.21) combined with (9.7.19) leads to the equation that 11,

Yo,+ % Vm- 2 ( u -2K:) 4'".'

= 0.

(9.7.26)

This can be regarded as a time evolution equation for ~#(X,I).The bounded soluuon for the nth eigenfunaion can then be represented from its asymptotic bchaviorin the form v"(x,r)-C.(t)exp(-K"x),

as x + + - ,

(9.7.27)

where the real constants cn are determined by the normalized condition (9.7.24). Substituting this asymptotic form (9.7.27) in (9.7.26) and imposing B major restriction u(x. I ) +0 as Ix I+ leads to tht equation fa ..(I) in the form

-

d $,(I) - 4 K: *(r) ~

dt

This equation admits solutions given by

=0,

(9.7.28)

Solirons and ,he bverrr Scamring Trmsfom

368

c.(t)=c.(O) exp(4r K.).

(9.7.29)

where ~“(0). n = 1.2,...,N ax the normalization con~tantsdetermined at I = 0. Thus, the asymptotic representation of ~ ~ ( 1 . is 1 )explicitly given by (9.7.27) with (9.7.29). As a passing remark, we note that the imposed restriction on u(x.1) limits the theory of the KdV soliton and eliminates solutions corresponding to other nonlinear WBVCS. For a continuous sprnmm of eigenvalues (unbounded states), the eigenvalue A = k’ > 0 , where k may take any real YBIUO. We visualize the wave function BE the spatially dependent part of B steady plane wave impinging on the potential u from x ++- for the time-dependent Schrddingcr equation. Physically. there will be an interaction of the plane wave with the potential allowing for the existence of sconering slates that can be mated as the linear combination of a wansmined wave exp(-ikx) at x =-- of amplitude o(k)

v

andareflccted wave e i p (i k x ) at x = + of amplitude b(k). In other words, the wave function for the continuous specmm has asymptotic behavior in the form exp(-ikx)+b(k)erp(ikx)

as x + -

a(k) erp(-ikx)

as x+--

1,

(9.7.30ab)

where ~ ( k and ) b(k) me called the fronsmirsion and refleclion coeffricnfs, respectively, and can be determined uniquely from the initial data. The term exp(-ikx) of unit ampliNde in (9.7.304 is a waveling wave h m I =+-. In general. both o(k) and b(k) are complex functions of a real variable k. A sketch representing lhe scattering by B potential u is shown in Figure 9.6.

Transmitted wave co ap(-ikr)

Incident wave exp(-ikr) t-

-

Reflected wave, b exp(+ikr)

Figure 9.6

Sketch representing the scattering by

a

potential

U.

369

9.7 The Inverse Scaifenng Tronrfom O S r , Method

We next consider two distinct eigenfunctions for the same potential U. so that Vz

-( Ki + u ) Vm= 0

and Vr-( K: + U ) V" = 0,

(9.7.31ab)

where the primes denote derivauves with respect to X . Clearly, (K:

- K:)

VmVa = ( VnV:

where W is the Wmnskmn of

- K:)

Vm),

(9.7.32)

vaand vm

Integrating (9.7.32) gives, for m (K:

- VmV 2 = dr w ( V.,

# n,

J vmvnd~ = [ ~ ( v v.)]r..

-

= 0.

(9.7.33)

because vm, vm+O as I x I + - . This shows that vm.vn are orthogonal. The continuous eigcnfunction IS also orthogonal to each discrete cigcnfunstion vm,Thus, the discrete and continuous cigenfunetionr together

v

form a complete onhogonal set, and hence. any function v E L' (--,-) can be represented as a linear combination of all the vm's plus an integral of v ( z , k ) over all k. If 4 and v are two solutions of (9.7.7) with the same value of 1 = k*(> O), then. equation (9.7.7) implies that the derivative of W is zero,that is, W(4, v) is constant. With an additlonal restriction that 4 is proportional to v. it turns out that W(#. I) = 0 for ell X . Denoting v' as the complex conjugate of the continuous eigenfunctlan v. we can compute W( v. v') at both x =f- and. then,use (9.7.30ab) to obtain

W(v,v') = 2ikno' = Zik(1-bb').

(9.7.34)

Inr+lbr=l.

(9.7.35)

Or, equivalently,

This is a statement of the come~alionof energy in the theory of scattering. A procedure similar to the case of discrete spectrum can be employed to determine the scattering cocfficienu. Integrating (9.7.21) once with respect l o x glVer

YV,

- VY,

=g(r;k),

(9 7.36)

where Y is defined by (9.7.19) and &;x) is an arbitwy function resulting from integration For the continuous elgcnfunctmn, v(,r,kf) is given by (9 7.30b), and hence. the asymptotic result for Y follows from (9.7.19):

370

Solitons ond rhr Inverse Scoranng Transform Y - ( O , + ~ ~ & ) C - ' " ' as

(9.7.37)

x+--DD.

Consequently.

(Yyf-vY,)+o

(9.7.38)

as x+--.

Obviously, p ( r : k ) = 0 for all time 1, and (9.7.36) can k integrated to obtain

(Y'/v)=h(r;k)

or

Y'=yh.

(9.7.39)

wh en h is mother arbivary function of integration. The use of asymptotic results for v and Y as x +--, to (9.7.39) yields the equation

d"+ 4 a 3 = ha. r

(9.7.40)

Similarly, we next use (9.7.30a) and the myrnptoticform of Y', m x the form p

- db r'"+4ik3(e'*'-br~'')

++-,

in

(9.7.41)

dr

in (9.7.39) to obtain

db c'*' +4ik3(e*'' -be'")= h(r"'+ Thus. it follows from the linear independence of

_dbr _4&b=hb

and

be"').

(9.7.42)

e'" and e-''

that

h=4ik'.

(9.7.43ab)

and therefore, equation (9.7.40) gives

(9.7.44)

!LO. dt Finally. the solutions for a and b are given by

a(kr)=o(kO) and

b(kr)=b(kO) erp(8ik'r).

f>O.

This completes the determination of the scattering data summarized ar follows: =constant:

cm(r)= cn(0)crp(4r,)r)

I

,

o(kr)=o(kO); where e.(O), o ( k 0 ) . and b ( k 0 ) the KdV equation.

b(k;r)=b(kO)exp(8ik't)

(9.7.45ab)

s(r). which are

(9.7.46)

an determined from the initial condition for

9.7 The I n v e m Scotming Tmrufom (ISTI Method

371

The simplest approach to solving the inverse scattering problem involves integral representation of solutions of the SehrMinger equation (9.7.7) with 1=k’

L v = k ‘v .

(9.7.47)

It is convenient to introduce the following integral representation of the solution of (9.7.47)

o,(=)=..p(ikr)+lK(x.r)erp(ilrr)&,

(9.7.48)

& ( x ) = exp(-i k . r ) + l K (qz) exp(-ikz)dz.

(9.7.49)

These arc called the lost soludonr which have the following properly:

._-

lim &(x)=exp(*

(9.7.50)

ikx).

As already stated earlier, when k is real. the eigsnfunstionr are continuous (unbounded) states. since 1 > 0 . On lhe other hand, if k is a purely imaginary. so that k = i k. where k is real, then 1 < 0, and the eigenfunctions cornspond to discrete (bound) states. Henceforth. we adopt k for continuous states and i K for discrete slates.

Substituting & , ( x ) in (9.7.47) leads to the inhomogenous wave equation J’K _ _ _ =J’K Jx’ Jr’

uK

(9.7.51)

and K ( r , r ) isrelatedtothepotential u ( x . t ) by

K(x,x), & when K ( x , z ) satisfies the following conditions:

(9.7.52)

u(x,r)=-2

lim K ( x . z ) = O r-*-

and

lim - = O . z+*-

&

(9.7.53ab)

Because + , ( x ) and &(x) are linearly independent, they can be mated as fundamental Solutions of the original equation (9.7.47). Hence, the general ~olutionof (9.7.47) corresponding to the eigenvalue k can be written as a linear combination of these two ~olutions.In panicular, we examine the solution v,(x) of(9.7.47)withasymptoticbehavior v,(x)-exp(-ikx). as x + - - . After some algebraic manipulation, it can be shown that v,(x) is given by

v,(x)=-i-

m+* #,,(.). a(k)

(9.7.54)

SubStiNting results for #,,(x) fmm (9.7.48) and (9.7.49) in (9 7.54) yields

o ( k ) W , ( x ) = e i p ( - i k x ) + l K ( x , z ) exp(-iki) dr

Multiplying this equation by (l*)-lexp(iky). with respect to k gives the following result

-1 -

when y > x , and integrating

n(i)exp(rky)dk

2* .-

(9.7 56)

B ( x ) = L jb(k)erp(ikx)dk

zn

(9.7.57d

.-

and 1 lexp[ik(y-x)]

z*

-

1 = S(y-.r) =0, y >

X.

(9.7.576)

Clearly. the function B(x) represents the Fourier transform of the reflection cocffieient b ( k ) and is a known quantity. The right-hand side of (9.7.55) has terms involving only an unknown K which is to be determined. I n order to determine K, we first simplify the left-hand side of (9.7.56) by using the theory of residues of analytic functions. Without any further details. it turns out from the residue computation from the simple pole of o ( k ) at k = i K with v,.(x)=c,#,Jz) that the left-handsideof(9.7.56)gives

There is a corresponding contribution to this equation from every bound state of the potential. Therefore. if there are N discrete states. (9.7.58) has to be modified by including a summation over the individual states. Finally. we combine this with result (9.7.56) and incorporate the I-dependence of each quantity to obtain !he integral equation for K(x,y;t) in the form ~ ( r , y ; , ) + ~ ( x + y ; t ) + l ~ ( r , r~;(t y) + r ; tm) = o ,

(9.7.59)

where B(r.f) is now given by ~ ( x , t ) = i c : ( t ) e x p ( - q x ) + l j b ( k , t ) erp(ikr) "4 27~

-

1.(9.7.60)

Fqquation (9.7.59) with the kernel B ( x , t ) given by (9.7.60) IS known as the Gelfond-LeviInn-Morchrnko (GLM) linear integral equation. ?he solution K ( x , y ; I ) of this integral equation is related to the potential U(X.I) by the following result:

+,r)=-zd ~(x,x;i). (9.7.61) dr This is the cxacr s ~ l ~ t of i ~the n original KdV equation. However, this does not admit any physical interpretation, which C M be achieved from the following asymptotre analysis and examples. For futun reference. it is convenient to write the final explicit formula for B ( 1 , I ) as

j b( t . 0 ) e r p ( 8 i k 3 r + i l u ) d k(9.7.62) . 2n In general, it is not possible to solve the integral equation (9.6.59) except for

B(x,I)= ic:(0)eip(8nb'r-x,x)+I

-

the reflectionless potentialr ( b ( k ) = O ) . For the general case, the longtime solution represents N SolitOnE traveling with various speeds to the right, and an oscillatory Wavewain with amplitude decmsing with time can be found to follow the seriesof solitons. are For a given initial potential energy function, the cigcnvalucs rm constants for all time. Phyacally. this means that bound-state energy levels are completely specifred and remain unchanged for varying the potential through the

Solitom and the hverrr Seottcring Transform

314

KdV equation. Another remarkable fact to observe is that the number of Solitons that eventually develop is exactly the number of bound states. Of course, this number depends on the initial state u ( x , O ) = u o ( x ) , which is sufficient for determining the number of developing solitons. This completes the description of the inverse scattering vansfom method for solving the KdV equation. I n summary, two distinct steps are involved in the method: (i) the solution of the Schredinger equation (the Sturm-Liouvillc problem) for a given initial condition, u ( x , O ) = u o ( x ) , from which we determine the scattering data S(r) and (ii) the solution of the GLM linear integral equation. Even though these two steps involved may not be technically easy to handle, however, in principle. the problem is solved. me effectiveness of the method can be best exemplified by many simple but nonvivial examples. The power and success of the inverse scattering uansform method for solving the KdV equation can be atuibuted to s e v e d facts. First. the most remarkable result of the method is the fact that the discrete eigenvalues of the Schiidinger equation do not change as the ptsntial evolves according to the KdV equation. Second,the meUlod has reduced solving a nonlinear PDE to solving two lineor problems: (i) a second-order ordinary differential equation: and (ii) an integral equation. Third, the eigenvalues of the ODE are con it ant^ and this leads to a major simplification in the evolution equation for v . Founh, the time evolution of scattering data is explicitly determined from the asymptotic form of v ,as . . So. this information allows us to solve the inverse scattering problem and, hence, to find the final solution of the KdV equation. The method is presented schematically in Figure 9.7.

II(+-.

u(x,O)=u.(x) I

DirectScat~eting

+,I)

Figure 9.7

S(0)

1

I

1""ersS Scatlcring

S(,)

The IST method for the KdV equstion.

We now illustrate the method for the reflectionless case ( b ( k ) = O ) with initial profiles given by the following examples.

9.7 The Inverse Sconering Transform ( I S T j Method

375

Example 9.7.1 (Single-Soliron Solution). We consider a particular case where the potential has a single bound State with the eigenvalue 1 = -K:. normalizationconstant co, and with zem reflection coefficient ( b ( k , r ) 0 ) For all continuous states k. In this case, B is obtained from (9.7.62) in the form B(x.r) =c,' ( 0 ) erp(8r:r-

K+)=

c,' exp(8K:

r-

K+).

(9.7.63)

I h e associated GLM equation (9.7.60) can be solved by assuming a separable

kernel

K ( x , y . r ) = z K " ( x . r ) cxp(-K,,y),

(9.7.64)

"d

which. in this case (N = I), becomes

K ( x . y ; i ) = K,(x.r) erp(-K,y).

(9.7.65)

The GLM equation (9.7.60) with (9.7.65) is given by K ( x . y : r ) + c i exp{sK:r- r , ( x + y ) ] +ct

e x p ( t r : r - r , y ) J K ( . r , z ; r )e r p ( - x , r ) d z = O .

(9.7.66)

Substituting (9.7.65) in (9.7.66) and performing the integration over z leads to an algebraicequationfor K,(x,r) K,(x,r) e r p ( - ~ , y ) +c,' exp{.sr: r - q ( x + y ) )

+ Ae x p ( 8 ~ :i- ~ , y ~- q x K,(x,r) ) = 0. 2 'c, This equation solved for K , ( x , i ) , can k written in the form K,(x,r) = - K , exp(4K:r

+ K&)

seeh[K, (I-,, -4x: r)].

where xa is defined by Xo

= (ZK,).' log(

51.

(9.7.67)

(9.7.68)

(9.7.69)

We substitute (9.7.68) in (9.7.65) to obtain the solution for K ( x , y ; f ) . Consequently, the ~olution(9.7.61) of the KdV equation becomes "(=,I) = -7.1~:

sech'

[K,(X - ( I t ) -

K,xo],

(9.7.70)

316

Solironr and the Inverse Scarrering Trmlfom

where U= 4 r:. Thbs represents a single-soliton solution of amplitude - 2 ~ :

traveling to the right with speed U = 4k: and centered initially at the point xa. Example 9.7.1 (N-Soliton Solution for n Sech‘ Potential). investigate the scallenng problem for the class of seeh’ potentials given hy u(x.0) = uI1( x ) = -Usech’

We

X.

(9.7.71)

y r_ +(I+U se c h ’x ) y=O.

(9.7.72)

where U IS a constant. The arswiatcd Schrdinger equation is

We introduce a change of variable by y = tanhx so that -I < y < I for < x < -. In terms of y. the potential (9.7.71) becomes

--

Uscch’x = U(1- lanh’ x ) = U(I - y’), and the second derivative in terms of y 1s given by

Consequently. equation (9 7.72) gives the associated Legendre equntion

yr=o.

r ; ” l ; + . [ + ]Y i ) ~y - * ( [ ;

(9.7.73)

We set U = N ( N + I). where N is a positive integer, and then, consider only the bound sLBtes ( b ( k . t ) = O far all k ) Physically, this means that the incident wave is totally reflected. end hence, the associated potential IS refemd to as reflectionless. If I =-K’ < O , then, only bounded solutions for -I S y S 1 occur when Kw = n , n = 5 2 , ...,N,andthecorrespondingdis=ctceigenfunctions

are given by

~ “ ( 1= ) An

Pi (tanhx). where the ossociotcd Irgrdrefunctions

p”, (y) ine definedby p“, (y )=(-I)”(l -~’)” ’~

P,,(y) and P,,(y) =

4“

IN !dZX“dy’( , ’-1)N.(9.7.74ab)

P”(y) is the Lcgendre polynorniol of degree N, and the e ~ n s t a n tof ~ proportionality A, are determined from the normalization condition (9.7.24). The asymptotic npreientauon of the discrete eigsnfunctions has the form y”(.)-c”(r)exp(-nx)

85 X + + ” ,

(9.7.75)

377

9.7 The Inverse Scattering Transform ffSTJMahod Where

= CJO) erp(4n't)

C"(t)

(9.7.76)

and ~ " ( 0are ) also determined from the normalization condinon. The function B(r:r) involved in the GLM integral equation (9 7.601, therefore, is given by

B ( r : r )= t c : ( O )

exp(8n'r-nr).

(9.7.77)

"4

Finally. the GLM quation can easily be solved. and hence, the N-soliton

solution assumes the asymptotic form u(x,t)

- - ~ ( 2 n ' ) s e c h ' { n ( x - 4 n 1 t ) r ~ " }a3 f-f-.

(9.7.78)

"4

are known p h m s. Clearly. a refleetionlesspotential with N bound states cornsponds

wherethequantities

E"

to p u n Nd m n s ordered according to their amplitudes. As 1 +-, the tallest (hence, the fastest) soliton is at the front, followed by a series of progressively shorter (therefore, slower) ones behind. All N solitons interact at t = 0 to form the single sech' profile, which was pnsctibed as the iniual condition at that instant. In particular. when N = 1, (I= 2 and k = K, = I , The corresponding eigenfunction is given by

y,(.x)=A, P;(tanhx)=-A, scchx. when the constant A, is determined fmm the condition (9.7.24). which gives 2A' = 1 or A =

* (I/&).

So. the asymptouc nature of the above solution is

V,(x)-&e-'

Therefore, e,(0)=&. B ( x ; r ) = 2 exp(8r-x).

and

as x + - .

c,(t)=&exp(4r),

and

the

function

Rnally, the associated GLM equation

1

K ( x , z ; I ) + 2 exp[8t - ( x + z ) ] + 2 K ( x ,y; I ) exp[St-(y+ z ) ] dy = 0 gwes the SOlUtlO" K ( r , z : t ) = F ( x : r )e v - 2 ) . for some function F ( x ; l ) that satisfies the quation

This can readily be solved to find the function F expressed as F(x;t)=-

-*p 1

+ 2-1'

Thus. the final solution is given by

This represents the single-soliton ~olvtionof amplitude - 2 and speed 4. Similarly. the two-solitoo solution can be obtained from the initial profile (9.7.71) with N = 2 , so that (9.7.80)

u(x,0)=-6scch1x.

This initial condition is !hen evolved in time according to the KdV quauon. and the solution consists of two solitom where the taller one catches up the shorler one. They undergo B nonlinear interaction according to the KdV equation and, then. emerge from !he interaction unchanged in waveform and amplitude. Eventually, the taller soliton reappean to the right and passes away from the shorler one as I increases. The wave. ~rofile"(=.I) . , with the initial condition (9.7.80) is plotted in Figure 9.8 as a function of x for different values Of tme: (a)I=-0.5,(b) I=-O.l,(e)r=O.O,(d) r=O.l,a nd( e ) I = 0.5. Similarly, the thme-soliton solution (N =3) with u(x,0)=-12scch2x is shown rn Figure 9.9 for different values of I : (a) I =O.O, (b) 1=0.05. and (e) I = 0.2, where evolving solitons have amplrtuder 18.8, and 2. respectively. Finally. to obtain the soliton solut~on.we examine the asymptotic behavior of the exact solution "(1.1) for large x and I by considering only B ( x , r ) given by (9.7.62). For large I and 1-4ki1, (K,c k2 <

where

kWis

the largest eigenvalue

... < K.),

the Nth term in the series in (9.7.62) dominates over all other terms. and the significant contribution to the integral in (9.7.62) can be found from the stationary phase approximation (see Segur, 1973) in the form I-''' BS I + -. Thus, it turns out that B(x,t)

- a enp(-rr,x),

whne

a = ei(0) exp(8KiI).

(9.7.81)

k* ,=-0.1

-8

0

-4

h(d)kJp ,=a1

0

F i y r e 9.8

4

.

8

,=as

0

4

8

The two-soliton solution. (From Drsdn and Johnson (1989))

(9.7.82) 'Ibis can be easily solved by writing K ( x , z , t )= F(x.1) erp(-x,r).

(9.7.83)

380

(0)

1 = 0.0

-5

Figure 9.9

0

5

x

The three-soliton dotion. (From Drazin and Johnson (1989))

Result (9 7.84) gives

Substituting (9.7.83) in (9.7.61) giver u(x.1)

- -2r:Eeeh2[K,(r

-xo)-4K~t],

(9.7.86)

when C : ( O ) / Z K , = eXp(2KNx0). Obviously (9.7.86) aSymptOflcally represents a soliton of amplitude I

-

4K:I

2 ~ :and velocity 4 ~ : . If. instead of large I. we consider

for any arbitrary n, the significant term in the series involved in

(9.7.86) is the nth term. and then. B soliton of amplitude 2 ~ and : sped 4 ~ : would emerge. Thus, the upshot of this analysis 1s that the initial (potential well) profile disintegrates into N solitons corresponding to discrete eigenvalues of the associated Schrtdmger equation.A further elaborate mathematical analysis of the integral in (9 7.62) reveals a complete solution consisting of a sene$ of N d i t o n ~preceded by an oscillatory uail of amplirude decnasing wrth time. This disintegration process of an initial profile into a series of solitons is usually calledfission.

Examples 9.7.1 and 9.7.2 illustrate thc rcflcctionlcss (b(k) = 0) initial profiles We now give examples for the nonzero reflection coefficient (W) 0).

Example 9.7.3 (A Soliron Solution Associated with Dirne's D ~ I O Funcaon InrIiol Profile). In this ease, we use the initial condition as " ( X . 0 ) = -u0 S(X),

(9.7.87)

where uo is a positive constant. For a discrete eigenstate, I , =-K:. k, =

uI) and the corresponding eigenfunction is given by

The continuous cigcnfunction for k = fiexists and can be written as

where

(9.7.89)

b ( k ) = - A (u. + 2 i k ) ’

We can then use the time evolution of the scattering data .,(I)=& erp(4r:r)

and b ( k . r ) = - e crp(8ik’r). (9.7.90ab) (%+2ik)

The function B(x.1) canberepresenledby

However. the function K ( x . y ; t) cannot easily be determined fmm (9.7.59). But the asymptotic solution for u(x.r) for the single soliton associated with the

d i m e eigenvalue K, =

uo and fix& ( x - u:

1)

is given by

where the phase change is given by erp(Zk,a,) = c i ( O ) / 2 r , ,

In fact, the asymptotic solution (9.7.92) is made up of the contribution to B(1.r) fmm the first term in (9.7.91). and the integral term vanishes when asymptotic solutmn (9.7.92) IS valid. The initial pmfile represenled by the delta ) a subsequent function is shown in Figure 9.10(a), and the solution ~ ( x . 1 for time is plotled in Figure 9.10(b).

Figure 9.10

(a) Initial profile

and (b) solution at

B

later time.

9.7 The Inverse Scomnng Tranfom (ISTI Method

383

Finally, sf the amplitude ug of the initial profile is negative. there is no discrete eigenvalue, and hence. there is no solimn. But the only dispesive w a r n exists in the solution for t > 0. As shown above, if uo > 0, there is only one cigenvalue k, = % and hence a single ~olitonwith amplitude 2r: =

u:

is

generated. Erantple 9.7.4 (Solitons Associated with Ncgalive Sech’ Initial Profile$). We use the method described by Crandall(1991) to examine the development of

solitons and a dispersive wave associated with B class of potenuals given by u ( x , O ) = u o ( x ) =-U sech‘x.

(9.7.93)

According to the general theory of solitons. the total number of discrete eigenvalues for general U is given by

(9 7.94)

where [

1 denotes the integral pan, and, if

{r :1’ :} U+-

--

IS an

integer. then.

the +I is omitted. For U = 4, X = 2. and hence, there are two discrete eigcnvalues and, therefore. a tolution with two solitons. Moreover. (I= 4 cannot be written as N ( N + I ) for integral N , and the solution also includes a dispersive wave. Finally, the solution of the KdV equation IS determined by numerical cnteptxon. The end result is that there are mnmbutmns from both the discrete and continuous states to the function B( r + I) in equation (9.7.60). The effect of the different contributions is seen m the time evolution of the solution which reveals the appearance of two solitons moving to the right and B dispersive Wavemain kaveling to the kt3 as shown in Figure 9.1 I. We close this ~eetionby adding a brief dmurrlon on the number of solitons associated with an initial profile u(x.O)=u.(r). . ” . The rolution depends on the discrete cigenvalues of the associated Schrklmgerequation.We discuss &is point by citing some specific examples. For the case u,(x)=-o sech’(bx). the n u m k r of eigenvaluer is given (Landau and Lifshitz. 1959. p. 70) by

384

Lx -12

-6

0

6

( b ) r.O.4 I2

,=lo

Figore 9.11 The lime evolution of the solution at three different times. (From Drszin and Johnson (1989))

where the number of eigenvalues is determined by the parameter P defined by

-

P = Jl"&)l"*dr.

(9.7.96)

9 7 The Inverse Scarering Trmrfom (ISTI Method

385

For the rech' profile, P = K b 6 . The number of solitons is given by

This shows that the number of solitons depend on the parameter P. which, in this case, is propamonal to b 6 . For the delta function case,

and P+ 0 as n + -, and hence, there IS only one soliton consistent with (9.7.97).In the other limit BE P + the formula (9.7.97)gives

-.

N-p.

(9.7.99)

K

Thus. result (9.7.97)shows that then is always one soliton for small P,and as P increases (P+ -), the number of solitons increases. Furthermore. when the initial disturbance IS large (P+-). there are many closely spaced sigenvalues that sausfy the famous Bohr-Sommerfeld rule

fp&=f[A-uJ.x)~

&=2x

(

n+

-)!.

(9.7.100)

Thuo. the number of solitons (the largest value of n for A = 0) ISgiven by

This ensures the validity of the result Obtained in the previous examples. Another approximate formula for the number of solitons with amplitudes in (o,n+do) was 6 n t obmned by K q m a n (1967)in the form N ( o ) 8=Kl f [ l u o ( r ) l - ~ ~ & .

This is over the range O< n c 2uOm= 21 uo

Imy, where uomis the largest value of

1 1 1fortheboundstatcsin(9.7.100),sincelherangcof Hence. the tow number of solitons is given by 2..

N = jN( n ) d o ='

(9.7.102)

n

jluo

I"&.

k

is O < r < G .

(9.7.103)

This is in agreement with (9.7.101).Thus. the dependence of P on the s i m (o and b) of a soliton suggests another general result

386

Soblorn m d the Inverse Scotrering Trmsfonn

(9.7.104)

Q=JluVdx.

Physically, this represents an interesting mesure of soliton shapes for a single soliton. and (9.7.104) E M easily be computed to obtain Q h L, which IS independent of the amplitude KN. Since solitons here are of unit size,

-

Q = h xN for a series of N solitons. This shows that there is a Planck's constant for solitons! It IS wonh noting here that he parameter P is the value of the integral in the initial drsturbancc. For large P. N P / z , and for large tlme 1,

-

p=h

L N -P.h

(9.7.105)

This describes the close connection between the initial P and the final Q.

9.8

Backlund Transformations and the Nonlinear Superposition Principle

Historically. Brklund vansformations were developed in the 1880s to study the related theories of differential geometry and differential equations. They occurred as an extension of contact transformations. which franrform surfaces with a common tangent at a point in one space into surfaces in another space. which also have a common tangent at the correlpondmg point. One of the earliest Bicklund transformations was found for the rme-Gordon equation. u,, =sin U . This equation onginally a m ~ e in differential geometry in connection with the theory of surfaces of constant negative curvature. However,the study of these transformations had been dormant until the recent work on ~olitonsxn the 1970s. It has been recognized that panial differential equations admit solitonlike solutions sf and only if they admit Backlund tranrformatms. Indeed. there IS a close relationship between the inverse scattering vansform (ISTj and a BBcklund transformation (BT), in the sense that the scattenng problem and the associated time dependence that constitute an IST also constitute a BT. In other words, every ev01ut~on equation solvable by IST has B corresponding BT, and. conversely, the existence of Bicklund transformations always or almost always implies integrability by the IST. Another approach to deriving conservation laws and the mverse scattering problem is through the use of the Blcklund uanrformationr. For second-order partial differential equations, the BT consists of a pair of first-order partial differential equauons relating the solution of the given equation to another ~ o b t m nof the same equation or to a solution of another second-order equation. In general, a Backlund transformation for a second-ordcr partial differential equationfor a dependent -able u ( X J ) is defined by the pair of equations

9.8 Bockfund Trmsfomtiom ond the Nonlinear Superposition Principle

w.=P(w.".+.U,,xJ)

1,

387

(9.8.lab)

~,=Q(W,U,U..U,,X.~)

where P and Q are functions of the variables indicated but not of the derivatives of w. A new equation for w is obtlined from the consistencycondition. One of the simplest Bzcklund transformations is the pair of the CauehyReimann equations in complex analysis u. = vy and uI =-v,, (9.8.2ab)

for the Laplace equations U _ + U , ~ = O and v _ + v n = O .

(9.8.3ab)

Thus. if v(x,y) = *y is B simple solution ofthe laplace equation, then u(x.y)

eanbcdeteminedfmm u , = x a nd u,=-y.Therefore,

u ( x , y ) = ~ ( x ' - y ' ) is

another solutionof the Lapla~eequation. In connection with the Miura transformation(9.6.10). it has been shown that. if v is a solution of the modified KdV equation (9.6.1 1). then, u is a solution of the KdV equation (9.6.3). Since it is possible to eliminate higher derivatives from (9.6.11) by using (9.6.10), we can Veal (9.6.10) and (9.6.11), written in the form uX=u-v2 and v,=6u1v,-v-,

(9.8.4ab)

as a BT for the KdV equation (9.6.3). Wahlquist and Estabrook (1973, 1975, 1976) have developed a more convenient and useful Blcklund transformation for solutions of the KdV equation. Since the KdV equation is Galilean invariant, we replace u in (9.6.1 I) by u - I . w h c n I isarcalparamctcr.Using u = I + u ' + v , , we r c wr i t e ! h c modified KdV equation (9.6.1 I ) in the form

v,-6(v2+I)vx+v,

(9.8.5)

= O.

u satisfies the KdV equation (9.6.3). Clearly. both v and -v satisfy (9.8.5). This leads us to wn~rmcttwo functions u, and +. comsponding to v SO that

and - v , in the form u, =

for B given

I

+ "2 +

"..

- v,

(9.8.6ab)

=2(I+ " I ) .

(9.8.7ab)

u1 = I

+ "1

I and v.

CoNeq"ently, u, -up

=Zg,

u, +u2

388

It

Sobtons and the Inverse Scomnng Transform

convenient 10 define B new potential function w,. such that u, = &,/Jx, i = 1,2. Thus. equations (9.8.7ab) nducc lo the following pair of IS

equations: w, -w* = 2u, (w, +wJ, =2A+$,

(9.8.8)

-w*p,

(9.8.9)

where any arbitrary function of I in (9.8.8) has been incoprated in w, without changing y . We next use (9.8.7ab) and (9.8.8) in order lo transform (9.8.5) in the form (w,

- 4, = 3 ( w t -wi)-(w, - 4 ,.

(9.8.10)

Thus, the uphot of this analysis is that equations (9.8.9) and (9.8.10) c o n ~ t i t ~ t e a pair of aulo-Bbklund transformations of t k KdV equation. when the former is the x pan and the lalter is the f pan. We next illusmte the method by solving (9.8.9)-(9.8.10) for w, with wl = 0 for all I and 1. The resulting equations for w, become w,=2A+iw:

and w , , = 3 w i - w , - ,

(9.8.11ab)

w h a t the former gwes w,~=$(w,w,.)=wt+w:w,,.

(9.8.12)

andhenee.(9.8.11h)canhewritten. byusing(9.8 1la)againand A = - x ’ . as

=o.

(9.8.13)

- 4K’I).

(9.8.14)

w,,+4r’w, Tlus equation admits the general solution wl(X,f) = .f(x

wherefis an arbitrary function. On Un other hand, (9.8.1 la) can readily he integrated lo obmiin the solution w,(x.f) = -2K bnh[ K x + a(f)]. when a(f) is an arbitrary function of

1.

(9.8.15)

For consistency of the solulions

(9.8.14)and(9.8.15), werequirethat a ( f ) = - 4 x ( x 2 f - x o ) . where xo is an arbitrary constmt. Thus, the BLklund transformations give the final solution w,(x,t)=-Z~tanh[r(x-x,-4r’l)].

and hence. il follows from uI = w,, that

(9.8.16)

38 9

9 8 Bocklund Tmnsfomasons and rhc Nonlmror Sup.rporrfron Pnnnph

u,(x,f)=

- 2 d sech' [ K ( x - z , -4K'r)].

(9.8.17)

This is the soliton ioluuon of the KdV equation (9.6.3). All of this illurtrates how the Bscklund vansformations can be used to obmn the soliton solution of the KdV equation. However. the procedure just demonstrated requires two integrations, one with respect to x and the other with respect to 1. Instead of the process of integration involving arbitrary functions. Wahlquist and Estabrook (1973) developed an elegant method. based on the theory of differential forms, for determining Solutions of the KdV equation to obtain a one-parameter family of Bscklund Uanrformations. In this method. two distinct solutions w, and wI are pneraled from Ihe Bscklund vansformauons by using the same given solution w with two different values 1,and 1, of 1. Thus, equation (9.8.9) can be written in two different forms,

-q, (w2 +w), = 21, +qw* -q. (w,+w)= = 2 1 , + q w ,

(9.8.18a) (9.8.18b)

now possible to construct another solution w , from ~ w, and 1, and. similarly, a solution w2, fmm w2 and I , , so that It

IS

(w,*

..(

+WJ.

=22, +L(w,*

+W2)'

=21,

-wJ.

++,, -.J.

(9.8.194 (9.8.19b)

We next use Bianehi's theorem of permutability for the BIcklund transformations in differential geomeuy, which states that w , =~w ~ ,Now. . we

subtract the difference of equations (9.8.18ab) from the difference of equations (9.8.19ab) and use the identity wI1 = w,,. so as to produce zero on the left-hand side of the resulting equation: 0=4(1,

-1,)+L[(y2- w , r

- ( w 2 , - w l r - ( w , -w)'

+(wx - w r

1

,

*,) 1s given by

whence the soluuon for w ,2 (-- w

'

(9.8.20)

w,2=w--,

(w,-w*)

Thus, 11 is now possible to find solutions of the KdV equation in a straightforward manner. Equatiun (9.8.20) is a p w d y simple algebraic expression, known as a nonlinear superposition principle, for constructing

Solitons ond !he Inverse Searmining Trmsfom

390

solutions. It is possible u) generalize superposition formula (9.8.20) and use h e procedure to EO~SVUCI multisoliton solutions. A pmcess Similar to the two-roliton solution can be generalized to obtain a thne-soliton solution of the KdV equation: Wll,

= w,--

4(L -2.

)

WL1 -w,s

- A~wt(w2

-w#)+L3w3(w!-w2)

-wi)+Lxw~(w~

(9.8.21)

A,(w~-w,)+A~(w,-w,)+",(w,-w*)

This represents the nonlinear supcrposirion principle for the three-soliton solution. Thus,the multisoliton can he produced by continuing this pmcess and sctung w, = - 4 r f seeh'

where

1, is

[fir , ( x -8r::t-

x,)].

(9.8.22)

constant.

In particular, when w = 0. w, and w2 represent hounded and unbouled solutions correspondingto bounded and unbounded solitons for Y expressed as u, =-4x'scch'

[fix(x-8x*t-x0)].

(9.8.23)

u2 = -4r'csch'

[ar ( x - 8 d t -

(9.8.24)

ad z~)].

Although solution (9.8.24) is not of physical interest, it is essential for the construction of a hounded solution for w , ~ which. . in the present CIEC. represents a two-soliton solution.

Example 9.8.1 (Two-Soliton Sofurion). We apply the nonlinear superposition principle (9.8.20) to derive the two-soliton solution of the KdV equation. We take wo=O. SO

w, =-2mnh(x-4r),

wl =-4coth(Zx-32t),

(9.8.25)

that A, = - I and A, = -4. Consequently. (9.8.20)becomes w I' =-

6 {Zcou1(2x - 3 2 ) - tanh(x- 4t)} '

The corresponding solution of the KdV equation follows from the result

(9.8.26)

391

9.9 The Lor formulotion

” ‘I

J

--w

- Jx

’’

=-

6 (4 cosech’ (21 - 3 2 ) + rwh’

=-

(Zeosh(2x

=-

(X

- 41))

{2coth(2x-32f)-tanh(x-41)}* 6(4cosh’ ( x - 4f)+anh’ ( Z x - 321)) - 32f) cosh(x - 4f) -sinh@x - 3 2 ) sinh(x - 41))’

lZ{3+ 4cosh(Zx -81)+ cosh(4z- ad!)} ’

(9.8 27)

{3cosh(x - 28f)+ cosh(3x- 36f))’

This represents the two-Jolitan Solution associated with the initial profile (9.7.80). The numerical values of (9.8.27) for five different times (a) I = - 0 . 5 , (b) I = -0.1, (c) I = 0.0, (d) f =0.1, and (c) 1 =0.5 plotted BS a function of X. give the same wave profile u(x,f) BS in Figure 9.8.

9.9

The Lax Formulation, Its KdV Hierarchy, and the Zakharov and Shabat (ZS) Scheme

In his 1968 seminal paper. Lax developed an elegant formalism for finding isospeetral potentials BS solutions of a nonlinear evolution equation with all of its integrals. This work deals with some new and fundamental ideas and deeper result^ and their application to the KdV model. This work subsequently paved the way to generalizations of the technique as a method for rolvlng other nonlinear partial differential equations. L a x developed the method of inverse scattering based upon an absuact formulation of evolution equations and eenain properties of operators on a Hilbert space. some of which are familiar ~n the context of quantum mechanics. His formulation has the feature of associating c e ~ a i nnonlmear evolution equations with linear equations which we analogs of the Schmdingerequation for the KdV quauon. To formulate Lax’s method, we eonrider two linear operators Land M. The cigenvalue equation related to the operator L corresponds to the Schrodinger equation for the KdV equation.The generalform of this eigenvalue equation is

Lv=Lv,

(9.9.1)

where 8y IS the eigenfunclion and L is the corresponding eigcnvalue. The operator M describes the change ofthe eigcnvaluer with the parameter I, which usually represenu time in a nonlinear evolution equation. The general form of this evolution equation is Vc=MV (9.9.2) Differentiating (9.9.1) with respect to f gives

392

Solitons and the Inverse Scomnng Transform

L, v + L v,= 1,v + v,. We next eliminate v,from (9.9.3) by using (9.9.2) and obtain L,

v+LM

= 1,v +d M

v = 1,y + M d 8y = 1,v +M L v .

(9.9.3) (9.9.4)

or. equivalently,

% v=d,yr+(ML-LM)v.

(9.9.5)

Thus, eigenvslues a m constant for nonzero cigenfunctions if and only if _" =-

( LM

- M L ) v = -[L, M I .

(9.9.6)

where [ L, M] = ( LM - M L) IScalled the cornmufaforof the aperaton L and M , and the derivative on the left-hand ride of (9.9.6) is to be interpreted as the time derivative of the operator alone. Equation (9.9.6) is called the L a equation and the opemton L and M are called the Larpoir. The problem, of course,is how to determine these operaton for a given evolution equation. Then IS no systematic method of solution of this problem. We consider the mitial-value pmblem for u(x.1) which satisfies the nonlinear evolutionequation of the form (9.9.7)

u, = N(")

with the initial condition

"

(9.9.8) (1.0) =f(XI> where u E Y for all f, Y IS a suitable function space. and N : Y + Y is a nonlinear operator that is independent o f t but may involve x or derivatives wlth respect to x. W e must assume that the evolution equation (9.9.7) can be expressed in the L a x form L, +( L M - M L) = L, +[ L, MI = 0,

(9.9.9)

when L and M are l i n w o p t o n in x on a Hilbert space H and depend on u and L, = u , is B scalar operator. We also assume that L is self-adjoint so that ( L @ . v ) = ( @ . Lforall y ) @and v e H with (.:)a~minnerproduct. We now formulate the eigenvalue problem for

v E H:

L v = d ( f ) v , 120. x s R . (9.9.10) Differentiating with respect to f and making use of (9.9.9), we obtain

1,v = (L-M v). The inner product of v with this equation yields

.qv,

(9.9.11)

9.9 The Lar Formulollon

393

(w.w)A,=((L-I)w.I,-Mv).

(9.9.12)

which is. since L - A is self-adjoint, given by (K V )A, = (0.V, - M v )= 0.

Hence. A , = O , confirming that each eigenvalue of L is a constant. Consequently, (9.9.1 1) becomes

L( V,- M V )= ( V , - M v).

(9.9.13)

This shows that ( yr, - M y ) 8s an eigenfunction of the operator L with the eigenvaluc A . It ir always possible to redefine M by adding the pmduct of the identity operator and B suitable function of 1. so that the onginal equation (9.9 9) remains unchanged. This leads to the time-evolution equation for y as ~ , = M v , 120.

(9.9.14)

Thus, we have the following: Theorem 9.9.1 If the evolution equation (9.9.7) can be expressed as the Lnrequnrion

L,+[L,M ]= 0 and if (9.9.10) holds. then k,= 0,and

(9.9.15)

sabsfies (9.9.14).

It is not yet clear how to find the operatonL and M that satisfy the preceding conditions. To illurvate thc L a x method,we choose the SchrOdingeroperator L in the form (9.9.16)

L y = A y becomes the Sturm-Lmville problem for the self-adjoint operator L.With this given L, the problem i s to find the operator M. Based OD so that

the theory of linear unitary operator on a Hilben space H. the linear operator M can be chosen as antisymmetne, so that ( M @ .y ) = - ( 0 , M y ) for all 0 and 0 E H. So. a suitable linear combination of odd derivatives in x is a natural choice for M. It follows from the inner pmduct that

provided M =J”+/Jx‘

foroddnand

0, y , and theirderivatives

withrcspcct to

I I+ -. Moreover. we require that M has sufficient freedom

x tend to ~ r oas. x

in any unknown constants or functions

to

make L,+[L.M] a multiplicative

Solitons ond rhs Inverse Scoifcring Trmsfmn

394

operator, that is, of degree zero. For n = I . the simplest choice for M is M = c ( d / d x ) , wherecisaconstant. Itthenfollows that [ L , M ] = - c u , . which IS automatically a multiplicative operator. Thus, L, + [ L ,M ] = u, -cu,.

(9.9.18)

and hence,the onodimensional wave equation u, -cu,

=0

(9.9.19)

has an associated eigenvalue problem with the cigenvaluer that BIC constants of motion. Tbe next natural choice is

M = d'- , - + A ~ A ++ B, L a?

(9.9.20)

ax ax

where a is a constant, A = A(x.r) and B = B ( x , t ) . and the third term on the nght-hand side of (9.9.20) can k dropped. but we remn it for convenience. It follows fmm an algebrae calculation that [ L M ]= nu, - A,

- B_ -2u,A

+(30u_ -4A, - 2 B X ) d + ( 3 a , - 4 A , ) -

J'

ax1

JX

'

This would be a multiplicative operator if A = ? nu and B = B ( t ) . Consequently, thc L a x quation (9.9.15) become

y--

3 2

n

an"*+- " =o. ,

Thrr ISthe standard KdV equation if reduces to the form M=-4d+3

4

LI = 4 .

(9.9.21)

The operator M defined by (9.9.20)

( d,+ d u u

+B(t).

(9.9.22)

J : Jx Hence, the timeevolution equation for y, can k simplified by using the SNrm-Liouvillc equation. w, - ( u -A) = 0 as

V, = 4 ( A V - u V ) , + 3 V. + 3 ( u V), + B V = Z ( u + 2 1 ) V, -ax V + B.I

(9.9.23)

We close this seclion by adding several comments Fint, any evolution equations solvable by IST. like the KdV equation. can be expressed in Lax form. However, the m a n difficulty is that there is no completcly systematic method of finding whether or not a given partial differential equauon produces a Lax

9.9 The Lax Fobnulorion

395

equation and, If so, how to find the Lax pair L and M .So far. guesswork has been used for determining Land M. Indeed. Lax proved that there is an infinite number of operator, M ,one associated with each odd order of J l h . and hence, an infinite family of flows u, under which the rpecvum of L is preserved. Second, it IS parrible to study other specVal equations by choosing alternative forms for L. Third, the nsmction that L and M should be limited to the class of scalar aperaton could be removed. In fact. L and M could be mot* oprrotors. The Lax formulation has already been extended for such operator^. FouRh. Zakhhamv and Shabat (1972, 1974) published a series of notable papers in this field extending the nonlinear SchrMingcr (NLS) equation and other ev~lutmn equations. For the fin1time. they have also generalized the Lax formalism for equations with more than one spatial variable. l h i s extension is usually known as the Zakhorov and Shobot (2s)scheme. which, essentially, follows the Lax method and recasts st in a masix form, leading to a mawx Marchento equation. Finally, we briefly discuss the ZS scheme for nonself-adjoint operators to obtain N-soliton solutions for the NLS equation. Zakharov and Shabat introduced an ingenious method for any nonlinear evolution equauon !$=NU, (9.9.24) to represent the equauon 80 the form

%=~[L,MI=~(LM-ML), (9.9.25) t when Land M an linear differential operators including the function u in the cocffiicientr,and L, refers to differentiating u with respect to t m the cxpnssion for L. We consider the eigenvalue problem L$=l$.

(9.9.26)

Differentiation of (9.9.26) with respect to Igives

i#(

2)

= ( L - l ) ( i O, - MO)

(9.9.27)

If @ satirficr (9.9.26) initially and changes in such a manner that

i$,=MO,

(9.9.28)

then, always satisfies (9.9.26). Equations (9.9 26) and (9.9.28) an the pair of equations coupling the function u(x,t) 8" the coefficients with B scattering problem Indeed,the nature of $ determines the scattering potential in (9.9.26). and the evolutionof 9 in time is given by (9.9.28). Although this formulation is quite general, the crucial step, still, is to factor L according to (9.9.25). Zakharov and Shabat (1972) inmduced 2 x 2 mavices associatcd with (9.9.25) as follows:

396

Soiitonr ond the Inverse Scoifcring Transfern

a].

l ~ o . ]+ ~[:

L=i[':

O]z + k c -'y [I+_

.=-a[' 0

-,ir], i";

(9.9.29)

(9.9.30)

1 ax'

I-a and the N U equation for complex u(x.1) is given by i u , +u,

+ yI u r u =o,

(9.9.31)

Whoe

Y = 2/(1- a').

Thus, the cigenvalue problem (9.9.26) and the evolution equation (9.9.28) complete the inverse scattering problem. The initial-value problem for u ( x , ~ ) C M be solved for B given initial condition "( L O). It seems clear that the significant convibution would come from the point spectrum for large timer (I+-). Physically, the disturbance tends to disintegrate into B series of solimy waves. The mathcmaticsl analysis is limited to asymptotic solutions so that u 1-3 0 as Ix I+ -, but a series of solitary waves is expected to be the end result of UK instability of wavetrains to modulations.

I

In gencra1,todate, there is no completely systematic method of determining the linear scatte"ng problem associated with a given nonlinear evolution equation. However, one systematic method is the ~ofalldprolongolionsructwc method, which WBS introduced by Wahlquist and Estabrook (1975,1976)using the knowledge of Lie algebra. This method w a ~found to be very useful for several nonlinear evolution equations. Dodd and Gibbon (1977)successfully developed the prolongation ~ t r ~ e t ~ofr ea higher order KdV equation. Subsequently, several authors including Dodd and Fordy (1983,1984). Fordy (1990). and Kaup (1980)used the prolongation method for finding the Scattering problem of B given nonlinear evolution equation. Finally, we close this Section by including another commulativc representation of (9.9.9). so that the theory can be extended to other nonlinear evolution equations. It has already been demonseated that the Schrodinger equati, Lyr-k'yr

(9.9.32)

plays a fundamental role in solving the KdV equation. It has also been seen that if the potential u in this equation varies in time according to the KdV equation, then, v satisfies a linear equation

9.9 The Lar Formulation

397 yr, = -4

vd +6u yr, +3u, yr.

(9.9.33)

A necessary condition for these equations to be compatible for all 1 is that, u ( x , l ) satisflct the KdV equation (9.9.1).This can be seen as follows.

We fint transform the Sehrodinger equation (9.9.32)into two fint-order equations by introducing B new function 0 so that yr’=ikyr+@

and @ * = - i k ) + u v

(9.9.34ab)

We can use (9.9.34ab)to eliminate all x-derivatives from (9.9.33) which, then, reduces to the form

~,=4ik3~+4k2@+2ik~-u,yr+ZuQ.

(9.9.35)

Using (9.9.34ab).equauon (9.9.35)leads to an equation for Q In the form @,

=-4ik3@+4k’.@+2ik.,~-2ik”@+(2u’-%)

Introducing a column vector Y =

y r + g @ . (9.9.36)

CI

, we can reformulate the System of

equations (9.9.34ab)-(9.9.36)in a compact form Y x=A(x,l;1)’f’

and ‘f’, = B ( x , 1 ; 1 ) I,

(9.9.37ab)

where 1 IS a complex parameter and A(x,l;1) and B ( x , l ; l ) are 2 x 2 matrices in the form

A=!’ 0

‘]+[I 11.

(9.9.38)

-il

B = d i A S [ ’0 -1 ‘]+4~2[:

:]+2i.~[;,

+[ Zu’-u,

J:

3

(9.9.39)

andA mdBdependonxandrthmugh u(x.1). Differentiating (9.9.374with respect to 1 and (9.9.37b)with respect to x leads 10 the compatibility equation J B ( n ) + [ A ( l ) . B ( l ) ] =O.

ax

(9.9.40)

This is a new commutative representation which can be used tn extend the KdV theory to other nonlinear equations. If we consider the CBSC where A and B arc

398

Solironr and the Inverse Scoriering Trmsfom

polynomial in 1,then the Id-hand ride of (9.9.40) is also a polynomial in 1. Since equation (9.9.40) is true for all values of A , all coefficients of thm polynomial equation must be identically zero. We next substitute (9.9.38) and (9.9.39) in (9.9.40) to obtain all coefficients which vanish except the constant term. This means that the left-hand side of (9.9.40) reduces to the matrix

(u,-6u:”-”_

:]

which vanishes, so that u(z.1) satisfies the KdV equation. We illusvate the above method by examples.

Eromplc 9.9.1 (Nonlinear Schr(Minger Equation). We consider the form of the 2 x 2 matrix as

A=r0 A

‘I+[: :I+[-,

-iA

:).

(9.9.41)

where q = q ( x . r ) and r = r ( x , i ) are complex-valued function o f x and 1. We choose the matrix B ( A ) so that (9.9.40) becomes cenain partial differential equations for q and I . We assume the matrix B ( 1 ) as

B=2iA2[0 ’ -1 ‘]+2iA(:

: ) - i r q0

- ro].(9.9.42) q

SubsUtuting these mavices A and B in (9.9.40) gives a system of equations

ir,+r.+~qr’=~,

Wenextwrite

r=$

iq,-qy - 2 r q ’ = 0 . or r = - q mtheahovccquationstoohtain ir,+r= * ~ l r r r = ~ .

(9.9.434 (9.9.43b)

(9.9.44ab)

These are h o w n as the nonlinear SehrMinger equations. Example 9.9.2 (The KdV and Modtfied KdV Equononr). We use the same A as m (9.9.41) and choose B as

+[

qr,-rq=

iqy+Zirq'

ir_+Ziqrl

-q,,+,q,

so that equation (9.9.40) becomes

1.

(9.9.45)

q,+6rqq. + q , = O .

(9.9.46a)

r, +6qrr, +r,

(9.9.46b)

=O.

We assume that q and I IVC n a l functions of x and f. When the famous KdV equation. When I = q 01 I = - 9 , equations (9.9.46ab) give r, f 6?r,

r

= - 1 , (9.9.46a) is

+ r , = 0.

(9.9.47ab)

These are the mcdified KdV equations.

Erample 9.9.3 (The rim-Gordon Equation). We choose

I

= q ='

u,, the

same matrix A in (9.9.41). and the matrix B as cosu

-isinu

irinu

--EOEU

B(A)=(4iA)-' so that equation (9.9.40) leads lo the sine-Gordonequation u,,

Example

9 . 9 . 4 (The

=sin u .

(9.9.48)

sink-Gordon Equolion). We substitute

.=-q=-'".,thesamemavirAin(9.9.41).and coshu

-isinhu

-isinhu

-coshu

B ( L ) = (4iA)-' in (9.9.40) to obtain the sinh-Gordonequation u., =sinhu.

(9.9.49)

400

Solirons and the lnvrrsr Sconrnng Trmsfom

9.10 The AKNS Method I n 1974. Ablowulta, Kaup, Newell. and Segur (AKNS) generalized the ZS scheme so that their method can be applied to solve many other evolution equations. We bnetly outline the AKNS scheme below without all the lechnical details. We begin this discussion by considering the pair of linear equations “.=A” and u , = B u . (9.1O.lab)

where u IS an n-dimensional vector and A and E are n x n mamices. Then differentiating (9.10 la) with respect to t and (9.10.lb) with respect to x and equating the results leads to the equation dB+[A,B]=O. Jt

dr

(9.102)

This is essentially equivalent to the Lax equation (9.9.15). It turns out that. given A , there is a simple deductive method to rind a B so that (9.102) contains a nonlinear evolution equation. For (9.10.2) to be effective. the operator A should have a parameter which plays the role of an eigcnvalue, say 5, which satisfies the condition 5, =O. Further, a solution of the related nonlinear evolution equation in an infinite interval can be obtained when the associated scattering problem 1s such that the inverse scatteringmethod can be carried out. Although there are many nonlinear evolutions which satisfy (9.10.2). at this time. a complete scattering method for many of the associated equations (9.10.la) has. not yet successfully developed. As an example, we consider a 2 x 2 eigenvalue problem far the pair of equations (9.10.3a) = - i t +4

v,’ v, v*. vx.=iC% + r w,.

(9.10.3b)

that is,

v,=Av. where

Bv=-i5v.

(9.10.4ab)

v

The bunded functions q ( x ) and T ( x ) . not necessarily real, arepotenennblsand 5 is the eigcnvaluc. It can easily been shown that there is a direct link between the pair (9.10.3ab) and the original Schrodinger equation. Dffferenuating (9.10.3b) with respect to x gives

v2,,= i5 v2,+ rr v,+ v,.

(9.10.5)

9.11 firreires

401

providedthat r. exists. Using (9.10.3ab). equation (9.10.5) can be simplified to obtain

v,,, = i C v2,+ r, v,++is v,+ q v,)

Or,equivalently,

v ~ ~ " - ( ~$~) ~ - [ q r - i C ( $ ) - C ' ] v=O. when

I

= - I . this equation redues to the Schredinger equation for L

v, +C' v>=o,

(9.10.6)

v, (9.10.7)

with q = --u and 1 = 5 '. Thus the system of equations (9.10.4ab) recovers the scattering equation required for solving the KdV equation. The choice r = - 1 furm out to bc a degenerate case, since, for all other evolution equations. we asnme that both q ( x ) and r ( x ) decay suffieienllyrapidly,as I x I + - . This ensures the existence of v for x E R .

We close this section by adding the following comments. Both the Z S end AKNS methods arc applicable to KdV and NLS equations and to mKdV and SO equations. Although the two methods overlap. the remarkable difference is that the ZS method is described solely in terms of operalo~s.whereas the AKNS method is expressed in terms of the scatteringtheory.

9.11 1.

Exercises forhe KdVeqwtion (9.7.1).

Show that (xu+31u2) isacon-eddensity

2. Find three conrcrvation laws for rhe mKdV equation (Miura CI 01. (1968)) u, -6"'".

+us

=0,

X E

R,

which involve u. u 2 , and u ' , respectively. 3. Show that the Uvee conservation laws for the BBM equation u,-ug

-",,, = o

Show that the mKdV equation u,+6u2u=+u,

=O

is invariant under the transformation

i=o’r. i = a - ’ u (o*o)

i=ox,

Henee introduce u ( . v ) = f - ‘ * J ( X I “ * ) nonlinear ordinary differential equation

f”-l< f + 2

f3

to

= 0.

show that

f(C)

satisfies the

C = x,-‘’’

3

pmvidedthat f -0 at mfmtty. Investigate a similarity solution of the cylindrical KdV equauon Zu,+!

=O

U - ~ U U , +.!urn

f

in the form u ( x , f ) = - i 3 the equation

* ’ f{x(2r)-’”}. (TI

Show that

u(<) satisfies

v”-
< = x(Zr)-’”.

Show that the NLS equation

i y +us

+y U

J U= ~ 0,

where y is a real constant, is invariant under each of the group vansfmations (i)

i=ox,

(ii)

i=x+a.

Show that

B

i=o*t,

i=o2u

(o*o).

i=r, i=u.

similarity solution of the form u ( x , f ) = f ‘ J ( < ) , < = x f ‘

exim for suitable valuer o f p and q. Hence,find the equation for f (5). Show that the Kadomtscv-Petviashwh equation (Freeman,1980)

( u , - ~ u u , + u - ) , + ~ u , ,= O

403

9.11 EX.rcircr

has the soliton solution u ( x , y , 1 ) =-!-I?

sech’

[i

(tx + ! y - w r)],

whne w = t ’ + ( 3 t ’ / k ) . 8.

9

Apply the Inverse Scattering Transform method to solve the initial value problem for the KdV equation (9.7.1) with u(x,0)=2sechax. Hence, determine the time development of the solution from the above initial data by numerical integration with a program given by Crandall(1991). Use the BLWund oansformations

u, = u - u z and u,=-u, +Z(uu,+u, u) to show that u satisfies the KdV qualion u, -6””’ +u,

=O.

and that v satlsfiesfhc mKdV equation

u, -6u‘ u, + u, = O . 10. Apply the BacWund transformations U. = - ‘ ( u v )

2”

to show that u and

and u, = v ( u 1 - 2 v u , ) 4”

u satisfy the Burgers and diffusion equaltons u,+uux=vn,

and u,=vu,.

11. Show that the Kadomtsev-Petviashvih (1970) equation ( ~ - ~ U U , + U , ) ~ + ~ U= ,O~

can k derived by Selectingthe Lax pair

12. Show that the operator M for the KdV equation is

*,

J’ M=-4 -+6ud+3 Jx’ Jr ax 1 3 . Using Markemariea,or othenvae, show that the Lax equation (9.9.6) with

reduces to the equation

For q = a r-I, and rnatnces A and B given by (9.9.41) and (9.9.45). show that (9.9.40) leads to the nonlinear equation r, + r- - 6 r r ,

+ 6or’ r, = 0

14. If the phase velocity of nonlinearwater waves is

c(k)= [ ( X I tanhkh

I’

=&

I-lk’h’+o(k’h’)} 6

as k h - 0 .

show that these waves are desenbed by the Whitham equation

where the kernel K is determined from the linear theory as the Fourier Vmiformof e ( k ) ,

Chapter 10

The Nonlinear Schrodinger Equation and Solitary Waves 10.1 Introduction It is now well known that the Korteweg.de Vies equation and the nonlinsar Schr6dinger equation are the lowest order nontrivial consequences of a perturbation approximation for the weakly and strongly nonlinear dispersive wave systems, respectively. This chapter is devoted to the nonlinear SehrOdinger equation and its solitary wave solutions. Several properties of the nonlinear Schrodinger equation including envelope solitons and recurrence phenomenon an discussed. Attention is given to conservation laws for the nonlinear SchrOdinger equation. Some examples of applications to fluid dynamics. plasma physics. and nonlinear optics are included. Based on an ingenious paper by Zakharov and Shabat (1972). the inverse scattering transform for the nonlinear Schrtiiinger (NLS)equation is described in some detail.

10.2 The One-Dimensional Linear Schrodinger Equation We consider the following Fourier integral representation of a quasimonochromatic plane wave solution @ ( x , r ) = j F ( k ) eip[i{Xx-w(k)r}]

-

dk.

(10.2.1)

where the spectrum function F ( k ) is determined from the given initial or boundary conditions and has the property F(-k) = F'(k) and a, = "(k) is the dispersion relation. We assume that the initial wave is slowly mdulaled as it propagates in a dispersive medium. For such a quasi-monochmmatic wave, most of the energy is confined in the neighborhood of a specified wavenumber k = ko. so that the spectrum function F ( k ) has a sharp peak around k, with a n m o w

The Nonlincor Schrdmgrr Equorion and S o l z t q Waver

406

wavenumberwidth (k-&)= Sk=O(&), and the dispersion relation m(k) can beexpandedaboutko as follows

+ (6 k)'I a,"+ ....

w = mo+ (6k)m; where mo=m(ko). mi=S'(k,)and

-

(10.2.2)

w,"=w"(ko).

Substituting (102 2) in (10.2.1) gives

+,r)

w ( x , r ) erp[i(kox- m,r)]+c.c.,

(10.2.3)

where C.C. stands for the complex conjugate and w(1.r) is the complex amplibde definedby

where it has been assumed that m(-k)=-m(k). (x-m;r)Sk

Since (10.2.4) depends on

and (6k)'r where Sk=O(&) i s small. the amplitude w(x,r) is

a slowly varying functm of

X'

= (x - m; I) and r. We next introduce slow

variables 5 and 7 definedby 5=c(x-m;r)

and r = e 2 t .

(10.2.5ab)

so that the wave field assumes the form )(x,r) = A($. r) erp[i(k,x

- m ~ )+c.c., ]

(10.2.6)

where the madulatedwave amplitudeis given by

(10.2.7)

6 and, hence,

mo = m(ko) are chosen to be constants, and all slow variations of the wavetrain are included in I n the expression for the wave field (10.2

6).

A(C.7). A simple computation giver Asa-LimtA

and A<<=-A

(10.2.8ab)

10.3 The Derivation of the Noofbeor Sehrodinger

(NU)Equation

407

so that the modulated wave amplitude A(5,r) satisfies the linear Schradinger eq&

10.3 The Derivation of the Nonlinear Schrodinger (NLS) Equation and Solitary Waves We show below that the nonlinear modulation of a quasi-monochromatic wave is described by the nonlinear Sehrodinger equation. To take into account the nonlinearity and the modulation in the far field approximation. the wavenumberk and frequency w in the linear dispenian relation are replaced by

k-i

d and

w+t

JX

2I respectively. It IS convenient to use the nonlinear

dirwmion relation in the form

We consider the case of a weak oonlineanty and a slow variation of the amplitude, and hence, the amplitude A is assumed l o be a slowly varying function of space and time. W e next expand (10.3.1) with respect to [All, -i L a n d i

a,

d toobtin m

where ulc fintterm D(k.w.0) = 0 due to ule linear dispmion equation. Introducing the transformation

X’

= x-C, I, I’ = I .

assuming that

A = O ( & ) , p = O ( & )and ~ = O ( ~ ’ ) , r e t a i n i n g a l l t e r m s u p tO(d),md o dx dropping the asterisks. we find that

,

(10.3.3)

and the following results

have b a n used. Equation (10.3.3) is known BS the nonlinrorSchrod,nger(orNLS)equnrion. More explicitly, if the nonlinear dispersionrelation is given by (10.3.6)

0 =o(k,n').

and if we expand 0 m a Taylor series about k = k, and'1.1

Replacing (0--om) by i ( $ ) .

k - k , by -i

= 0 , we oblain

(i), and assuming thatthe

resulting ~pcratomact on the amplitude function a ( x , I). it turns out that (10.3.8) Where

(10.3.9) Equation (10.3.8) is known as the nonlinear Sehrodingrr equation. If we choose a frame of reference moving with the linear group velocity 0;. that is. < = x - o i r a n d r=r,thetcrminvolving a, willdropoutfrom(10.3.8),and therefore. the amplitude n(x.1) satisfies the nomaltzed nonlinear S e M i n g e r quation (10.3.10)

TIE correspondingdispersion relation is given by m=

L mtk'

-yoi.

(10.3.11)

10.3 T k Derivation of the Nonbnem Schrdingw ( N U )8quolron

409

According to the stability criterion established by Whitham (1974). the wave modulation is stable if y w i < 0 or unstable if y u i > 0.

To study the solitary wave solution, It is convenient to use the NLS equstion in the standard form rv,+v,+~l~’v=O.

f>O.

--oo<x<-,

(10.3.12)

We then seek waves of permaem form by assuming the solution v=f(x)e““X””,

x=x-ut,

(10.3.13)

for ~ o m cfunctions fand eonstant wave speed U to he determined, and m,n

ilre

mnltants. Substitution of (7.3.13) in (10.3.12) gives

f”+i(Zm-U)f’+(n-m*)f+ylfll f = O .

(10.3.14)

Weeliminate f’ by setting 2 m - U = 0 , and then wnte n = m 2 - a , so that f can be assumed real. Thus. equation (10.3.14) becomes

f”-af + y f ’

(10.3.15)

= 0.

Multiplying this equation by 2 f’ and integrating, we find that (10.3.16)

f”=A+af’-~f‘~F(f), where

F(f)=(a,-a, f’)(p,-p, f’),

so

a =- (a,p, + a,p,). y = - Z ( a , & ) , and nls and and datinct. Evidently, it follows from (10.3.16) that

that

p’s .we

A=alp,,

assumed to be reat

Setting (a2/a,)uz f = Y in this integral, we deduce the following elliptic integral of the first kind (see Dutm and Debnath, 1965):

when u=(a2b,)’* and .=(a,&)/(D,a.,)

Thus, the final solution can be expressed in terns of the lacobian elliptic funcum

The Nonirnrar SchrMinpr Equotion m d Solitary Waves

410

U

=m ( 0 X . K ) .

(10.3.19a)

Thus, the solution for f ( X ) is given by

,(.,=[$)"

(10.3.19b)

sn(0X.K).

In particular. when A = O . a > O . and 7 > 0 , we obtain

B

solitary wave

solution. In this case, (10.3.16) can be rewitten

Substitution of (y/Za)" f = seche in this integral gives the exact soltuion f ( X ) = [ ~ ~ ~ ~ h [ 6 ( ~ - " ~ ) ] . (10.3.21)

This represenls a solitary wave that propagates without change of shape with constant velocity U.Unlike the solution of the KdV equation,the amplitude and the vclociry of the wave an independent parameters. It is noled that the solitary wave exists only for the unstable case ( y > O ) . This means that small modulations ofthe unstable wavewin lead 10 B r e r i s of solitary waves. The well-!aown nonlineardispersion relation for deep water waves is

w"--s

m;="* 2&

and y = - $ m , k i .

(10.3.23)

4 k,'

and the NLS equation for deep water waves is obtained from (10.3.8) in the form

i(,

+s ax] - - 2k0

8ki

a_

-.!m,ki 21.1

o=O.

(10.3.24)

2

The normalized form of this equation in a frame of reference moving with the linear group velocity w; is

10.3 The Demotion of the Nonlinear Schrodiinger 1Nt.S) Equation

411

Since y m ~ = ( m ~ / 8 ) >this 0 . equation confirms the instability of deep water waves. mihis

is one of the most remarkable recent results in the theory of water

Wayel.

We next discuss the uniform solution end the solitary wave solution of the NLS equation (10.3.25). W e look for SOIUUO~Sin !he form a ( r , t )= A(X) exp(i y't), and substitute this in equation

X = x - mi 1 ,

(10.3.26)

(10.3.25)to abLlin the following equauon

A~=-~(y2A+fmok~A3).

(10.3.27)

ru, We multiply this equation by ZA, and,then,integrate to find A: = - ( x m "

.I

+By'k,'A'+%k@A =(&-A')(A'-m'* mo

where &m"

IS

& = 4J/mOki (m'

&).(10.3.28)

an integrating constant, 2ki=1. m"=I-m',

and

- 2). which relates 4. y , and m.

Finally. we rewrite (10.3.28) BJ

Or, equivalently.

'

ds I

= (A/%).

(10.3.29b)

Thts can readily be expressed in terms of the Iacobi dn function (see Duttl and Debnath, 1965): A=%dn[%(X-X,).

m].

(10.3.30)

where m IS !he modulus of the dn function. In

the limit, m +0, dnz + I , and

9 + -I m o k i 4 . Hence, the solution

becomes a ( x , f )= A(t) = %exp{-f

imoki&

t}.

(10.3.31)

411

The Nonlinear Schrddinger Equation und Solitary Wovcl

On the other hand, when rn + 1. dnr + sech L, and y'

+ -iw&&.

Therefore, the solitary wave solution is

+r)=

n, exp( -iO J & ~ 1) sech [ n , ( x -

0; i - X 0 ) ] .

(10.3.32)

10.4 Properties of the Solutions of the Nonlinear Schrodinger Equation We discuss several important pmpelfies of the oonlinear Schrddinger quation in the form i u , +pu,,

where

p

+ y lull u = O

and y are real constants. W h e n 3! =

(10.4.I )

1w,". this equation reduces to

(10.3.10). Simple solutions of equation (10.4.1) can always be obtained. However. the nature of the solutions depends on the sign of 0 and y . If they are of the Same sign, thst is, if B y > O , the solution of (10.3.35). which lends to zero as reprerents a solitary wave soluuon

lc1+

-.

u(<,r)=

-2 L [E51 sech

eip(-ivr).

(10.4.2)

where Y ISa consmnt parameter. It follows from the Galilean invariance of the Schr6dinge.r equation that equation (10.4.1) is also invanant under thc following tranrformations

where (I is another constant. Thus equation (10.4.1) admits solution in the form

B

solitary wave

where Y and U ine independent consmano. There solutions decrease very rapidly as 151 so that they may be considered to represent solitary waves which are similar to KdV solitons. However, unlike the KdV equation, the nonlinear

10.4 P r p m i c s ofthe Solurions ofrhc Nonlhcor Sehrddinpr Equation

4 13

Schrcdinger equauon does not admit B soliton soluUon cornspanding to a steady wave propagating with a eon~tant velocity. I n fact. the plane-wave pan represented by the exponential function and the amplitude of the sech profile propagate with different velocities. In this sense. the solution 110.4.4) IS called the envelope solrlon as shown by Figure 10.1.

Figure 10.1

An Envelope Soliton,

It was first shown numerically by Yajima and O a i (1971) hat, like the KdV soliton, the envelope s d m n behaves like a panicle. This was also proved analytically by Zlkhamv and Shabat (1972) by the inverse scattering melhod. An excellent experimental agreement with the rech-soliton ~ ~ l ~ t(10.4.4) m n was found by Hammack as reported by Ablowitz and Scgur (1979) Hammack's experimental findings shown in Figure 7.2 (see Debnath p. 363 (1994)). On the other hand, if u tends to B constant value ua at infinity. then, the plane wave solution exists in the form u(

5 , ~=)q, erp[i(lS - 6 4 .

(10.4.5)

where

6=(flC' - y u i ) .

(10.4.6)

However, the plane wave solution is modulationally unstable. This can be shown by introducing two real functions p and u through the expression (10.4.7)

Substituting this exprcorion m (l0.4.l)gives

(10.4.8)

Since B y > 0 , in the long wavelength approxrmation, equations (10.4.8)(10.4.9)are equivalent to a hydrodynamic System with negative pressure. In other words. the system bccomcs elliptic. Thus, we consider that the perturbations modulate B eonstant amplitude po and phase 4, $0 that (p, u)= (p*. u,)+(6p, 60)exp (i(r<-R7)}. (10.4.10ab) Substituting these perturbations in equations (10.4.8)-(10.4.9) leads to an equation for gmwth rate R in t m s of the wavenumber c as

f2 =-uo I( f (-2p yp0); x + O ( x ’ ) . This shows that the perturbation grows exponentially as t + wavenumben C . provided that Po is finite.

-

(10.4.11) for small

On the other hand, if B y < 0 , that is. p and y are of opposite signs, then erisu a stable plane wave solution. In this case, the system of equations (10.4.8)-(10.4.9) can be reduced to the KdV equation. This follows from the exparrsjhl of p and u in t e r n of the small parameter B about the con~tant state, so that we Fa” write p = p a + € p a+ €‘p2 +-., %se

and u = uo+€a,+ €‘a, +-..

(10.4.12ab)

resulu combined with the Uansfmation

nducc the system (10.4.8)-(10.4.9) to the KdV equation for p, and 0,

Thus, when B y > O . the above solution is often called

brrght soliton (or

envelope soliton). The Wght pulse mses when luf incrcases fmm a finite value at infinity and. subsequently, returns to the same value. When P y c O , the

solution is of the expansive type in which #2 decreases fmm a finite value at infinity and subsequently returns to the same state. This is often called a dark soliton (or dark puke). So far, the self-modulation of I\ single mode governed by the dispersion relation is described. However, when different modes with amplitudes of the

10.4 Pmpenier ofrhr Solurionr offhe Nonlimor Schrddingrr Equorion

4 15

same ader O(s) coexist and undergo mutual interactions and self-modulation, it is possible to split the system of equations into independent nonlinear Sehrtidingcr equations similar to the case of long waves (Oikawa and Yajima (1974)). Consequently, envelope solitons associated with various modes, moving separately at the beginning. preserve their identities and. then. propagate as envelope solitons, even after the mutual interactions. Zakhorov and Shabat (1972) discovered that the inverse scattering method gives an exact solution of the initial-value problem for the nonlinear Schrrjdinger equation. Based on the initial data. which tend to zero rapidly at infinity, their remarkable analysis can be summarized as follows. (i) An initial wave envelope pulse of arbiuary shape eventually disintegrates into a number of solitons of shorter scales and an oscillatory tail. Each soliton is a permanent. progressive wave. The number and ~UUctureof these solitons and the nature of the Id are completely described by the initial data. (ii) These soliton solutions are definitely stable in the sense that they undergo nonlinear interaction and, then, emerge from the interaction without permanent changes except for a possible change in position and phase. (iii) The oscillatory tail is relatively small and disperses linearly with . . amplitude which decays like t ', as I+ . Another significant property of the solutions of the nonlinear SchrSdinger equation is the recurrence phenomenon. In a conservative system, the existence of invarianu plays B fundamental role in establishing rec~mneephenomena. as shown by Gibbons el al. (1977) and by Thyagaraja (1979, 1981, 1983). As demonstrated by these authors. the recurrence phenomenon is typical only for bounded or periodic domains. Thyagyagarajaconsidered the initial-value problem for lhe nonlinear S c W i n g e r equation

-

iv, = vI,

+ Y 1812 v

(10.4.14)

in the periodic domain 0 L x L 1 with periodic boundary conditions and y B real constant, whish can be positive or negative. In many physical problems, y is positive. We assume the periodic boundary conditions v(O,t)=O=v(I.t) for l > O .

(10.4.15)

or v*(O,I)=O=v,(l.t)

for t > O ,

(10.4.16)

and ule initial condition v(x,0)= Y(x)

forall

XE[O.I]

(10.4.17)

Without prwf, we assume that this inilisl-boundary-problem for equation (10.4.14) poslesser a smooth solution which is uniquely determined by the initial data 'f'(x). To investigate the qualitative properties of the solution. Thyagaraja derived certain a priori bounds involving the integral invariants

Thc Nonlinror S e h d i n g r r Equorion ond Soizmy Waves

416

associated with equation (10.4.14). Two in van ant^ of the problem which are constants of the motion are given by

l(r)=

jlv(.r.t~~ dr,

(10.4.18)

d

J ( t ) = j l v , ( x , r ) r dr

-

jlv(x,t)r dr. 2 ,

(10.4.19)

However, Zakharov and Shabat (1972) provcd that this problem has an infinite set of integral invariants, provided r(x,r) is sufficiently smwth. We consider m y function ~ ( x J ) ,not necessanly a solution of (10.4.14). which is defined for I f I > 0 and sufficiently smwth. W e assume that v(x.1)satisfies the given boundary conditions and evolve in time, so that functionals I(r) and I ( [ ) . defined by (10.4.18)-(10.4.19), an constants in time. We then find bounds of the Rayleigh quotient Q(r) defined by (104.20)

(10.4.21)

10.4 Pmprmcr of the Solutions ofthe Nonlinaor Schrdrngrr Equation

417

J~vvz~~~~(~)&R. Multiplying (10 4.23) by Ir ( x,t ) r and mtcgmting gives

Jlw(w)r

dr

<

(10.4.24)

1’+21’&R.

Substituung in (10.4.19) yields the quadratic inequality

a(:)<

I;[

(3

-+ -

fO+Yl0&R.

(10.4.25)

Finally, (10.4.23) can k reorganized to obtain

~ w ( x , r ) ~ ~ 1 ~ + ~ 1 ~ & R .

(10.4.26)

Since Q(r) 5 0 by definition, it is c a y to solve the inequality (10.4.25) and obtain the following resulrs:

Q(r)

5, M’(fo.J0.Y).

(10.4.27)

I l y [ ~ - m a i I ~ ( x . r ) r I,(I+ZM), <

(10.4.28)

OULl

where M IS the positive mot of the quadratic equation

M2-yIoM-

{[-;] +(i) - .)I

-0.

(10.4.29)

To examine the mqor implications of the a priori bounds in (10.4.27)(10.4.28). we introduce the concept of Lagrangion srbili,y. A solution w ( x . r ) of (10.4 14) IS said to be Lagrangion stable. if there exists a constant K independent of r, but possibly dependenton the initial data such that Iw(x,t)llt=~~~:831w(~.:)12
(10.4.30)

forall 1r15o. Clearly. 11 follows from (10.4.28) with K=I,(I+ZM) that any solution w(x,t) is LagrangIan stable.

To give an interpretation of the inequality (10.4.27). we expand v(x.r) Founer series in X. so that

3”

a

In analogy with quantum physics, we may interpret Q(t)/4z1 as the insmmneous average of n'. and ( M / Z z ) can be interpreted as an upper bound to the rms value of". the number of modes carrying !he "wave energy" la. The quantity

($)

is called the number of cflcctivr moder and is denoted

by N < . The inequality (10.4.27)says that Nef is hounded by

(2)

which

.

.

depends only on the initial values of loand J, of the integral invariants and on the interntion strength y . lblhur. we have proved that every mom" described hy the equation (10.4.14) with given la and J , can be assigned N# as a measure of the number of effective modes. It is then suaightfonuard to apply BirkhofFs theorem (1927)to conclude that. in general. dynamical systems with afinite number of degrees of M o m , hounded motions are 'generically' recurrent. It is interesting to compare the results of this analysis with numerical simulations and experiments. Yuen and Ferguson (1978)investigated spatially periodic solutions of (10.4.14). They numerically evolved initial data for which the linearized form of (14.4.14) is unstable to the modulational instability of Benjamin and Feir (1967). lbe long time behavior of these solutions reveals that the energy is shared effectively between afinire number of wave modes. The Dumber depends on the initial dam. Moreover. the s ~ l ~ t mreconsmeled n itself after appearing to show a tendency to break up due to linear instability (Yuen and Lake (1980)). Both the failure of the energy to thermalize and the tendency to recur have been known since the classic work of Fermi. Parta, and Ulam (1955). They observed a remarkable recumnce phenomenon, known as the Fermi-PartoUlam (FPW recurrence. Initially, if energy IS given to the lowest mode with the lowest frequency, then, several higher m o d e are excited by the nonlinear effects, hut after a suitable lapse of time during which flowing hack and forth among several low-order modes takes place, then, the energy eventually returns to the mode comprising the initial state. In other words, the system docs not approach thermal equilibrium by the principle of the equipartition of energy due to the nonlinear effects, but the FPU reeumncc IS observed, and consequently, the nonergodicity of the system can be demonstrated. Subsequently. Zabuky and Kruskal(1965)and Fornherg and Whitham (1978)reported similar observations. From a series of computational and experimental works by Yuen and Lake (1975)and Lake et al. (1977).i t has been confirmed that recurrence is a genenc phenomenon for any physical problem described by the nonlinear Schrodinger equation. At the same time. the above mathematical analysis is now accepted as ~ o n ~ l u ~evidence ive of the FPU recumneephenomenon.

10.5 Conse~alionk w s f o r the N U Equalion

419

For a long time it was mistakenly thought that the recurrence property of the NLS equstion is atmbutable to the fact that it admits a multisoliton ~olutionand can be solved cxaerly by the inverse scattcnng method. However, the above discussion shows, beyond any doubt, that the reeorrence of solutions has nothmg to do with ~olilonsolutionsor exact integrabilrty.

10.5 Conservation Laws for the NLS Equation Zlkhamv and Shabat (1972) proved that equation (10.3.12) has an infinite

number of polynomial eonsewation laws. These haw Ihe form of an integral. with respect to x, of a polynomial expression m terms of the function v(x,r) and its derivauves with respect to X. These laws M somewhat similar to those already pmved for the KdV equation. Ihenfon. the pmofs of the conservation laws are based on similar assumptions used in the context of the KdV equation (see Chapter 9, SecUon 9.7). We pmve, here, three c o n ~ e ~ a l i olaws n for the NLS equation (10.3.12):

J l~l'dr=constant=C,,

We multiply (10.3.12) by and its complex eonjugale by the latter fmm the former to obtain

(10.5.1)

v and substract (10.5.4)

-

when the bar denotes the complex conjugate. Integration with nspeft to x in < x < gives

--

This proves result (10.5.1). We multiply (10.3.12) by and its complex Conjugate by add the resulting expressions to obtain

v,

and. then.

i(v,F. - W , V , ) + ( ~ ~+~",n)+Ylwl'(~,+vv,)=o. V~ (10.5.5)

The Nonlinear Sehrodrnger Equalion a d Solimy W o w

420

We differentiate (10.3.12) and its complex multiply the former by This leads to the resnlt I

conjugate with respect to x,

I and the latter by v , and then, add them together.

(Iv,, - vI.,) + ( v - l + lmv )+ Y [ I(ldv),+ ~(14' l)=] = 0 .(1 0.5.6)

If we subtract (10.5.6) from (10.5.5) and. then. simplify, it turns out that i

(W ,

-

w,) = d ( v ' ~ .+ )d ( ~ v+-w-) dr dr -;(V,I,,+y$

(VI)'

;

= d( I V Y + WJ+7 Ivl'.

dr Integrating this result with respect to x yleldr

d dc This proves result (10.5.2). We multiply (10.3.12) by m d t i n g equations to derive

-I.-

- ' ( V v . - W,) dr =o.

I,and its complex conjugate by v, and add the

(I,V" + I"v,)+Y ( vxm, + I'w,)= 0 .

or. equivalently, (I,V. + V,l.)

-$ (v'I,,+;;

dr Integrating this result with respect to x gives

d I[Iv,r di

-

( v2V )= 0.

-I Id4]&=I$(l,v.+ v,I.)

&=O.

(10.5.7)

This gives (10.5.3). The p c d i n g t h conser~atlonintegrals have a simple physical meanmg. I n fact, the eonsiant~of motion C,, C , , and C, are related to the number of panicles, the momentum, and the energy of a system governed by the nonlinear Schrbdingcr equation. An analysis of this section reveals several remarkable features of the nonlinear Sehr6dinger equation. 'Itis equation can also be used to investigate instability phenomena in many other physical systems. Like the various formsof the KdV equation, the NLS equation arms in many physical problems, including nonhnear water wwes and mean waves, waves m plasmas, propagation of heat

10.5 Consawtion Lowsfor :ha

NU

421

Equlion

pulses in a solid. self-trapping phenomena in nonlinear optics, nonlinear waves in a fluid-filled viuoelastic tube. and various nonlinear instability phenomena in fluids and plasmas. Whitham's equations far the slow modulation of the wave amplitude (I and the wavenumber k in the ease of two-dimennonal deep watei waves are given by (10.5.0)

(10.5.9) where oo=

.& is Ule first-order appmrimation for the wave frequency 0 and

c = (*/Zoo)

is the group ve~oeity.

Chu and M a (1970, 1971) observed that certain terms of the dispersive type. neglected in Whitham's equations to the same order of appmrimation. must be included to extend the validity of these equations. Whitham's theory is based on the direct use of Stokes'dispersion relation for a uniform wavetain.

o = oo(I + + &'a2k'),

(10.5.10)

whereas Chu and Mei added terms of higher derivatives and dispersive lypc to the

expression for

0 , so

that

They used the expression (10.5.11) to mansform (10.5.0)-(10.5.9) in a frame of reference moving with the group velocity C and obtained the following nondimensional equations (10.5.12)

where we have used Chu and Mci's result W =-2@. and 9 IS a mall phase variation. Integrating (10.5.13) with respect to x end setting the constant of integration to be zero gives @,

+'

$1

-I aa - n _ = o .

2 ' 0

320

(10.5.14)

422

The N o n l m m Schradingrr E q u t i m a d Solitmy W o e s

A transformation %'=a erp(4i0) 1s used to simplify (10.5.12) and (10.5.14), which d u c e lo t h e n o n l i n e o r s e h r ~ ~ ~ ~ q ~ l i ~ "

+t

i Y, + Y,, Y ~ Y=*0. (10.5.15) 8 2 This equation has also been denved and exploited by several authors including Benney and Roskes (1969), Harimoto and O m (1972). and Davcy and Stewamon (1974) to examine the nonlinear evolution of Stoker' waves on water. An alternative way to study nonlinearevoluuon of two- and hedimensional wavepackets is to use the method of multiple scales in which the small parameter E is explicitly built into the expansion procedure. The small parameter E characterizes the wave steepness. This method has been employed by several authors in various fields and has also been used by Hasirnoto and Ono (1972) and Davey and Stewartson (1974).

10.6 The Inverse Scattering Method for the Nonlinear Schrodinger Equation Zakharov and Shabat (1972) developed an ingenious method for solving the nonlinear Schrainger equation in the form

+ ~ l u$= o

(10.6.I)

by considenng the spectral problem u,, = - i S u , + q y ,

(10.6.2)

iu, +u,

v2. = I Cv, + r v , .

(10.6.3)

In mavix notation, they take the form

what v

IS the

I:[

vector

nal) enporenrials. and

, bounded functlono q ( x ) and .(X) (not necessarily

C is the eigenvalue.

The most general linear time dependence is given by v,, = A vI + B v l ,

(10.6.5)

ui, =Cv,+Du,.

(10.6.6)

424

The Nonlinear Schrodingrr Eqmrion ond Sotimy waves

Since the Schradinger equation is second order in X. we may expect that series expansions in for A. B , and C include only terms through <', and so, we write an exact rmncated power series

<

Substituting these expressions in (10.6.10)-(10.6.12) and equating coefficients of powers of 5 gives Am, B., and C, where n=0,1,2. The coetlicienuof

<'1n(10.6.11)and(10.6.l2)give

B,=C,=O.For~ocffielents

of <',equation (10.6.10) yields 4 =at =constant. Equations (10.6.11) and (10.6.12) give ( E , , C , ) = i ~ ~ ( q For , r ) the coefficient of <,equation (10.6.10) gives A, =(I, =constant. For simplicity, we set (1, = O . However, if a,10,a more general evolution equation can be obtained. Then, (10.6 1l)-(l0.6.l2) gwe

(Bo. C o ) = f + (-9,.

4= i a , q r + a ,

r,). Finally, for the coefficients

where a@ is

B

<', (10.6.10) gives

constant which is set equal to zero.

Consequently, equations (10.6.1 I ) and (10.6.12) for coefficients of

5'

give

-la,q_=q,-o,q'r,

(10.6.14)

i n , r" = r, +a* 4'2.

(10.6.15)

This represents B coupled pair of nonlinear evolution equations. In particular. when r = f 4'. equations (10.6.14) and (10.6.15) are compatible provided that 0, = i a . where a is real. If we take (I = 2 , we obtain the equation iq,=q,i~q'q'.

(10.6.16)

This equation with positive sign is called the nonlinear Schrodingcr cquanon which admits special soliton solution, while the equation with the negative sign doer nor give any soliton solution for potentials decaying rapidly. as x + m, since the spectral operator in (10 6.2) and (10.6.3) is self-adjoint (or Hermitian).

10.7 Examples of Physical Applications in Fluid Dynamics and Plasma Physics Example 10.7.1 (The Nonlinear Schrhdingcr Equationfor Deep Water and the Benjamin-Feir Imtobiliry).

415

10.7 Erornplrs of Physrcol Applicorionr

One of the simplest solutions of the nonlinear Schriidingcr equation (10.3.25) is given by (10.3.31). that is,

A(!)=& erp(-f

o,k,'&t),

(10.7.1)

where & IS a constant. This represents, essentially. the hmdamental component of the Stokes wave. We consider a perturbation of (10.7.1) and express it 80 the f m a(.r,t) = A(t) [I+ B ( x , ! ) ] .

(10.7.2)

Substituting this result in (10.3.25) gives

i(l+B)q+iAB,=

(3

As.,

i wokiAi [(1+ B)+ BB'(I+ B ) + (B+ B')B+( B + B')] A,

(10.7.3)

where B'(x,t) is the complex coopgate of the function B(x.t). Neglecting squares of B. equation (10.7.3) reduces 10

We look for a solution for perturbed quantity B ( x . t) in the form

B ( x . t ) = B, exp(Ot+itx)+ Ba(12'f -i!x),

(10.7.5)

where B, and B2 are complex constmts, !is a real wavenumber. end I2 is a growth rate (possibly a complex quantity) to be determined. Substituting the solution for B m (10.7.4) yields a pair of coupled equation$

[

iR+

s]

4-

wok,'&

(4 + 8;)= 0.

(10.7.6)

(107.7)

+s]

We t&e the complex conjugate of (10.7.7) to Vansform it in the form

[

-iR

B; -

f wok:&

(B, + B;) = 0.

(10.7.8)

7 h c pair of linear homogenous equauons (10.7.6) and (10.7.8) for B, and B; admits a nonhivial eigenvalue for I2 povided

which is equivalent to

f11=i[9]'bi&--$]. 2 2k0

(10.7.10)

The growth rate fl is purely imaginary or real (and positive) depending on whether I' > 8 k i A i or I' < 8 k t I i . ?be former case represents B wave solution far B , and the latter corresponds to the Benjomin-Feir (or modulmionnl) insfobiirfy with a criterion in terms of the nondimcnsional wavenumber i = ( t / k m ) as

i'< 8C.4.

(10.7.11)

Thus the range of instability is given by O
(10.7.12)

Since fl is a function of 2, the maximum instability occurs at =, 2 = Zk, 4. with a maximum growth rate given by

(Re

='m&&.

(10.7.13)

To establish the connection with the Benjamin-Feir instability.

we have to

find the velocity potential for the fundamental wave mode multiplied by cxp(kz). It turns out that the term pmponional to B, IS the upper sideband, whereas that pmponional to B% is the lower sideband. The main conclusion of lhe above analysis is that Stokes water waves are definitely unrfobie. Example 10.7.2 (The Nodinem Klein-Gordon Epafion).

We consider UK nonlinear Klein-Gordon equatlon in the form

l u . - ( ~ ~ + l u , , ) + m ' y r + y y ' = O , x e R , O S y S n , (10.7.14) where m and y an constam. We assume the boundary condition

v=O

for y = O and y = a .

(10.7.15)

For the linear case, we a w m e lu(x.y.r) = Y ( Y )e x p [i (b-

nu)]

(10.7.16)

421

10.7 Eromples of Physical Appbcolionr

to obtain

Y(y) = A sin (q) + B eos(ay), a’ = a*- k’ - rn’ .

(10.7.17)

Since B = 0 by (10.7.15). from the boundary condition, we obtain = W l -k’ - m a =

(7)’.,.... n=1.2,3

(10.7.18)

so that the group velocity C=ru’(k)=E and o = o ( k , n ) . Hence, there exist various modes dcpcndent on n. We consider a wave witb n = I and its amplitude varier slowly due to nonlinearity. so that slow variables 5 and r defined by

(10.2.5ab) can be used. We next assume the following form of the solution: y=

c&”

y‘”, y‘” =

yy’ ( 6 . 7 , ~ erp[it(k. ) - at)]. (10.7.19ab) <--

1-1

Retaining only terms of thc first order in

E , we

and the r-d-order

terms in

&

y‘2)

@+-+(t’d

J5

It I # I .

(10.7.20)

give

a‘

Zi!(k-&)

obmn

vr’ = 0,

yf“ = #(<. r) sio(ay).

-t2k’ -m’)

vy’ =O.

(10.7.21)

J’Y

where the first term vanishes bceausc C =

a

so that

!$’= 0 for II I # 1. For

I = 1. the terms of the third order in E yield

- 3 y l $ r # sin’(ny)=O.

(10.7.22)

Multiplying this equation by sin(cry) and integrating with respect to y from 0 to a causes the first and second terms to vanish. Consequently. equation (10.7.22) gives

#(C,T). Since a’-k’ > 0.a modulational instability occurs when y > 0. Denoting the initial amplitude by 4. the solitary w w e solution is given by This is a nonlinear Schrodinger equation for the amplitude

[[

Eromple 10.7.3 (Motion of Schrdingcr Equnfion).

0

]

-2 5 (10.7.24)

x sech A o

,

8(02-k1/

Vortex Filament and the Nonlinear

The motion of a very thin isolated Yonex filament X = X(x.1) of radius

&

in

an incompressible unbounded fluid by its own induction is described

asymptotically by HaSimOtO (1972) in the form E=Gkb,

(10.725)

I

where s is the length measured along the filament, t is the time, k IS the ~urvature,b is the unit vector along the binormal, and G is the coefficient of local induction

G=

(5)

[loP(€-')+O(l)].

(10.7.26)

which is proponional to the cmculation r of the filament and may be mated BS consmnt if the slow vmatmn of logarithm compared with that of its argument. The local motion is approximated by that of a thin circular ring with the same curvature. Under this appmrimation, the tangential motion along the filament due lo suetching 1% neglected, even though I[ is very imponant in many cases. InValucing a suitable choice of the units of length and time transforms (10.7.25) tnto the nondimensional form

X=kb.

(10.7.27)

when a dot denotes b partial derivative with respect to time. We next include the equations of differential geometry. that is. the Semt-Frenet formulas for the tangent vector t, the principal normal vector n, and the binormal b ( t = n x b and I, n and b form a right-handed system) L'=KII,

n'=rb-rt,

and b'=-rn.

(10.7.28)

where a prime denotes a partial denvative with respect to arc lengths, k is the r is the torsion of the curve. When k and T are specified at each point of the curve, the shape of the curve, except for its location. is uniauek detumined. We prove that the complex function C U W B ~ and ,

429

10.7 &ler of Physical Applieononr

which contains both K and T and. hence, completely determine the shapc ofthe filament, satisfies the nonlinear SchrMinger equation. We first combine the last two of the Semt-Frenet formulas(10.7.28) into the complex form

(n+ib), + i r ( n + i b ) = - ~ 1 .

(10.7.29)

Inmducine N = (n+ i b ) exp[ i

j Td$'],

v = K exp[ i j r ds'].

(10.7.30ab)

we obtain

N'=-vt

(10.7.31)

and the first ofequation (10.7.28) becomes =

i (v'N + V N ' ) ,

(10.7.32)

where the astenrk denotes a complex conjugate. The displacement of the position vector along the filament gives a unit tangent vector to the curve,that is, x,=I. (10.7.33)

To find the time derivatives off and N, we proceed as follows: _=

-( X , ) = ( K b ) , = K , b - K r n .

m a

(10.7.34)

Sl"W

K, b - K

m =Re [ K, ( b + i n ) + ; K r ( b + i n ) ] =Re (i(K,+iKT)(n-ib)]

In general. the time derivative of N is a linear combination of N . N'. and I , and hence,we can write

(10.7.37)

( z ) = a N + p N ' + yl.

where a, ,9. and y are constantsto be determined.It follows from the definition of N and the fan that 1. n, and b are onhogonal that N = O end N . N ' = Z .

N N=t

(10.7.38)

We take the dot product of (10.7.37) with each of N , N ' , and 1 to denve

Zp = N

N ) =0,

.* = ' ( N .

a t 2

2 a = N'

.

*.

(10.7 39)

at

Consequently.

2(a+a') =

2t ( N . N') = 0

(10.1.40)

and hence, a must be purely imaginary. so that a = i R . where R is real. Finally,sinee d ( N . t ) = O , y=t I 0 that

.

* at

= -N

. %=

a

-'

N ( y , N ' - !y: N ) .

(10.7.41)

2

y = -i y,. It turns out that (10.1.42)

*=i(RN-y,t). 'lhe timedenvativeof(10.7.31)and thesderivativeof(l0.1.42)givc

N,,

='(

Ww: N - wP,N')-l*.

N , , = N ( iR,

-' y,

v')-i

yt: N - i ( R

(10.7.43)

+I

y!,,)l.

(10.7.44)

Equating the components of lhcse two expressions yields aR 1 =-( P' w, + P V : ) . as 2

ad

Integrating (10.7.45) giver the value of R as

(10.7.45)

10.7 f i m p l a of Phyrtcol Applrcotronr

R=

431

1 [ v I‘

+A(!)],

(10.7.47)

where A(r) arises due to Integration. Substituting the value of R in (10.7 46) leads to the nonlinear Sehr6dinger equation

i v,+ v,,++ where A(f)

CM

[Iv f + W ]I= 0.

(10.7.48)

be eliminated by the transformauon u=-yrexp

[

--jA(r‘)dr’

].

(10.7.49)

This transformation i s nothing but a shift of the origin of integration in (10.7.30ab). Therefore. without loss of generality, we may set A = O in (10.7.48). Conrequently, (10.7.48) reduces to the standard nonlinear NLS equalion

i y, + v,,+

1 Iv

(10.7.50)

1y = 0.

We obtain a solitary wave solution which propagates steadily with a constant velocily c along the filament which is B suaight line at infinity. that is, K+O Bs $+-. (10.7.51) l k s , in h e h

e ofreferenee.

K

and I a n functions of

5 =$-el,

1.

V =K (<)e x P i j r ( < ) h ’ ] .

that

ts,

(10.7.52)

The real and imaginary pans of (10.7.48) give. respectively, -CK[C(<)-

I ( - c t ) ] = k”- K T 2

++ K ( K * + A ) ,

(10.7.53)

and C K ’ = ~ K ’ I + KC’.

(10.7.54)

Integrating (10 7.54) and using (10.7.5 I) to determine the constantof integration gives ( c - Z r ) K = O which yields r = r O = ~since e ~ t 0 This . means that the torsion i s constant along the V O ~ C Ifilament and the velocity of propagation

along the filament is twice the torsmn. The shape of the Y O R ~filament X corresponding to the single solitary wave of the NLS equation, r = I . K=Zsechs, IS shown in Figure 10.2. Several other figures have been presented by Hasimoto(l972).

432

Figure 10.2 (a) Shape of the vortex filament, (b) Projection of the curve on the (x,r)-plane and (e) Projection of the CUme on (he (x,r).plsne.

I0 7 Eurnplcs of Physical Applicotionr

433

Finally, we use (10.7.51) to integrate (10.7.53) 80 that the Solution becomes

k({,r)=2v sech ( v C ) = Z v sech [ ~ ( s - c r ) ] . provided that A is a ConsLBnt related to r, and

Y

(10.7.55)

by A = 2( 7,' - v ' ) . The actual

shape of the filament is obtained from (10.7.55) by seaing r , = i e . Physically, the solution represents B helical curve of constlnt torsion ro with a euwat~rer that decreaser; from B maximum value 2v at the point s = c t to zero,BE 1s -. This single loop of helical motion moves along the vertex linewith a veloeity e = 2 r o .

I+

Example 10.7.1 (Lnngmuir Solirons and "keir Insrability) We conrider one-dimensional Langmuir waves propagating in a plasma of cold ions and hot electrons. We assume that Ihe eleevonr respond adiabatically to high frequency motions. Using the standard notations for various quantities. which are similar to those involved in Example 9.3.1. the equations of motion for the eleeuonr are given by (10.7.56)

-+u.

JI

-[:

" .= Jx

-=4m(q JE

$1 *.

-..).

(10.7.57)

JX

(10.7.58)

JI

We decompose the varioui quanuties into B high-frequency pan and a Iowtiqueney pan characterized by the time scales mi: and 0;:.respectwely. $0 that ne=n,+n,+n,,. ",=no+"*, E=Eh+E,. (10.7.59) where and E, refer to the high-frequency perturbations and n, and E, denote the low-frequency perturbations. Assuming charge neuvality in !he lowfrequency motions, equations (10.7.56>(30.7.58) reduce to the form

-+he, Jt

J Jx

((a+",)u.]=o,

(10.7.60)

(10.7.62)

3=-4nr".,. JX

Eliminating u. from (10.7.60) and (10.7.61) gives

which is, by (10.7.62).

Equation (10.7.64) describes the Impping of a Langmuir Wave packet in density cavities. We assume that Eh can be decomposed into a fsst time-varying component and B slow modulation. that is. we seek B solution in the form

E,(x.I) =%(x.I) e x p ( - m ~ ) ,

(10.7.65)

where %(x,I) has a slow modulation in time I. Substituting (10.7.65) (10.7.64) gives an equation for the modulation function %(x,I)

I"

This equation is based on the hypothesis that the effects of nonlinearity and finite tcmpcrature appear as corrections of the same order to the linear dispersion relation of cold plasmas near cutoff values of the wavenumber. Based on the assumption that clccuons respond isothermally to low-frequency motions. the law-frequency motions of plasma are given by

- E< -- - A % - v e * , (no+"() Jx

m,

-+-[(a+.,) J5 Ji

Jx

JI

m,

Et = O ,

where elecuon inema is ignored

4=0.

Jx

(10.7.67) (10.7.68) (10.7.69)

435

10.7 Elomplcr of Physical Appfualians

Eliminating a, from (10.7.68)-(10.7.69) yields

(10.7.70)

We next use (10.7.67)to eliminate E, from (10.7.70).SO that equation (10.7.70)becomes

3-c:

Averaging

oytr the

$L=[y]:(ue%).

(10.7.71)

time scale 0;:. the low-frequency contribution of

= (I6L meno)-'

d(l% f ).

(10.7.72)

ax

This represents the fast-time-averaged low-frequency pondcromotive force generated by the self-interaction of the high-frequency part of the elecuic field This force moves the electons out of the plasma regiom where the eleeme field has a local maximum. In view of (10.7.72),equation (10.7.71)takes lhe form

-_a%, &'

a%, axz

c.--

I

a'

16nm, ax2

(1%r).

(10.7.73)

This equation describes the formation of density cavities due to the pondemmotive forcerelated to thc elecme field. It IS convenient to use the nondimcnsional quantities defined by . x x =-

1,

,

.

f =Wncf,

and

,'=;n "a

E

.

4

=

W

i

'

In terms of these dimensionless quantities, dropping the asterisks, equation (10.7.66)and (10.7.73)represent Zokhorov's sqwtionr

(10.7.74) (10.7.75)

Eromple I0.7.S (Motion of an Electron Fluid). b lme basic equations of motion for the onedimensional adiabatic motion of an elechunfluid are h J -+-(nu)=O, (10.7.82) & Jx

*+u*+L'(nT)-L*=O, Jr Jx mn Jx m Jx

(10.7.83)

"(i T )'= O . r

(10.7.84)

-%+4ae(n-n,)=0.

(10.7.85)

JX

where n and u are the density and the flow velocity. respectively, electrostatic potential, and T is the electron temperature. Eliminating T from (10.7.83)-(10.7.84) gives Ju

Ju

Jr

Jx

-+u-+

[-

n h -L*=O, J x ) m Jx

0 is the

(10.7.86)

where 4 and To are the average values of n and T, respectively.

We intrcduce the mnsfomation of variables < = ~ ( x - C r ) . z = & * t ,and O = k - m t ,

(10.7.87)

whcrc B is a small parameter and Cis B constant to be determined. We expand n. U. and 0 in powers of E as

n = 4 + ~ " " ) ( 5 , r , ~ ) + & ' ~ ' ~ ) (. ~ , r , ~ (10.7.88) )+... Substituting

+"'= 1 ( 5 , 7 ) e x p ( i ~ ) + c . c .

(10.7.89)

from the first-order equation, we obtain the dispersion relation (10.7.90) whe

m9 =(4xe'n,/m)i

me compatibility condition of the second-order equation gives

438

The Nonlinear Schrbdinger Equoaon and Solimy Waves

&!=

dk

3T,lr

(10.7.91)

( m w ]

Simildy, the compatibility condition of the third-order equation yields

+ where R is an arbitrary function of (10.7.92) reduces to the form

a$

-

R ( < , ~ ) ~ = o . (10.7.92)

5 and c. If R is independentof 5 , equauon

i*+a*+BIvr

ac

tl

v+iyyr=o,

(10.7.93)

where q = & ( ~ - C t ) = & C - C r ,~ = & ' l = & and r ,a.O and y me known e o n~ t a n lPWCD ~ bv Sanuki et al. (19721. I n this case. a B < O . and Y is a

439

10.8 Applicorionr to Nonlinear Opscs

VxE=-Z, V.D=p.

VxH=z+j, V .B=O ,

(10.8.lab) (10.8.2ab)

where p an d j an elecuic charge and c u m t densities. respectively. The law of conservation of charge is associated with the Maxwell equations. Adding the time denvatwe of (10.8.2a) with the divergence of (10.8.lb) giver

*+v

j=o.

(10.8.3)

I

Integrating this equation over an entire volume Vyields

2 JpdV = - j V . j Jr

"

"

dV = - j j

.D

dS.

(10.8.4)

This expresses the fact that an increase or decrease of charge in a given volume V IS effected by the Van~porlof charge, that is. flow of current. through the boundaries of V. Since we deal with dtelecuics in which there are no free charges andin whichnocurrcntflows,lhatis. p=O and j=O.Equations(10.8.lab)(10.8.2ab) combined with constitutive relations connecting B and H and D and E represent Be field cquationr. me constitutive equations an B=m, D=$E+P=€E. (10.8.5ab) where f l =

( E ~e-')

and

E

(both of which are constanu) a n called rnogncrie

penneabdify and dielectric eomIonls.respectively. and e is the speed of light in a vacuum. The vector P is called the polarization. and it is z r o in a vacuum. When light propagates in a dielectm medium, the eleclric field causes distortion in the atomic ~ t m ~ t ~generating re. local dipole moments, and thereby induces a polarizatatlonfield P which depends on the elecvic field E. For small to mcderate values of the elecVic field amplitude and if there is no resonance between the electric field and the medium. P is a linear function of E . In an isouopic medium which responds instantaneously 10 the electric field, P = Eo x E , where 1s called the rlecrric susccptibilify.

n

Taking the curl of (10.8.la) and utilizing (10.8.lb) and (10.8.2a) gives

, J'P

v'E-~~-Vp.E)=(E,c

-1

Jt' JI' ' (10'8'6) This represents the propagation of the electric field. valid for both linear and nonlinear wave propagation. Since P = E , x E and x is a constant, then, (10.8.6)reducestothc wave equation c

V2E-(!!)1c

dJr' 'E=O,

(1 0.8.7)

440

where n =

The Nonlmaor Schrbdingsr Equanon ond Solitary Waver

fi (> I) is B nondimensional quantity called thc refmcrive indu

of the medium. Equation (10.8.7) shows that the electric field P propagates with the velocity ( c / n ) . We seek singly polarized. plane, linear wave rolutions of equation (10.8.6) in

the form (10.8.8)

E ( x . r ) = ; [ A exp(ie)+c.c.],

2 is B unit vector in the direction of polarization of E . A is a complex amplitude. @ = k . x - o r , k = ( k . t . m ) . and C.C. stands for the complex conjugate. They an called monoellrowtic waves because 'hey contain only one wavevector k and one frequency w . Such solutions (10.8.8) exist for A 10. where

pmvided that the dispersion relation

[

D(k,w)A= (k*+!*+m')-

[UI ? !

A=O

(10.8.9)

is satisfied. For nonlinear dispersive media, D also depends on the wave amplitude. In general, D=O can bc solved for any one of the variables k,C,m, and w in terms of the other three. In optics. 11 is customary to solve for the wavenumber in terms of frequency befause the refractive index n depends on the frequency. that is. n = n ( w ) . For linear or nonlinear waves. the dispersion equation (10.8.9) is algebraic.For wavepackets in a linear dispersive media, (10.8.9) is a linear panial differential equation for A that desenbes how the wavepacket envelope A moves with the group velocity and is modified by dispersion and diffcaetion.However, for wavepackets in a nanlineardispcrsive medium, (10.8.9) repmenu a nonlinear panial differential equation for the complex wave envelope A . For the simplest nonVxvial case. A satisfies the nonlinear Sehradinger equation which is a very basic equation in nonlinear optics. Moreover, the dispersion relation is one of the most fundamental concepts in the theory of wave propagation. The dispersion relation (10.8.9) is also valid when the refractive index of the

(2)<<E.

medium varies slowly, that is, V n = 0

1

where L IS the distance over

which n changes. We assume that the electric susceptibility x . and hence. the refractive index n an slowly varying functions of the space variable 1. and we also suppose that E = j E is polarized in t h e y direction. This implies that equation(10.8.7)holdsforthemalarficld E ( x , z , t ) . We thenseekasolution E ( x , z , f ) = A erp(ie)+c.c.+&E,,

(10.8.10)

441

10.8 Applicalroni to Nodimor Optics

when k = ( k , m)= VB and w = -0, denote the wavevector and tkquency of the

elecmagnctle wave and

&=

.z << I

Since n varies slowly with x, that is, n = n ( X = a ) . when & = & < < I , L hence, k and w also vary slowly. If 0 IS assumed to be constant, and since k = 8, and rn = B,, then. kx = m, holds. Since k is independent of I we q u i r e that m also be consLBn1, and hence, only the wavenumber k can change in the direction of variable refractive index. Substituting the solution (10.8.10) m equation (10.8.7) and assuming that n. A , and k are slowly varying functions of X. that is, functions of X. gives the dispmion nlanon (10.8.9) to leading order

We also find that O(&) : V’E,

-(!I

9

= (-2ik*

-idxe

A)

JX

ela

+ c.e.

(10.8.12)

To eliminate solutions of E, that grow like z exp(fiB), we must set 2ke+* A=O. dxdx

(10.8.13)

or (kA2)=cons1ant = ( k o

4).

(10.8.14)

Tknfon. k ( x ) end A ( x ) are given by k(x)=[n’(X)$-m’],

A(x)=(p)’A.,

(10.8.15ab)

pmvid.4 that k # 0 Equation (10.8.14) is eallcdtheei~nolequarion,and (10.8.14)represenuthe conservation of wave actmn. This method of analysis was first proposed by Wenuel, Kramers, Bdlouin. and Jeffrey and is known as, the WKEIl method.

Initially. with

(?)<

(y)>m.

the wave cannot propagate, and k > O . However, if

m ,then exists a point x, called a causrie. at which n ( x , ) , = ’

m

and k = 0. At this point, the light rays turn amund and the deeaomagnctic wave

/m.

T k Nonbneor Schrodinger Equorion and Solitary Waves

442

is reflected back to the medium with k =

-

The exact "at"=

near x = x, cannot be determined by the WKBl method because k +0. the

[ -:I

amplitude A increases like 0 k

and the WKBl method fails. In this ease. we

canapprorimate n2 locally by { n ' ( x , ) - a ( x - z , ) } and seek a solution of the

fnm ~ ( x , t ) = ~ ( exp[i(~r-wt)l+c.e.. x) It

N r n S out

k=%.

(10.8.16)

e

that B(x) satisfies the Airy differential equation

When x > x,. we obtain a solution of (10.8.17)which decays exponentially. On the other hand, when x < x,. the solution of (10.8.17)has two components -one corresponds to an incoming wave with the wavenumber (-k.0.m). where

k=-[(yr-m'}i

and the amplitude is given by (10.8.15) and the other

represents the reflected wave with the wavenumber (k,O,m). This reveals the fact that light rays turn toward regions of higher refractive index and turn away from regions of lower refractive index. This fact is widely used in the theory of the propagation of light in diclccuic materials and has tremendous ramifications. First. the correction to the refractive index is an increasing function of the field intensity, that is,

If the medium is linear, then 4 = 0. The electromagnetic waves are governed by the standard linear wave quation. So the quantity S = ".IA I) measures the strength of nonlinearity. and producer the $0 called Kerr effect. When light propagates in a Kerr dielecmc medium, Ule light continues to focus in regions of p a t e s t intensity This instability IS called the selffocusing inrtobility m optics and it IS widespread in nature. I n certain cases, a wavetam of amplitude. which is uniform in space and time. becomes unstable. and light intensity mncenmtes in collapsing filaments. If there is only one tansverse direction, B balance

10.8 Appbcalionr to Nonlinear Optics

443

between diffraction and nonlinearity leads to the formation of solitons. Hasegawa and Tappert (1973) proved. mathematically, that an optical pulse in a dielectric fiber forms an envelope soliton. that is, the pulses of light either do not change shape or change shape periodically. Mollenauer el aI. (1986) demonstrated this effect crpe~mentally.An interesting discussion of these developments has been given by Hasegawa (1990). In two transverse dimensions, the collapse is no longer confined.and the light filament continues to intensify. I n reality, light waves are not exactly monochromatic. They usually consist of a linear combination of plane waves whore waveve~torsand frequencies are confined to B narrow band of order & about some central values k, and E,, where 0 c & i 1. Such plane waves are called wavepockets represented in the f m ~~

E=i[A(x,r)eip(i(k,.x-ml)]+c.c.]+O(E).

(10.8.19)

combined with the dispersion relation D ( k o , t o . m ~ , w=o 0) .

(10.8.20)

where the wave amplifude A(x,r) is a slowly varying function of space and time, that is. (10.8.21)

We intmduce the notions of local wu~vev.%lor k and local frequency 0 . and write Ule complex amplitude in the polar form A = D erp(i9). where a and Q are real. Then the phase of the leading order eantnbution to the electric field is given by

0 = ko . x - mo I+~ ( x J ) ,

(10.8.22)

so that

k=VO=ko+V#

and

w=-O,=mo-@,.

(10.8.23ab)

Evidently.

dc+vw=o.

(10.8.24)

I

This represents the conservation of wavenumbem. We next consider the evolution of the wavepacket (10.8.19). with kQ =(O.O,m,) and ;=(O.l.O) of an eleevie field E that satisfies equation (10.8.7). For simplicity. we first consider the case of eonstant n. Substituting (10.8.19) in (10.8.7) gives

T k Nonlinear Schmdrngcr Eqmtim ond Salirory Wovrr

444

where m, =

[?).

Since each derivative is of order 8 , the dominant terns in

(10.8.25) are the fmt and the fourth, so, to the leading order, (10.8.25) yields

Ac +( f ) 4 = 0.

Thus, to a first-order approximation, A is

{ -(a-)

1

(10.8.26)

a function

{

of t -

(3- I I

end is

z is constant, that is, m a hame of reference moving

constant when t

with the group velocity

c, = v,

+ r).

(10.8.27)

0.

In view of (10.8.26). it turns out that A, =

equauon (10.8.25) beeoms

(3)’

A,. and so, to order

JA

&’,

(10.828)

This is usually known BS thcpnmriol equation bccaue the rays. the normal$ to the planes of constant phase, of ell the plane waves in the wavepacket are nearly parallel. h o k i n g the change of variables to a frame of reference moving with

, equation (10.8.28) assume$the form

4-- A_=O.

(26.1

(10.8.29)

This equation desnibcsthe dispersion of B wavepacket in the tran~vemedimtion. Such a phenomenon in optics is called W W I diffraction. We next recall the dispersion relation (10.8.9) with constant refractive index which WBS originally derived by using A e ip{i( k.x- ot) } in (10.8.7). If. instead of substituting B plane wave for E, we insen the wavepacket (10.8.19) with k, = ( O , O , q ) toobtain amodtfieddispenionrelalion D -id,

[ j x

-id,

aY

mo-id,

rn

I:

Q$+i-

A(x,y,z,t)=O.

(10.8.30)

10.8 Applications

'Ibis

10 Nonlinror

445

Opllcr

IS BD equation for the slowly v q i n g wave envelope A.

We now utilize the frequency dependence of the refractive index. and use

',sorhatequaUon(10.8.30) becomes

To first order. we recover the dispersion nlation for the plane Waves (10.8.32)

To the wdcr of 8 , we obtain the equation (10.8.33)

m i s is consistent with the fact that, to t k leading order. the wave envelope A of B

wavepacket propagates with the p u p velocity C,. where C;' = Next, we expand (10.8.30) lo order

8'.

and. retain all terms up to second

We now incorporate the effects of small variations of the refractive index to the order E' in (10.8.35) by replacing n tn (10.8.31) by (n+Sn). Consequently. an extra term e-'(Znm; S n ) A will be included (10.8.34),

I" the

equation

and, if we divide across by (Zinb), an additional term

-iSn(:)A=-i(:)

-i(

% A isgcneratedin(10.8.35). andthisterm becomes

S n A . Thus. equation (10.8.35) reduces to the form

$)o

oJ$(

4+

4

+k[($]@

A=

+($3+($3 A,,

-;($)(an)]

A”

A=O.

(10.8.36)

This equation is widely used in nonlinear optics.

Finally. we derive the nonlinear SehrBdinger equation for lhe evolution of B wavepacket envelope A ( x , t ) in a dispersive medium with a nonlinear refractive index n given by (10.8.18). The nonlinear dispersion relation is then given by

ForawavcamplitudeAoforder hence, the variation of

E , sothat

A, =O ( E’ ) . l A l ’ A = o ( ~ ’ ) ,and

S n in refractive index is of order

E’, the resulting equation for ule wave envelopeA is the same as (10.8.36) wiul S n = 4 IAr. To include h e effects of slow variations of space variables and amplitude. we nplaee 6 n in (10.8.36) by S n ( x , y ) + n , l A r . $0 that dropping the subscripts zero on m and the cocfficicnu. equation (10.8.36) becomes.

441

10.8 Applicorrons to Nonlinear Oplm

IftheassociatedcarrierwaveIsproponionaltoexp(i(k.r-wf)],where k=(O,O,m) and m=*,

(10.8.41)

then Jm _ =__

do

c

do

(no)and a=B=-.!..

(10.8.42)

m

Thus, the resulting nonlinear Schmdinger equation becomes

-(:)(Gn+n,lAr)

A = 0.

(10.8.43)

The above analysis reveals that the NLS equation (10.8.39) or (10.8.43) describing the self-focusing of light beams is the fundamental equation in nonlinear optics.

In particular, if A IS independent of X . y. and t and the resulting equation (10.8.39) IS given by

S n I 0, the solution of

A ( r ) = A , er.p[im(:)IAorz].

(10.8.44)

This solution can be used to determine the wavenumber of the electric field E = E [A(L) erp{i(mr-ol)}+c.c.].Th~*alucofthiswavcnumberiscquol to or E ( n + n l l A f ) . According to the theory of the NLS

m(l+:lAr)

equation, it also reveals that. if the produet of ( m n , ) with any one of three dispersive

mn

(

coefficients

4J

) -.!. =-n2)

a, p, or y

(that

is,

a=S=-'

and

m

is negative, then, the amplitude-modulated solution

(10.8.44) is definitely unstable.

In particular. if the wavepacket envelope A is independent of

1

with

a=B=-l-.equation(10.8.39) becomes m

-L(A= + ~ , , ) - ( + rm )(&n+n,IAr) 2m

A =O.

(10.8.45)

T k Noniincor SchnMinger Equation ond Solirory Waves

448

Funher. if we assume that 6 n = 0, then this equation nduces to the from i A, +-!-(AP

+A,)+(

2m If, initially, the field envelope A=A&)

E)

",(A(' A = 0.

(10.8.46)

IS independent of x and y.

then, a

small permrhation 6 A confined at Some point causes the refractive index n, (A(' to increase there. The light pays from the neighboring regions continue to focus toward this region. enhancing the intensity more and more there. As more light focuses there, the refractive index becomes even higher. and even more light eonverger. Thus, the effect of an increased refractive index leads to focusing instability in nonlinear optics. However. this instability is not confined to tran~ve~ modulations. e We now denve the nonlinear Schrminger equation for a eoncenvated w a ~ modc e traveling in a waveguide. and, for this model, there ISno transverse dispersion but only dispersion in the direction of the wave propagation. So, if we suppose that A(z,r) is independent of x and y with the factthat S n = 0, then, (10.8.39) becomes

Obviously, the focusing instability occurs provided that the product

[$](-?)-? =

):(

is positive. Far the diffraction case. the product is

whose sign depends on that of n2, In the case of wave pmpagalinn in an

optical fiber. it can be shown that ",> 0 only if y c 0. However, in general, y can be positive or negative in the neighhorhwd of resonance where lhe real pan of the susceptibility depends on frequency. These situations are referred to as normi dispersion and mornalous dispersion,respectively. According to the general theory of the NLS equation, there exists a stable soliton solution of (10.8.47) in nonlinear optical fibers. Two Solitons of different frequencies have different group velocities and pass through each other without distortion. However, if two solitons have the same frequency and, hence. the same velocity, they undergo a p e r i d e oscillation. There is a remarkable change in the propenies of solutions of (10.8.39) when two transverse dimensions and time are included. In the defoeusing ease n2 O ) , uniform wavetrain solutions are neutrally stahle. In fact, deformations in the uniform wsvetreins. in which intensity diminishes loeally, neither grow nor decay, but evolve as solitons of a different nature. These solitons are called dnrk solitons because they have intensities lower than the ambient wavemains.

Show that the NLS equation in Exercise 2 has the M A solitary wave iolution (Ma, 1979: Peregrine. 1983) of the form u(x,t) = n exp(io’t){l+

Zm(rncos e+ insin e)/r(x.r)}

for all real a and r n , w h e r e n ‘ = ( l + m ‘ ) , f(x,t) = n cosh(ma&x)+em

8=2rnna’l, and

8.

Seek a traveling wave solution of the NLS equalion in Exercise 2 in the iorm u(x,t)= I

cxp ( i ( e + o r ) } ,

w h e n +--el) CODS~BIIB.

and B(x-et) Prove that

arc real funcitons and c and 0 are

when R = r 2 ,and F ( R ) = R’ -2(

0)

- 1c’) R’

real

+ BR+ A , and A and B are

arbitrary real constants of integration. Hence, show that there exists a solilary wave solution of the form u(x,1 ) = n

exp

K 8

- c(x - c t) + 0 t}] sceh

[%

(1- c t)]

,

for all a ‘ such that

2 ‘ = 2( 0-’ 2 ) > 0. Following the same method as given in Exercise 4. show that t h e n exile a solitary wave solution ( Z a k h m v and Shabat, 1973) of the NLS equation iu,+u” -(ur u = 0 with r’(<)= m - 2 x ’ sech’ ( K<) and e bm[S(<)) = - 2 k tanh( K<) for all c,where < = x - c t ,

w=-m,md r=1(2m-c2)t>0

Show that an autodbeklund transformation (Konno and Wadati, 1975) for the NLS equation i”, +U” IS the

followingpair of equations:

+ 21ur u = 0

whenuand u satisfy the equation. Hence. find the solitary wave solution of this NLS equation by setting u = 0 for all x and I. 7. Derive the nonlinear Schmdingerequation for the local amplitude A ( x . y . 1 ) from the resonant interaction equations (Phillips, 1981) in the form

when the derivatives ate computed at k = I = 0. m = mo = n ( mo) 9. Show that the solution

w = % ev(iy I b r z ) of the nonlinearScWinger equation equation Az -ia(A=+A,,)= i y A’A’

is unstable Also show that the maximum growth rate occurs for

k=l%lm.

Chapter 11

Nonlinear Klein-Gordon and Sine-Gordon Equations 11.1 Introduction 'Ibis chapter desk with thc theory and applications of nonlinear Klein-Gordon and sine-Gordon equations. Special emphasis is given to Y ~ O U Emethods of solutions of these equations. The Green's function method combined with integral vansfoms is employed to solve the linear Klein-Gordon equation. The Whitham averagrng pmeedure and the Whitham averaged Lagrangisn principle are used lo discuss solutions ofthe nonlinear Klein-Gordon equation. Included me different ways of finding general and pulicular solutions of the sine-Gordon equation. Special anention is given to solitons, antsditons, breather S O I L I O ~ S . energy associated with them. interaction of solitons, Biicklund Uansformations, similarity salutronr. and the inverse scattering method. Significant features of these methods and s ol ~ t i on me ~ described with other ramifications.

11.2

The One-Dimensional Linear KleinGordon Equation

The onedimensional. inhomogeneous. Klein-Gordon equation is given by u. -e'u,

+d'u = p(x. I).

x E R, f > 0

(11.2.1)

with the initial boundary conditions u(x, O ) = O =

&.O)

u(x,r)+O

ss

for x

(XI+-.

ER,

1>0,

(11.2.2ah) (1 1.2.3)

whne c and d are consmu.

The Green function C(x, f ) associated with the above problem satisfies the

esG,,-c'G_+d'G=S(x)S(t),

with the same initial and boundary conditions.

XSR, f>O,

(11.2.4)

454

Nodmaor Klein-Gordonand Ine-Gordon E q u f b m

Applicationof the joint Laplace and Fourier uansform (Debnath, 1995) giver

(I 1.2.5) when a = ( c * k ’ + d ’ ) i .

% inverse Laplace transform yields (11.2.6)

Finally, the inverse Fourier vansfom leads to the solution

(11.2.7)

I n the limit as d + O , this result IS In perfect agreement with that for the standard wave equation.

If the source is located at

(5, r). the Oreen’s function assumes the form

.(I 1.2.8)

In ems of the Heaviside funnions, the Gnen‘s function can be expressed BS

X

H[S+cr - ( x + c T ) ] .

(11.2.9)

In the limit as d + 0, this result Educes to that for the wave equation. me solution of the Klein-Gordon equauan (I1.2.1) with the initial condttionr

455

I 1 2 The One-Dimensional b n u r Klcm-CordonEquation

u(x.O)=f(x), u,(x,O)=g(x).

XER.

(11.2.10)

5. O)] 65.

(11.2.11)

can be expressed in terms of the Gncn's funcum as I

-

u ( x . f ) = j ~ P ( C r)C(x, .

f;

5, r)dC dT

0-

-

+I[g(C) G(x.f; <.O)-f(C)

G,(%

f;

Itshouldbenoted that, in thedoubleintcgralin(11.2.11). G = O f o r r > r , and hence. the limit in the r integral extends only up to f . Also, 11 follows from (1 1.2.8) that G IS nonzero when Ix 51c c(f - T ) , which IS equwalent to x - c ( f - T ) < < < r + e ( t - r ) . Consequently, the double integral in (11.2.11) komes

-

I

Idr 0

-

IP(<.G(x.f.5.7)dC T)

--

Similarly, since the product of the Heaviridc functions vanishes outside the interval (x -cf, x + cf), we find that

It follows from direct inteption that

+ 6(4 - x+cf) H ( r + cf - C)). In view of this result, the second tern within the square bracket of (11.2.11) beeomes

456

Nonlinear Klein-Gordon ond Sinc-Gordon Equoriom

If d = 0,this solution reduces inhomogeneous wave equation.

to

that far the Cauehy problem for the

11.3 The Two-Dimensional Linear KleinGordon Equation The two-dimensional.hnear, inhomogeneous,Klein-Gordon quation is

457

11.3 The T w o - D z m ~ ~ i o mlinear I Klein-Gordon Equrrion u,, - c’(u= +u,”)+ d’u = p(x, y. I).

-- <

X.

y<

-.

> 0. (1 1.3.1)

f

The mud and boundary condiuons are forallxandy,

u(x. y , O ) = O = y ( x , y , O )

u(x,y.t)+O

as r = w + - ,

(11.3.2)

1>0.

(11.3.3)

The Green’s function G(x. y . I ) for the two-dimensional Klein-Gordon equauan sausiies the equation G ~ - ~ ’ ( G , + c , , ) + d * C = 6 ( ~ ) ~ ( y -) -6<( x1,)y,c - .

1>0,

(11.3.4)

with the conditions C ( z , y , O ) = O = G,(x, y . 0 ) G(x.y,f)+O

for all x and y .

as ( x ’ + y ’ ) + - .

f>O.

(11.3.3 (11.3.6)

In ems of polar cwrdinates ( r , 0 ) without @dependence,equation (11.3.4) becomer

Application of the joint Laplace and Hankel transform of order zero gives

z ( k . S2 Z) 1 =l

(11.33)

(s*+C’K’+d’)’

whence. ule ~nvemeLaplace m s f o r m yields. ,

a=(e’r’+d’)~.

Finally. the inverse Hankel bansform leads to the solution

which is, by using the standard table of Hankel banrforms,

(11.3.9)

If the Source IS located at BIiSUrneS the form

(5, r). r) instead of the origin, the Green’s function

,A2 [

C(r,y,t;~.q.r)=---

T

$ [j -. ] ( );

--

H T-- .(11.3.12)

cos d T

where T = f - r and R ’ = ( ~ - t ; ) ~ + ( y - q ) * ,

11.4 The Three-Dimensional Linear KleinGordon Equation The Green’s function C ( x ,t) for the three-dimensional Klein-Gordon equation satistics the equation

C, -c*V’G +d’G = S ( x ) S ( y ) S ( z ) G ( r ) ,

-- < x. y. L < -,

t > 0,

(1 1.4.1)

with the conditions G(x,O)=O=G,(x,O) C(x,t)+O

as

forellx,y,z.

I.\+-.

( I 1.4.2) ( I 1.4.3)

where x = ( x , y , z ) . Application of the joint Laplace and the triple Fourier transform (Debnath. 1995) lo Lhe above problem gives

e(K, .) = I . ; (2 +&* + d ’ ) .

(11.4.4)

(27r)V

whnr

x=(t.f,m).

The join1 inverse Vansforms gives lhe solution C(x,t)=l

(W’

where a = ( e ’ ? + d ’ ) i

ljr c x p { i ( u . x ) }

-

dr,

a

(11.4.5)

( I 1.4.6)

which is, bccaure the lntegral is

Bn

W e n function of K, and

J,

can k written

outside the Integral,

-

-

=

4nrc

Jr

/f(c'f'-r')

(1 1.4.7)

in which the standard table of the Founer transfoms is used. Finally. we carry out the differentiation and use the identity J i ( k r ) = - k J , ( k r ) and J,(O)=I toobtain theonen's function

In the Iimjt as d 0. this result is in perfect agreement wtth that for the threedimensional wave equation.

11.5 The Nonlinear Klein-Gordon Equation and Averaging Techniques Whitham (1965ab) has successfully extended the averaging pmcedun for the study of nonlinear ordinary differential equations associated with oscillation problems to partial differential equations rnvolved in nonlinear dispersive waves.

Nonlinrar Kim-Gordon and Sine-Gordon Epvltions

460

The pmedure that he developed was to average over Iwal oscillations and then, to obtain partial differential equations describing the slow variations of the hasic wave parameten. such as, wavenumber, frequency, and amplitude In all problems where the equations admit laeally uniform periodic wavetrains as solutionsofthe form u ( x , t ) = u ( E ) , w h c r e e = ~ - r - o r is thephase function, " ( 8 ) is B periodic function of 8. and k and 0 are the wavenumber vector and frequency, respectively. In h e a r wave problems. u(x.f) =acos8, which is sinusoidal. According lo the Whitham theory of slowly varying wavemains, the S O ~ U U O ~ of the form u "(8,a) is maintained, where o is not a constant and e is not a linear function of 1 and f. The parameten 0 ,I ,and 1) are assumed to he slowly varying functions of .z and f corresponding to the slow modulation of the wavetrains In other words. we allow gradual changes in a,k , and a assuming them to he noticeable only at distancs much larger than K? and afier times much longer than 0.'. This is usually known BE the geomefricnl optics oppmximrion. Even though the wave parameters evolve, the general funeuonal form of u is retained in this approach. We consider thc (I + I)-dimrmioml.mnlineor. Klcin-Gordon equation ~

""

-!Im

+ V'(") = 0,

(11.5.1)

where V(u) is a general nonlinear function of u, hut not its derivatives. and the pnme denotes the derivative with respect to u We now multiply (1 I .5. I) hy u, and u, and reorganize the tams to derive the following co"Servati0" eq"BU0"S

[;(",?+".')+"I +[-.,I,

,I.,.-[

+[f(u:

++v],

=o.

(ll.5.2)

=o.

(11.5.3)

We look for a stationary wave solution of (I 1.5.1) in the form

.=.(<).

<=.-.I

(I 1.5.4ah)

so that intcgrauon of the multing equation gives

i ( c ' -I)=; +V(u) = A ,

(11.5.5)

where A is an integrating eonitant related to the amplitude of the waves. So, it follows from equation ( I 1.5.5) that (I 1.5.1) admits stationary wave solutions of the form u; = f ( u , c , A ) .

(11.5.6)

11.9 The Nonlinror Klcin-Godon EqwtiDn and Avcmghg Techniques

4 61

Based on the phase-plane analysis with the phase diagrams of u,(<), it can

bc shown that periodic wave solutions exist for all V ( u ) with a minimum. So we assume that (11.5.6) admiw a solution of the farm u = u(& c , A ) so that (11.56) gives

in which we assumed that c2 > I and A > V(u). Or, equivalently, (11.5.8)

Clearly, the solution is oscillatory betwan hvo consecutive zeros. say u ,(c, A ) and %(e.A ) with uI > Y, of the function f ( u . C. A ) between which it is positive definite. lbe positive definiteness between u, and u2 is a necessary condition to

ensure that

(2)

is nal. We suppose that

5, and 5,

are the valucs of

5

at

which u(<,, e. A ) = a , ( c . A ) and u(<,.c, A)=t+(c, A ) .

Using the analogy of the linear nonlinear waves by

WBYCI.

we define the wavelength

I for

Similarly, the wavenumber k = k ( e , A ) , the frequency 8 =m(c,A) and the periodic time T = T(c.A ) are &fined by

11.5 Thc Nonlinear Khn-Gordon Equmion and Avemgmg Techniques

.'[ k W c ] + d [ c k W c - A ] = 0.

465

(11.5.30)

dr Jx where, in writing the second term in (11.5.29), we have added kcAW, and subweled its equivalent expression cA. We next calculate panial derivatives of various terms I" ( I 1.5.29) and collect the coefficients of c. A. and W to obtain c [ d ( k rW c a) + dx( c k W c- A ) ] + A [ ddr ( k W A ) + dax (ckWA

-C)

-W[$+&(ck)]=O.

(115.31)

The first term in this equation vanishes due to (11.5.30), the second term a150 vanishes since kWA = 1. and hence, finally. we oblain

2+ d ( c k ) =0 ,

( I 1.5.32)

Jx

which is. intraducing the frcquatcy

0 = ck

~

I+da)=O.

(11.533)

?+

(11.5.34)

m ax

Or, equivalently,

m

*

m ' ( k ) 2 =0 .

ax

'l'his is the fundamental kinematic equation of the wave. Thus, along thc charactensties

= m'(k), the wavenumkr k (and hence, frequency 0 ) is

t

consnved.

Subslitmion of k = W;' in (11.5.32) yields

3W" dc = 0 . ax ax

?%+c at

(I 1.5.35)

Similarly, expanding (11.5.30) combined with (1 1.5.24) leads to Ule result dw-+,2!!-w"dA=o. (I 1S.36) JI ax dX Finally. the two independent averaged conservation equations (11.5.35)(11.5.36)canbeexpnssedinfheform

5!!Lw Dt

&=o A

ax

( I 1.5.37) *

_DW* _ w,dA=o,

(11.5.38)

21

Dt

where Ule total derivative is defined by

To determine the charactenslxs and characterislie speeds,we we (11.5.23) and (11.5.28) and. then, replace W i n terms of G to express (11.5.37)-(11.5.38) in terms of c and A as G " A , + c G " A , + ~ c , + L c, =0,

(11.5.40)

+ C'A. - G c -L c, =a.

(I 1 5 41)

( A I )

CC'A,

(2-1)

( A I )

'

(2-1)

These equations admit the following two charactenstics C,, so that C, :

;=[%)

and C. :

$=(=),

(11.542ab)

whse = (-")i.

( I 1.5.43)

G" The associatedcompatibilityrelations are given by

A-(-5)id4=0 along C,. (&I)

(11.5.44)

11.6 The Klein-Gordon Equrron

461

Equations ( I 1.5.42ab) reveal that there are two characteristic speeds ( I + _ c a ) / ( c + - aIf) . we cmsidw a wavetmn which is initially umform with c = co and A = 4 outside some finite region, then. after Some interaction period, the disturbance splits into two simple waves separated by a domain of constant values of e and A. In one simple wave, the characteristics C, are straight lines which carry conitimt vslucs of r, and the other Riemann invariant is constant everywhere. On the other hand, in the second simple wave, r has the s m e value everywhere, and s is constant along the characteristics C., w h e h are straight lmes. Between these two simple waves, c and A assume constant values. Since the wavenumber and amplitude are expressible in terms of c and A, the qualitative features similar to those as stated above are also applicable to them. The energy propagation speed is then obtained from (I 1.5.29) as the ratio of energy flus to energy density and has the form

kC(twc + AW, - w)- CA - kc(cwC - w)+CA - CA

Y(eWc+AW, - W )

k(cWc- W)+A

by (11.5.24). '

whichis,hyrerultJ(11.5.19)and(11.5.21). CG

= -

(11.5.48)

G+(c*-I)AG'

This n another imponant neu s p e d which IT dcRniicl) nor one of the 1%" characterst#. speed\ Ihn new ?peed daer nut wem to have a simple ph) w a l inierprctation Th w in the prc~mtfully nvnlincar cmc. there e x 1 4 three tm(un*nt -peed? - two eharacrerirtic y e d r and an energy pmpagaton sprd Cledh. one of them should br. intcrprctcJ as the proup \ e l ~ . i l y

11.6

The Klcin-Gordon Equation and the Whitham Averaged Variational Principle

The Lagrangian density of the Klein-Oardon equation (1 1.5.1) is given by ( I 1 6 1)

We seek a progressive, periodic wave solution of the nonlinear Klein-Gordon equation(11.5.1)intheform

u=u(e),

e=lrr--t.

(11.6.2)

Nonlinear Klein-Gordon end Sm-Gordon Equotionr

468

where the local wavenumber and local frequency are given by k = 0. and w -43%respectively.Substituting (11.6.2) In quation (11.5.1) gives (a’ - k‘)u,

+ V’(u) = 0,

(11.6.3)

!(w2- k’)u:

+ V(u) = A .

( I 1.6.4)

which gives the integral

whereA is a con~tsntof integration. This equation can b solved by B quadrature for a suitable form of V(u) and has a solution. +odic in @, given by

Since u is periodic function with period

Zn,we q u i r e that

where f denotes integration over a cmpletc period. Result (11.6.6) reprerenu a relation between w , k, and A, and hence. it reprercnts the dispersion relation. According to the whitham theory. the wave parameters k. 0 , and A are slowly varying funnions ofx and f corresponding to the slow modulation of the way.?. These pamnelua can then bc duemined from the averaged Lagrangian defined by Whitham (1965n.b) as

( I 1.6.7) where L is given by (11.6.1) which is simplified by using (11.6.2) so that L = L ( w ’ - k ’ ) ug* -V(u) = ( o x -k’)u: - A . 2 Substiluting (11.6.8) in (11.6.7) gives

I($)-

Ir

ie(o,k.A)=L(m’-k’)

2n

2

d@

2=

(w’ - k ’ ) S ( * ) d u - A

=

zz which is. by integral (ll.6.4),

0

a@

0

(11.6.8)

69P

470

Nonlinear Kim-Gordon and

Smr-Gordon Equariom

nonlinear waves. However, the variational method is applicable to weak nonlinear waves and also to all nonlinear wavetrains.

In particular, for the linear Klein-Cordon equation, V'(u)=u so that V ( U ) = ~ U ' Hence . the function G ( A ) defined by (11.5.20) can easily be computed by using the linear solution for u =-a cos B when 2 A =a'. It turns out that G ( A ) = Z n A so that G ' ( A ) = Z n . Consequently, the averaged Lagrangian8canbeobt~inederplicillyfmm(lld,9)as

so that the linear dispersion relation is

8, = 0.that is.

m' -k' = 1. (11.6.18) This is independent of the amplitude A, as expected in the linear theory. This also confirms the fact that the Whitham variational method is applicable to both linear and nonlinear wave pmpagation problems.

11.7 The Sine-Gordon Equation, Soliton and Antisoliton Solutions The sine-Cordon equation has a long history that begins in the latter pan of the nineteenth century when this quation was discovered in differential geometry in connection with the theory of surfaces of constant negative c~wafutt. Various methods for finding panicular solutions of this equation were developed at that time. One of the methods is known as Blcklund transformations that will be considered 8" a subsequent section. Other methods include traveling wave solutions, the similarity method, the inverse scattering method. and the method of separation of variables, which deals with the representation of ~dutmnsas funeuonr of independent variables. All of these methods will be discussed in subsequent secuoni. It has already been indicated in Chapler 4 that the sine-Gordon equation is one of the basic nonlinear evolution equations and has recently been employed to describe various important nonlinear physical phenomena. The standard form of this equation is _a'._ _ _1=a'"

&*

c*

m'

S'"U.

In terms of the characteristic coordinate wnsformations

(11.7.1)

11.7 T k Sme-Gordon Equairm. Solmn and Antisohion Solurions x =f

1( X- cT).

r=

1 ( X +cT),

equation (1 1.7.1) lakes the following form ux, =sin".

471

(I 1.7.2ab)

(1 1.7.3)

Thus. the two forms (11.7.1) and (11.7.3) of the sine-Gordon equation are interchangeable The equation is invariant under u + u + Z n r , where n = f I , f Z , f 3 , . . . This transformation u + u + ( Z n + l ) s leads to the replacementof sinu by --sin" in theequauon. Ihelineanzedfformoftheequahon(11.7.l)sg1ven by u,

-c

=u .

A plane wave solution of the form u(X.T)=Aexp[z(kX-mT)] provided the dispersion relation m'=c~(l+l~)

(I 1.7.4) exists (1175)

is satisfied. This shows that 0 is real for all real k and the equilibrium solution u = O of (11.7.1) is stable. On the other hand, if u = i + n for a small perturbatton a, (1 1.7.1) em be linearized to oblain

-

u,-e

-2-

uw=-u

-

( I 1.7.6)

Aplanewavesolutionoftheform i(X,T)=Aexp[i(W-wT)] also exists. pmvided the dispersion relation mz = c* (1' - 1)

(11.7.7)

issatistied.Hencc m ' c 0 . if O S k 2 < l , and B(X,T) willgrowexponentially in time T. that IS. the equilibrium solution u = IT is definitely unstable. Physically. this ISquite natural because, when the term u, is absent (that is when k =O), the sine-Gordon(11.7.1) reduces to the equation of a simple pendulum representing finite oscillations. Thus, the question of stability or instability is an important feature of the equilibrium solutions of the sineGordon equation.

We next discuss the single-soliton salutianr of the sine-Gordon equation (11.78) We reek a soIution of the form u ( x , r ) = $(I- Ur) = $(<) which corresponds to a wave traveling with a velocity 0 and, then. substitute in (11.7.8) to obtain an ordinary differential equation for @(<)in the Cum

where V'

=$.

Dividing both sides by ( I - V ' ) and multiplying by

@'({)

gives (11.7.10)

or.equlvalcntly,

that IS,

(11.7.12)

Solving for

@'(t) gives a first-order,adinary differentialequation for @(c) (11.7.13)

We can reparate the variables and. then, integrate to obtain

where A = B m . This rcsult depends on the two parameters V (or (I= c V ) , the veloeity of the soliton and B an integrating constant. Thus, the stable solutions represent solitary waves. periodic W ~ V ~ or E , a monotonically increasing function of 5 (Scott, 1969) depending on the sign and magnitudes of 0 and A. When A = I , a solitary Wave solution exists for any velocity. so that O < IVI < I (or Oc 1U1 c e ) . In this case. we use the trigonometric identities 1 - ~ ~ @ = 2 ~ i ~ ' ( and i @ $) [ l o g m ( f @ ) ] = ( l a i n f @ ) both sides of (lI.7.l4), so that

to

slmpllfy

413

11.7 The Sins-Gordon Equation, Soliton and Antidifon Solutions

(11.7.15)

This leads to the solution for

$(e)as ( I 1.7.16)

where a=tan(a$o) and

5,

are constants of integration which can be

incorporated in several ways. We choose a = I and solitary wave solution for u ( x , r ) in the form

5, = 0

to obtain one simple

This is called the soliron (or kink) solution of the sine-Gordon equation and represents a continuous profile with u + O , as x+-=-, and u-,2n, as x++-, shown in Figure 11.1. The soliton propagates in the positive xdirection with velwity V. Another solution can be oblained from(1 1.6.15) in the f m

0 Figure 11.1

The soliton and antisoliton solutions.

This is called the ontisoliron (or ontikmk). This Solution profile Uavels in the negative x-direction with velocity V. It represents a continuous profile with u - 2 2 . as x + --, and u + 0, as x ++-, also shown in Figure 11.1.

-

The denvauver of u ( x . t ) . that is, u, and u,, also represent solitary waves which are given by u,(x,r) = #'(x - a) and u,(x.r) = -Op'(x- Ur), where

@'(<)i s o b t a i n e d ~ ~ ~ ( l l . 7 . 1 6 ) a s 4un(t$)

9°C)

Zscch[(x-Vt)/dl-Y')

=

q l + t a n f + 9 ) }

inwhich m(f@)irreplaeedby

(1 1.1.19ab)

e x p 1-v [ e ] .

If A t I , other types of solutions of the sine-Gordon equation can be obtaned. Inpanicular. when V > l and I A l < I , the integral in (11.1.14) with r& = O and 5, = 0 can bewritten as

' [Fli1 which is. substituting

dv

d(l -cos y ) - (1- A )

=

1 (9

I=-

sm

,

'

I '

=(v2-i)i J

,dl-sl

ds

W'

(11.7.20)

This can be expressed in terms of lacobi'r elliptic function (outu and Debnath. 1965)

(11 7.21)

475

11.7 The Ins-Gordon Egutlon, Soliton ond Antisoliton Solurions

where m is the modulus of lacobi's elliptic function sn(r,rn). Thus the final

11.8 The Solution of the Sine-Gordon Equation by Separation of Variables The form of the solution (I 1.7.17) suggests that the Vansformation

"(x. can be used to wnsform

t)

= la"

(i .)

the sme-Gmdonequation u, -urn = sin"

(11.8.1)

(1 1.8.2)

into the form ( I + u')( u, - v,, - u)- Zu( u: - u: - u') = 0

in which the trigonometric identity s i n ~ = 4 v ( l - v ' ) / ( l + u 2 ) '

(1 1.8.3)

has been

utilized. We seek mlutions by the separation of vsriables in the form

u(x,t)=tB"(~")=O. (11.8.4) w(0

for some functions $(x) and (11.8.3)

"(1) to

equation

sin4R = 4 t m R(1-1m2 R)/(I+m'

and

be determined. Substituting (11.8.4) in using the identity

8)' with R = is, we obtain

Differentiating this equation with respect to x and I enables us to Separate the vanablcs so that

whcre -4x' is B separation constant. Each of there ordinary differential equations (1 I 8.6ab) em be integrated twice to find

Nonlrnror Klein-Gordon and Sine-Gordon Equofion.

476

4’’ = -k2$‘

+a$’ + b and

*I

= r*I ‘ +cv2+d.

(11.8.7ab)

Substituting these equations in (11.8.5) reveals that a - c = l and b + d = O . Setting o=m‘and b=n’,weobwm $: = -X’$‘

+ m’$’

v’ - n’.

+ n 2 and y: = K’V‘ +(m’ -I)

(I 1.8.8ab)

where m and n are integrating con~tsnts.I n general, these equations can be solved in terms of elliptic functions. However, we solve these equations far the following special cases of interest: Case 1. r = n = O a n d m > l In this case, equations ( I 1.8.8ab) take the form

$,=*m$

and

I,=*=

,I

(11.8.9ab)

fir).

(11.8.1Oab)

which give erponenual SOIUUO~S )(x)=a,exp(i N)

where

(I, and

where a =L!

and yt(r)=o,cxp(?

are integrating constants. Thus. the solution ( I 1.8.4) beeomes

and U =-

m

(or m = ( l - U ’ $ ) m

constants.

Evidently. one of these solutions(11.8.1 I) U(X,r)=4tan-’[acxp{m(x-ut)]]

(II.~.IZ)

is identical with (11.7 17) for a = I , and hence. it represents the solifon (or kink) solution of the sine-Gordon equauon. On the other hand, from (ll.8.11). another solution can be found in the form u(x,t)

= 4 d [ a crp{-m(x- Ur)]]= 4cot-’ [ i c x p ( m ( x

- Ur)}].

11.8.13)

This is also identical with (1 1.7.18) for a = I and represents the antisoliton (or antikink) solution of (11.8.2). Funhermore, .N and u, also represent solitary wave solutions given by u,(x,f)=f2mscch [ m ( x f U f ) + l o g a ] ,

ad

(11.8.14)

11.8 The Solution of the Sm-Gordon Eqmrion by Seprolion of Voridkr

q(x.r) = * 2 f i sech [ m ( x + Ur)+ l o g a ] .

4 77

(1 1.8.15)

Two other choices in sign m (11.8.11) represent antisoliron (or antikink)

solutions of the smc-Gordo"equation. Case 2. ~ = O , m ' > l a n d n + O In this case, solutions of (11.8.8ab) me obtained by integration in terms of hyperbolic functions as

(

)(x) = f :)sech

whm

0,

(m+a,),w(r) =

dml-l

cnsh( f i r + + ) . ( 1 1 . 8 . 1 6 )

and a2 integrating cons!mts. Thus. the solution is given by

(11.8.17) Since this result is made up of the ratio (O/w), it does no1 depend on n. In panieular,when a, = a 2 =O,solution(ll.8.17)~snbccipns~edinlhcform u(x,r)=*4m-'

[0

o
(11.8.18)

corh(mUr)]'

This solution was first discovered by Perring and Skyrme (1962) based on numerical calculations for two interacting solitons. Then, they verified analytically thnt (11.8.18) is. indeed, the exact solution. and It describes the interaction of two solitons similar to those of the KdV solitons. where solitons emerge from the interaction only with a slight change of phase. To show that the solution describes the interaction of two equal solitons, we investigate it asymptotically as r + f -. Expressing si n h (m ) in terms of B sum of two erpmentials. we first note that cosh(mUr)-le,p(mU)rl).

as

r + f -.

10 vicw of there results, it follows at once that

u(x,r)- Uexp{m(x+Ut))-(lerp{-m(x-Ur)},

as

I+--. (11.8.19)

Solution (11.8.19) is uniformly valid for all X. and represents two distinct Solitons approaching one another at equal hut opposite speed U,as shown rn Figure 11.2. More precisely. m the limit as x + and r + --, the solution

becomes

--

u(x,r)--4tan~'[~exp{-m(x-~~)]].

(II.S.ZO)

> .U,+!V

zI ’:. 41.4

U,-!r

-2n

Figure 11.2

Interaction of two equd solitons (11.8.18) for t+-m, (b) I=O, and (e) L--t+m. (From Drazin (1983))

U > O , (a)

11.6 The Soluiron of the Sina.Gordon Equalion by Sepamrion of Variables

479

Solution (11.8 20) represents B pulse moving in the positive x-direction, as + , I ) increases fmm -20 to 0, as x passes through the value Ur. As x +

+-

and

I

+--, u(x.1)

the solution nduces to the form

- 4 t m ? [Uexp{-m(x

+ Ur)}].

(I 1.8.21)

This also represents a pulse vaveling in the negative x-direction. 8s u(x.1) increases from 0 to 2n,as x passer through the value -Ul. At I = 0 , two pulses suffer from an interaction. All of these results are shown in Figure 11.2. Similarly, as t + -, the asymptouc solution is given by

In other words, as x

+

--

u(z.1)

As x

++- and r ++-,

and

1

++-,

- -4tan?[,,exp[

-=]I.

this asymptotic solution is

the asymptotic solution for

(11.8.23) U ( X . 1 ) IS given

by ( I 1.8.24)

Since u varies from -2x to + 2 x as x changes from

--

to

the

+OD.

correspnding solution is called a 411 pulse. Finally. the only visible widenee of the interaction remaining. as I + + I , is longitudinal displacement of each soliton. To show this. we observe the exponential form of the solution

B

[ (

f U e r p [ f m ( x + U l ) ] = f exp f m x + U t f i y , 2 1 1

what 7 = 2

m log(;)

(11.8.25)

represenfs the displacement of each soliton which

is relarded by the interaction. The above soliton solutions produce u, and u,. Clearly. for U > 0, we find

+,)

=

4Ucoshh)coshWf)

[m h ’ ( m )+ cosh’(mut)]

(11.8.26)

This repre~entsthe inleraeuan of two h u m p - s h a d solitons as shown in Figure 11.3. This also shows, more clearly. the interaction of two solitons for I+*-.

480

Figure 11.3

Interaction of two hump-shaped solitons for (a) t+-ro, (b) I =0, and (e) I + + - with

amplihldr A = Z U / m . (From Draein (1983))

\

11.8 The Solutcon of the doe-Gordon Eqution by Sepwrarion of Variobler

481

Case 3. K#O,n=O,and m ' > l

In this case, the solution u(x.1) is given by

This result is somewhat similar to that of (l1.8.17), and can be interpreted physically with a given boundary condition u,(O.t)=O for all timer I. It represents B soliton moving toward the boundary and reflected back as an antisoliton, and hence. solution (11.8.27) represents an interaction of a soliton with an antisoliton. In the limit as m + I , solution (11.8.27) reduces to the form ~(~,r)--4lan-'[(I+4)sech(x+o,)].

Csac

(11.8.28)

4. K#O.n=O,and m 2 < 1

This ease gives a new solution which can be obtained from

F i y r e 11.4

Breather solution 01 (11.8.2) lor m = 0 . 8 .

482

Nonlinear Klein-Gordon and Sine-Gordon &mrrons

It may be noted here that a variety of solutionr of the sinsGordon equation have k e n studied b y Lamb (1973, 1980).

Example 11.8.1 (Energy of. soliton, an ontisoliton, and n breather). We consider the energy associated with the soliton and the antiSoliton of the sincGadon equation up-u,=anu. (11.8.33)

The Lagrangian density of this equation IS given by d(",X-":)-(l-cos.).

The Hamiltonian density has the form

(I1.8.34)

which is, due to (11.8.34).

=

i($ + u:)+

( I -cosu).

(I 1.8.36)

ur)}].

(11.8.37)

2

For the single-rollton solution u ( x , r ) = 41an-l [exp{m(x-

wkre m ~ ( l - U ’ ~ ~ , w e o b t a thcHamiltoniandensity in

L = 4m’sech*{rn(x-

L andenergyEas

Ut)],

(11.8.38)

-

E= IL&=8m, For the antisoliton solution

n(x,r) = 4tan.’ [exp{-m(x the energyE BE also equal to 8 m .

(11.8.39)

- ur)]].

(11.8.40)

This shows that the solution npresenU the

elastic int~r~ction of B soliton and M antisoliton,each of which has energy 8m. For the soliton-antisoliton solution ( I 1.8.27). we find that

%= I+rsech m)

where x1 = m ’ - l > O . r=(:)’smh2(xl).

,=,’+(l+x’)~eeh’(rf),

In this case, it

IS

(11.8.41)

q=m’dnh‘(rr).

and

easy to check that energy E=16m.

when m i l .

Finally, the breather soluuon is given by (11.8.29). The associated energy is E = 16m,when m > I . The energy is, thus, less than that of two free Solitons (or antisolitom) and leads to the physical interpretation of the breather as a boundsrore composed of a pair of soliton and antiso111on.

11.9 Backlund Transformations for the SineGordon Equation We consider the characteristic form of the rine-(iordon equation U" =s,nl(. In 1880. BaEklund discovered the transformation as t

(11.9.1)

k pair of equations

i ( u + v), =Lsin . !( u - u). (11.9.Zab) a 2

L ( u - v), = a sin i ( u + v).

where (I is a nonzero arbiwry real constant. This pair of equations is called the

B&k/&

rmnsfomrion of (1 1.9.1). The crossdifferentiation of (1 1.9.2ab) gives

i ( u - u),,

=a(.+u),cos L ( u + v)=sin '(a-v)

CM L ( a + v ) .

(11.9.3)

i ( u + v), = i ( u - v), cos i ( u - v)=sin '(a+ v ) cos L ( u - v ) . (11.9.4)

Z'?

Adding and subhscting there equations,we obtain u*==sinu and ",,=sin".

( I 1.9.5ab)

Evidently, UK BLklund transfomatiors ( I 1.9.Zab) relate two solutionsu and v lhal satisfy the same equation (11.9.1). Since bath u and v satisfy the same sine-Gordon equation, the pair (11.9.2ab) is referred to as an auto-Bdcklund rronrformarion for (11.9.1). The sutc-Blcklund uanrfmation C M be used to CO~SVUEI a sequence of solutions of the sine-Gordon equation beginning with any given solution. We next use (11.9.2ab) to solve the sineGordon equation (11.9.1). Obviously. u ( x , r ) = 0 is B vivid solution of (11.9.1) for all x and r. We use this solution to C O ~ J I ~ U C I a nontrivial solution. We set v=O, 10 that the Bkkhnd oansfomatlon (11.9.2ab) kcomcs u,

= Za sm(+ u).

u,

=

sin(+ u ) .

Intcgrsung these equations gives

2ar = jcoscc(+u)

du = *log1 tan[

t

u ) l+A(r),

~ = ~ c o s (ei cu ) du =2log)tanful+B(x),

(I 1.9.6ab)

(1 1.9.7)

(11.9.8)

11.9 Backlund Trms/omiionsfor the Sine-Gordon Eguotion

485

when A(r) and B ( x ) are arbitrary functions involved as a result of integration. It follows from (11.9.7) and (11.9.8) that

la"(+I'"p("+t).

where a is a CODS~M~.Or. equivalently.

.I):

u(x.1) = 4m-3[a cxp( ax+

( I 1.9.9)

(11.9.10)

This npnsenU B new solurion of the sine-Gordon equation (11.9.1) and it describes a soliron (or kink) solution. Returning to the original sine-Gordon equation (11.7.1) with x = i ( X + c T ) md t = L ( X - c T ) , weobtain

Funher simplification of this result gives

Or.equivalently.

This gives B new solution in terms of given Solutions u0. u , , and 4 and can be regarded BS the nonlinear superposition p m c i p l e for the sine-Gordon equatton. It is impanant to point out that the same ~olulion(11.9.17) can be denved from ( I 1.9 Zb). I1 is also possible to continue this procedure many timer, and hence, this leads to an n interacting soliton solution urnfrom an (n-1)-soliton solution v".,. I n panicular, the solution for two interacting solitons can be obmned m the form

11.10 Solution ofrk SineGordon E ~ m t i o nby t k Inverse Scamin8 Method 4 8 7

andsolutions(ll.9.11)and(ll.9.16) havehen used. We next set Ul = -U, = U to reduce the solution (I 1.9.18) to the form

U sinh(mX) cooh(rnUt)

(11.9.19) '

This is exactly the same result (11.8.18) obtained by Perring end Skyrme (1962). It has already been investigated as the asymptotic behavior of (ll.9.19). as I -, in Section 11.8. The asymptotic ~olutiondescriber the interaction of twrrsoliton solutions

+*

11.10 The Solution of the Sine-Gordon Equation by the Inverse Scattering Method We use the characteristic form of the sine-Gordon equation (11.9.1). Following Lamb (1980). we describe the method of solution of (11.9.1) by the inverse scattering approach. The essence of this appmach IS to relate the equation (11.9.l)toapairoflinearscatteringequations for v, and u,

These equations, together with the conditions that v, and vI are bounded as x + *-, may be considered as a problem to determine the eigenvalue for a

<

given function q =

-1,

called thepotentiolcomspondIng toa ~olutionU ( X , I )

of (11.9.1). We note that. if v, and v, are bounded and q -+ 0 sufficiently rapidly as x +f-. then. v, - 0 , erp(-t<x) and v1 - n, exp(i<x). as x + k-. So, there exists an unbounded statc if and only if is real. If 5 is taken as a fixed. real number, then. 5, =O. Differentiating (1l.IO.lab) with respect to t integrating with respect to x , and then, combining with equation (11.9.1) gives the evolution equations for the vector cigenfunctionr (v,, v,) in the form

<

488

Since the solution u is related to q through q = - i n x so that *(r,9=-2

-J*(S.t)dS.

(11.10.3)

Itfollows that "-0, as x~-~.Henee,equations(ll.10.2ab)grvc

v,,-Lv,,

vI,--~q.

45

m

x+--.

(11.10.4ab)

45

Thc form of the equations (11.10.2ab). as x++-. depends on the value of u ( x . t ) BS x +-. The only cases that can be solved without my problem are thoreinwhich u(x,r)+Znn.as x + + - .

-,

[

I;: .

It follows from (11.10.4a) hat v, is either zero or lends 10 erp -x+--oD.

[ :*I

eip --

as

Similarly. (11.10.4b) shows that v1 is either zero or tends to , BS x

+ --.

We then consider the solution that is proponionalto

the fundamental solutionsgiven by O,(x.k)=crp(-ih)+ IA,(x,S)erp(-iXS)dS,

-

(11.10.5)

ad

-

A ( x . k ) = I4(x.S)e+*S)dC.

where A, and

(11.10.6)

4 represent the scattering (or wake) and the reflected panicles.

Then.

The form of the solution can bc represented as a linear combination of two linearly independent solutions. that is.

Consequently,in the limit as x +

+OD.

whns the timedependenceof e,, and c,, IS found from (11.10.2ab) in the limit rn x~+c-.Sinceweeonsidertheca~ewhenu-+Znzas x-++c-, V,,-LUl, as I++-. (11.10.1hb) 45 45 Following the method used for the modified KdV equation. it turns out that

II;:

c,,(l.t)=c,,(Co)eXp --

Using

the

notation

I

c,,(<.r)=c,,(6.0).

R,(~,f)=~~,(~,r)/e,,(5.r)

mtt(kt ,t) = -ic,,(kt,t)/c,2(ke ,I), we find the following result

and

Nodineor Kldn-Gordon and Sine-Gordon Equations

490

where I is a unit mavir of order N, N is the number of zeros of c,*(<,O) in the upper half plane, and M is an N x N mauix with time-dependent elements ML,(k,,r)= m,,(k,,O) exp

[-i:,] .

(11.10.14)

Here, we consider !he simplest case which deals with a single pole on the so that imaginary axis at k, = i h L

end

Consequently. the single soliton solution is derived from (1 1.10.13) as

where , = I o g [ i .

].

m (k 0 )

,,

This solution is identical to (11.8.ll) which

was found earlier by a different but simpler approach.

Finally, the solution of the sine-Gordon equation representing the interaction of two solitons can also be obfaned from two poles on the imaginary axis. 'Ibis ease is similar to that of the multisoliton solution of the KdV equation. Here, wesct k, =:ia,.

r=I,Z.Th~nIheexpression l/-iMLltakestheform

Iml/-iMLI=- [ ~ . , p ( ~ , i ) + ~ e x p ( o , x ) ]

(11 10.18)

Consequently. the two-soliton solution takes the form u(x,r)=-4tK1

c m h i ( u --u

)

1

[(=IA anh&(u,+u,)

, a , > a ! ,(11.10.19)

I 1 . 1 1 The Similoriry Merhod for the Smr-Gordon Eqmrion

491

Where

u,=(o,x+$+y,),

erp(y)=(~)[*].

(11.10.20)

The above solutions are those special solutions whrch evolve into pure multi~oliton~ o l ~ t i o n s .

11.11 The Similarity Method for the SineGordon Equation To obtain a similarity solution for the characteristic form of the sine-Gordon equation (11.9.1). we assume that this equation IS invariant under a group of transformations :=a"*, for suitable valuer of constants

iij-sin; =

i = a e t and i = a ' u

a.0.and

(11.11.1)

y . It follows that

u,, -sin (a'") = u,, -sin",

( I 1.1 1.2)

provrdedthat y = O and a=+. This shows that equation (11.9.1) admits similarity transformations given by u(z.1) =

"(a),

q =X

~ P = X'I .~

( I 1.11.3)

Using ~ = t d . ~ = x d , t h e ~ u a t i o n ( 1 1 . 9 . l ) r e d u ctotheordinary es ax dg JI dg differentialequation

q v"(q)+ ~ ' ( 7=sin ) v,

(11.11.4)

where the prime denotes differentiation with respect to q . We next invoduce a new dependent vanable w=erp(iu) in (11.11.4) to derive an equation for w in the form (11.1 1.5)

Equation (11.11.5) is a special c u e of the equauon that defines the Painlev6 equation of the third kind (Davis, 1962, p. 185). However. it is normally not easy to find a solution of (11.11.5). Hence. it is necessary to use numerical integration to find a solution of (11.11.5). An example of such a solution IS given by Lamb (1980) with a phase-plane analysis and graphical representauom

492

Nonlinear Klcin-Gordon ond Sm-Gordon Equotionr

"t *+ : -20

I, 0

FIgure 11.5

of functions

40

20

60

80

100

Similarity solution of (11.11.4). (Fmm Lamb (1980))

u and u * = - % = s by (11.11.3). t

r

I h e numeneal solutions for

u and v' shown in Figure 11.5, satisfy the conditions v(O)=O.l and "'(0)=sin " ( 0 ) which gives B solution that is finite at q = 0.

11.12 Nonlinear Optics and the Sine-Gordon Equation McCall and Hahn (1967. 1969) discovered a remarkable DCW wave mode which propagates in a twolevel atomic SyStem without attenuation. When a sufficiently intense elecuomagnctic wave is incident on the system BE a Short pulse, then, the number of atoms in the excited state in the leading pan of the pulse exceeds that in the ground state due to the absorption of the clcctmmagnetic wave. However, h e elefmmagnetic w w e in the rear pan of the pulse is again emitted due to induced emission so that Ule system uansmiu a pulse wilhout absorption. This is called se@iduced r m p r e n c y (SIT). In SIT.a medium Vammits a light pulse at the resonant frequency without resonance absorption. When an electromagneticwave is incident on a two-level atomic system. an ensemble of atoms has a ground state of energy E, and an

11.12 Nonlmaor Optics and the Sins-Gordon .!?quation

493

excited stste of energy E, . If the frequency of the wave is equal to the transition

-

frequency wo = (E, E , ) / n , the transition from the gmund state to the excited state occurs due to the re~onaneeabsorption of the eleemmmgnctic wave. In the present system composed of large number of atoms. the atoms excited to the higher level lose energy by collision with atoms and the slow imvmiblc decay of the number of excited atoms to lower levels. However, the remarkable fact is that the pulse propagate as a soliton without attenuation of energy. To dc-be self-induced transparency. we may regard the cleetmmagneticwave as a modulated. errcularly polarized plane wave traveling along the x-axis so that Be electric field E(x.f) is represented by

B(x,f) =%(x,r)(j c o s @ + i sin 8 ) .

(11.12.1)

whae 8 = ( k x - ~ r ) + ~ ( x k ) . = z isthercfraetivsindcidcterminedby the

linear dispersion relation of the medium, the amplitude %(x.f)and the phase ~ ( x vary ) slowly compared with that of the carrier wave. that IS,

and similar results hold for ~ ( z ) .

The elccVic field E(x.r) can be obtained from the Marwell equations and satisfiesthe wave equation

where P(x,f) is the polarization of the medium caused by the electromagnetic wave. If the equation for P(x,f) is given, this quailon combined with (I 1.12.1)

forms a closed system for two unknowns 8 ' and

I.

We assume that the medium is made up of N two-level atoms and the tranrition frequencies oo of individual atoms are not all the same but are dismbuted about Be frequency 0) of the incident wave field in accordance with the spectral density function f (A 0 ) which is normshred by the conditions A 0) =(om - 0))

and

If (A

0))

d(Am)=I .

(I 1.12.4)

Consequently. the polarization of the medium reduces to P(x.f)=

-

N I f ( A 0 ) ) g(Aw.x.f) d ( A 0 ) ) .

(11.12.5)

494

Nonlinear Klern-Gordon and Lnr-Gordon Equtionr

where p(Aw,x,t) is the polarization of a two-level atom induced by the elecbomagneticwave and p(A o,x.t) satisfies the slowvariation condiuons as those of ( 1 1.12.2). Thus. when the polarizationp consiss of two components, one IS in phase with the carrier wave ofthe elecme field E(x,t) and the other is out of phase, so that it can be expressed as p ( A o . x , r ) = u ( A o . x . l ) ( ~ cos@+k^ a n @ ) + V ( A W , X , I ) (s~i n e - i

ease),

(11.12.6)

where u and v correspond to the modulations of the elecvie field and kcome slowly varying functions o f r and r. Therefore, the problem reduces to determining the equations for u and v , which can be obwned from the Schr6dingcr equation for a two-level atom in the incident electric field. Invoking the electric dipole approximation and neglecting relaxation effects of the excited states, McCall and Hahn (1969) denvcd the following equations for u and u

au

-=

VAW.

(11.12.7)

I

-=_

'"

UAW-[$)~W.

(11.12.8)

-= aw

UZW,

(1 1.12.9)

where u and u an usually called the dispersive and absorptive components of the elecuic dipole respectively, K = (Zplk),p IS the magnitude of the c l c c ~ dipole c moment. and W is the energy of the two-level atom, that is. W = the ground statc and W =

-in om~n

1 k w in the excited state. Moreover. we assume that

all two-level atoms are in the ground state until the dipole uansitions OCEUT. Thus. equations (11.12.3) and (11.12.7)-(11.129) constitute aelosed systcm for U. u. W ,and E . Since u , v , and E are slowly varymg functions corresponding to the mdulation ofp, we denvs equations for the modulational quantities 8 and from (11.12.3). Substituting (ll.12.1), (11.12.5). and (11.12.6) in equation (11.12.3) and using the slow-variation conditions (11.12.2). wederive the followingequationsfor 8 and v : -+--=-ar: "ar:

dr

at

2 n w ~j u-f ( d m ) d ( A o ) , nc

-

(11.12.10)

496

11.13 Exercises

with the Cauchy dam

u(r.O)= f ( x )

and

L,(x,O)=g(x)

forall r E R”

U(k,I)=F(k)cos(tr+d‘)+&

r i n ( t d m )

..1.) and k = (k,.k *,__..kn).

when U(k,t) = S { u ( x ,I ) ] . x = (x,.x2.. .

Show that the energy integral associated with the Klcin-Gordon equauon in Exercise I is given by

12

I

E(r)=J(lu, +c* v u r + d +

I’)dr.

Show that E ( t ) is independent of I. Seek B similarity solution of the Klcin-Omdon equation U “ - u , =* ’

in the form u ( x , t ) = I”f ( r t ” )for suitable valuer

of m and n. Show that

f ( z ) satisfies Ihe equation (z’ - I)f”+

4 r f ’ + Z f = f’, where 1 = X I - ’ .

If V(u) involved in the Klein-Gordon equation (I 1.5.1) is given (Whitham, 1974) by V(u)=:u‘

+ bud+ ....

11.13 Exrrei,r,

497

denve the result

u(e)=ocoss+lo1bcos3e+...,

Bs

o+o.

where the nonlineardispersionrelation is given by m*=l+k2+30'b+o(n'),

as 0 - 0 ,

and the nonlinearamplitude is as a-0

A=~n2+%rb+o(n4)

2

8

verify that

is a solution of the sineGordon equation ( I 1.8.2) where a and U arc CO"Ota"ts. Show that !+.I)

= 4tan-1

is an exact antisoliton solution of the sine-Gordon equation, where O
5=

+(

(;

x-

J),

q=

u,

- c'q,

= sinu

U~

=sinu.

x

+J)

transformule equation into the form

(a) Show lhat the sine-Gordon equation urn-un =sinu isinvariantundcrthewnsformations i = m ( x - U f ) and

w h n c m = ( l - U ' ~ ~and - 1 < U < 1

i =m(x-Ut)

Nonlinear Kl&-Gordon m d Sznr.Gordon Eguaiionr

498

(h) Henee or otherwise, denve that

is B solution of the sine-Gordon equation, where constants, - I < @ < l and - 1 < U < I . 9.

f l # O,r,,r,,

and U are real

(a) Use the Bbklund vansfomation (11.9.2ah) to show that four solutions u, ( r = 1.2.3.4) satisfy the relation

where 0, and 9 are nonzero constants. (h) Apply the relation in Exercise 9(a) to obtain a soliton solution of the sine-Gordon equation by setting u, =0, u1 and u, different soliton solutions. 10. Examine the asymptotic nature of the two-soliton solution (ll.lO.l9), as I+*-.

I I. Derive the solitary wave solution of the sinh-Gordon equation u" =sinhu.

12. (a) Use a similarity solution of the sine-Gordon equation (11.9.1) of the form "(.GI)= f(zf") for suitable" to show that w ( z ) = e r p { i f ( r ) ] yields a Painlev6 equation for w ( r ) .

(b) Use the same method in Exercise 12(a) to discuss the smh-Gordon equation in Exercise 11. 13.

vedry the following consenration laws (Lamb, 1971) for equation (11.9.1)

(L"?) -(l-cosu), =o. 2

=o.

'

*(;: -+";":

+)':.'

+((in;

-~":)cosu],

=o,

499

w h a t V'(u) = sinu, derive the sine-Gordonequation u_-u,=sinu. 15. Show that the bilinear form of the ane-Oordon equation In Exercise 14 IS the pair of equations (D: - D: -I)(@. where u=4ta"-l(;)

w )= 0

-

and (D: - D f ) ( $ . e V . V )= 0,

Chapter 12

Asymptotic Methods and Nonlinear Evolution Equations 12.1 Introduction Many physical systems involving nonlinear wave propagation include the effects of dispersion. dissipation. andlor the inhomogeneous property of the medium. The governing equations are usually derived from conscwation laws. In simple cases, there equations are hyperbolic. However, m general, the physical pracerses involved are so complex that the governing equations are very complicated and hence. are not integrable by analytic methods. So. special attention is given to seeking mathematical methods which lead to a ICES complicated problem retaining all of the impmant physical features. In recent years, several asymptotic methods have k e n developed for derivation of the socalled evolution equotionr that describe how some dynamieal vanabler evolve in time and space. So. we begin this chapter with one simple method of consrmctionof the linear evolution equation from a given frequency-wavenumber dispersion relation ofthe f n m a)= f ( k ) .

(12.1.1)

-

This relation is multiplied by - i U ( k ) c x p [z (h -m) ] and integrated with respect to the wavenumber k fmm to to obtain h e equation

--

-J=UL ( u ) = - i

I f ( k ) U ( k ) c x p [r(h- m) ] dk,

(12.1.2)

t

where L(u) represents an operator and U ( k ) is an arbitrary function which is related to the function u ( x , t ) by the Fourier wnsform

"(sf)= l U ( k ) e x p [ i ( h - m ) ] 1.

-

(12.1.3)

Equation (12.1.2) represents the fundamental result that determines the srmctun of the evolution equation. In panicular. if f ( k ) is a polynomial in k. then. L(u) is a diffsrenual operator. More sxplmtly. if f(k)=.k-iyk'-a,k'

a"k",

(12.1.4)

502

Asympeiic Uethodr and Nonlinear Evolution Equations

where c.y,a,;-,an ace arbitrary constants, then, (l2.l.2)gives the explicit evolution equauon

If a,= a 4=...=an =0,(12.1.5) reduces tothelinear Burgers equation. When an10 for n24,equation (12.1.5)reprrsentsthelinearKdV-Burgerrequation. On th eo th crh a n d .~f y = - i v fo rre a l v a n d an=O for "23, then,(I2.1.5) reduces to the linear Schmdinger equation. The &bove analysis reveals that the polynomial dispersion relation cornsponds to systems governed hy evolution equations of the differential type. On the other hand. if the dispersion relation has the transcendental form. the corresponding evolution quation is of Integmdifferential type. However. there is no such simple method of consmetion of evolution equations from a given nonlinear dispersion relation. For nonlinear problems. asymptotic methods are employed to construct evolution equations. The linear p a t of the associated e v o l ~ t i ~equation n can he recovered from (12.1.2). So, the rest of this chapter is devoted to deriving of evolution equations for various nonlinear systems by asymptOtiC methods. The idea of a far-field asymptotic behavior of a physical system originated from the p ro p ti c s inherent in a given evolut~onequation which do not depend. in any sensitive manner, on the details of the initial conditions. Thus. such propenies are generally ohserved after a sufficiently long period of time for a large class of initial conditions. In this sense. it is not m l y useful to study the initial-value problems in detail. Rather. it is more useful to lnvestlgate the asymptotic behavior of a physical system in the limit, as time goes to Infinity. In many Situations, these far-kield EolUtiOnS merely represent the Simple waves described in Chapter 6. Included here are reductive perturbation methods for describing weakly dispersive systems. strongly dispersive systems, and quasi-linear dissipative systems. The pertwhation method due to Osvovsky and Pelinowky (1971) was developed for a general system of first-order nonlinear partial differential equations that can be applied to both conservative and nonconservative systems. Ihc method of multiple %ale is intmduced in Sections 12.7 and 12.8 and used to derive both the nonlinear Schriidinger and the KdV equations from nonlmear dispersive systems

12.2 The Reductive Perturbation Method and Quasi-Linear Hyperbolic Systems Historically, this method was first formulated by Taniuti and his collaborators (Taniuti and Wei (1968), Taniuti and Warhimi (1968). Kakutani et al. (1968).Taniuti and Yajima (1969). Asano and O m (1971). Taniuti (1974))

12.2 Thr Reduciive Perrurborion Method and Quori-LineorHyperbobc Sysrems 5 0 3

in a more general form applicable to bath weakly dispersive and weakly dissipative systems. We first consider a first-order, quasi-linear totally hyperbolic system of

equations

U,+ A (U)Us= 0 .

(lZ.2.l)

w hen U i s acolumn vectorwilhn components u,,u,, ...,urnandA is a n x n matrix. Such a system of equations has already k e n investigated in connection with simple waves. We expand the vector U about a constant state solution U" of (l2.2.l) in terms of a small parameter E which represents the smallness of the amplitude of U in the form

u = u"

+&d'l +&l

(I"

+....

(12.2.2)

Since the amplitude is small, the solution of (12.2.1) can be approximated by the linearized form of the system in which A ( U ) is replaced by A(U"'). As thc system IStotally hyperbolic. then are n real distinct Pigenvalues A"' ofA(U),

of A(U") where r = 1,2. ...,n . In the linear care. the characteristics are straight lines. On the other hand, for a nonlinear system. as time evolves, characteristics will no longer be parallel lines, and hence, after a sufficiently long time. characteristics inteiSeCt and lead and hence, there are" conrranteigenvalues A:'

to the brealtdowun of

differentiability. To investigate the process more explicitly.

we expand thejth family of characteristics of (12.2.1) m powen of E

and, then, ntain only terms of order

where x ' = x - r A : ' ,

At) =A''(U1O'),

&

to obtain

so that the coordinate system moves with speed At', u!' is the n h

element of the vector U"', and V" is the

gradient operator with respect to ujl',

"1'' ,.... a!'.

It follows from result (12.2.4) that the characteristics suffer from deviation from parallel straight lines by an amount of the order E . Using the time scale & I , the phase

velocity

x. 44

moving with the speed At' in the reference

frame. reduces to the order of unity. Hence. for small t << c-', the phase

504

Arymprotrc Mcrhodr and Nonlinear

Evolution Eqmrionr

v e l d l y can be well approximated by the phase velocity for the linear case. However. for large time scales, t - & - I , the charactemties deviate significantly from those found m the linear case, even if the amplitude IS very small ( E << 1). Consequently, the far-field solutions can be described by using the smtched time scale t 6.1.

-

It also follows from (12.2.4) that

V uAt’ is constant, and hence, U

IS B

function of x’ and & t .Consequently. the time denvative of 0“’ is of the order & and becomes small. This means that x’ is constant and U“’ varies slowly with time along the linear pmllel equiphass straght liner. All of these lend to

the use of the following uansformations for the far-field solutions of (l2.2.l) XI

a a ax ax’

= x -At’t.

~a

I’= E t .

J

-=&--A!’-.

Clearly, -=-and

JI ~ t * ~i together with Ule expansion of Ule matrix A as

A = A,

where

4= A ( U “ ) .

(12.2.5ab)

Hence, weuse(12.2.1)-(12.2.2)

+ E U“’ Vu4+ ...,

(12.2.6)

Thus. we obtain the following system of perturbation

equations:

O(&):

[A;’I-&]uj?=o,

O(&’):

[-a!’

I + % ] rJ!?+.~!’+(U“’V”AJ

(12.2.7)

u!?=o.

(12.2.8)

Equation (12.2.7) shows that U!! must be proportional to the right

eigenveetor R:’ of 4 corresponding to the eigenvaluci Lt’.Integrating (12.2.7) with respect to x’ gives the solution

U“’”’’ where

@’ :

= @:’(x’.t’)

,q+v;’(t’),

(12.2.9)

is a function of x’ and t’ and V;’(t’) are vector

functions of t’ only that arise from integration and so, can be determined by the boundary conditions. However. functions #:’(x’,t’) cannot be determined fmm (122.9). and we have to consider (12.2.8) to find them. Equation (12.2.8) is an algebraic equation for U:’, with a multiplying factor ( - A t ’ l + 4 ) . If At’ is

an ergenvalue of

4.so the eigenvalue equation IS given by (12.2.10)

12.2 The Reductive Penurbation Method and Qmi.Limear Hyperbolic Symms 5 0 5

If

$’ is the lefi eigenvector of 4cornspanding to the eigenvalue it’, then, $’ (-At’, + A o ) =O. (12.2.11)

Thus. a necessary and sufficient condition for the existence of the solution of (122.8)is $1

[US’ +(u“’V”4)I!;’]

=o.

(12.2.12)

Without loss of generality, we set Vy(t’)=O in (I2 2.9) so that the resulting equation (12.2.9) can be substituted m (12.2.12) to obtain the’ mnlineorpmial differentialequation for @y’(x’.t’) in the form

(12.2.13)

d @ ‘ l ’ = O . I

when

ax*

’

a, IS given by a,= $1

[$’V”41$I/$’

$1.

(12.2.14)

which. after some simplification, reduces to,

a,=

[.. A:’] $’.

(12.2.15)

Thus, the present asymptotic analysis allows us to derive (12.2.13). which npments the basic sealnrcvolution equation describing the far-field behavior of the jth (column vector) mode of wave propagation of the hyperbolic system (12.2.1). Further, to the order of c. the so l ~t m nof (12.2.1) can bc wittcn as

U = U‘O’ + f @I: Rt’, whne

9’:

ISa ml~uonof

(12.2.16)

(12.2.13)

Evidently, the characteristic quation of (12.2.13) is

dr’

@I”

’‘

(12.2.17)

which ensuns the existence of the far-field behavior, provided that a,# 0. The solution (12.2.16) represents simple waves already discussed in Chapter 6 that describing the far-field behavior of (l2.2.l) until the discontinuity is developed as of shock waves.

The present method of analysis is called the reductive perturbation method because it reducer the determination of the far-field behavior of a system to the Solution of a scalar. nonlinear evolution equation, such as (12.2.13).

506

Asymplolic Meihods and Nonlinear Evolution Epvlrionr

Finally, we close this section by adding a remark on the nonuniqueness of the far field for hyperbolic equations. Equation (12.2.13) IS invariant under the transformation

:==',

j=

&-It'

$"I,E$lll,

(12.2.18)

However, this vansformation is nor unique because equation (122.13) is also invmant under the following transformation

i = E - . Xi,=&-l.*'l,,,

$I"

= &,p

(12.2.19)

where a is an arbitrary constant. This transformation can be found from (12.2.5ab), and the comsponding vansformation

5 = E.

(x- 1,t)'

?=&(**')I

(12.2.20)

leads to equation (12.2.13). This Vaniformation was first intmduced by Gerdner and Monkawa (1960) for a solitary wave in a plasma propagating normally B C ~ S S a magnetic field and is known as the Gordner-Morikmuo tronrfomtion. Since a is arbifrsry, solutions vary more slowly over space and time for large values of a than they do for smaller values. Thus, the far-field solutions are no1 determined uniquely for hyperbolic equations. The breaking time ta after which smooth solutions cease to exist IS determined by the initial conditions. so that, m the case of hyperbolic systems, the time scale for lhe far field solution must be less than rd. Thus, h e concept of a far-field solution for the hyperbolic ease is not meaningful unless rB is sufficiently large. It is possible that a discontinuity (or shock) is developed before an initial disturbance breaks up into simple waves. The above asymptotic method can be applied to physical systems with dissipative terms which prevent breaking. So, smooth solutions exist for all time for dissipating systems, so that la + O D , unlike the ease of hyperbolic equations. This leads us to consider an arbitrarily long time evolution of B solution. For example, in the case of a perfect gas, the viscosity begins to act effectively as the sound wave steepens, and these two effects. the nonlinearity leading to bnaking and the dissipation that C ~ U S ~smoothing, E then. balance one another to pmducc smooth solutions. As a result, shock waves BIT developed and propagated as already described by the Burgers quation in Chapter 8.

12.3 Quasi-Linear Dissipative Systems We now apply the reductive perturbation method to solve a general dissipative system, which becomes totally hyperbolic when the dissipative t e r n arc neglected. Using the same notations as 8" (lZ.2.1). we consider the system of dissipative equations described by the system of equations

12.3 Qu~si.LmearDtssipoave Sysrmtr

507

U,+ A ( U ) Us+ K, ( K2U,)== 0.

(12.3.1)

where K, and K2 are n x n mstrices which are functions of U end the cigcnvaluesof A (U)are all real and distinct because of the tint-order hyperbolic system embedded in the system of equations (12.3.1).

We now adopt a frame of reference that moves with a constant velocity I, which is equal to one of the eigenvalues of the matrix

4= A(U'") associated

with the constant state U'O'. We assume that dispersion relation for small wavenumben k of the lincarilcd system about U"' is given by a,= I, k + i p k'+

...,

(12.3.2)

where p < O . It can be anticipated that equation (12.3.1) can be reduced to Burgers equation by the Gardner-Morikawa transformation (12.2.20) with a = 1. I n view of this transformation. quation (12.3.1) admits approximate solutions whish vary slowly over space and more slowly with time due to weak nonlineariryandweaLdirsipatingeffects. With { = & ( x - l , t ) a n d r=&'t,we apply the same perturbation solution (12.2.2) and use the following results

-= a

ax

&-

~ a ay' axa

'

ay'

L . & ld + & d (12.3.3)

at

ay

ar'

in the original equation (12.3.1). Then, equating the lowest order terms O(&') gives

( I oI -

4)up = 0,

(12.3.4)

so that integration of this result gives

u"'=$"'(y.r) 4,

(12.3.5)

where the scalar function $"'(C.r) is to be determined, 4 is the right corresponding to the eigenvalue Lo. and the boundary cigcnvcctor of 4, condition (I")+ 0, as 5 + -, has k e n utilized. It is noted that (12.3.5) IS not affected by the dissipative term to this order of approximation. Using the same argument as before. we obtain the following cquatlon to the nda O ( E ' )

(-Io I +4) U r + Uy)+ U"'V* 4U p + KOU : = 0. KO = ( K , K,),,,

.

(12.3.6)

508

Asymprrrc Meihodl md Nonlmeoar Evobaon Equorions

Multipliying from the left by the left eigenvcctor 4. corresponding to the eigenvalue 1,. and using the same argument as in Section 12.2 yields the necessary and sufficient condition for the existence of solutions for

U p as

Substituting U"' from (12.3.5) in (12.3 7) gives the following quasi-linear. second-order equation for r)I r):

$"'(c,

@(c,

0, + a @ @< + B

@,, 0, =

(12.3.8)

whac

Thus, the scalar function @(<.r ) satisfies the Burgerr equation (12.3.8) that provided that f l < 0. This equation governs the far-field behavior of the system (l2.3.l). corresponding to rhc characteristic field 1, of the hyperbolic pan of the system. Finally, the solution of system (l2.3.l) to the order O ( E )ISgiven by I J = u " + & @ ~ " ( ~ . T ) 4.

(12.3.10)

Thus. $'"(c,r)=$(<, r) satisfies a nonlinear evolution equation (12.3.8). which determines the far-field behavior corresponding to the characteristic field Lo.

12.4 Weakly Nonlinear Dispersive Systems and the Korteweg-de Vries Equation For nonlinear dispersive waves, the disprrsron relation represents an algebraic equation for the frequency a. wavenumber k , and amplitnde a', that is, a = a ( k , n * ) . If the amplitude is small, the dispersion relation is u~ually determined by rhe linearized equation. SO that both the phase and group velocities are functions of the wavenumber k only. The dispersion ISs a d lo be weak if the p u p velocity of the wave is almost constant in a certain range of hquenetes,

I$I 1" 1

>> The KdV equation is B elasie example of weak ak' ' drrpersion. However, if the above inequality is not valid. the dispersion is said to be strong. The NLS equation is a famous example of svong dispersion.

and such that

509

12.4 Weakly Nonlinear Dispersive Syramr

We develop the reductive penurbation method nonlinear dispersive system m

to solve a

B homogeneous medium

general weakly

given by

where U is a column vector of n components w,.u2 ....,un(n 2 2). A, H f . and K f are n x n rnauicer. which are functions of U. For suitable H t and Kt, the nonlinear system can be dispersive kcause the highest spatial derivative is of order three and it is weakly dispersive because it does not contain a ( n x I) column vector of the form B(U)on the left hand side of (12.4.1). Further, we can take into BCEOY~Ithe nonhomogenity of the medium by adding an exva term BV. on &cleft-handadeof equation (12.4.1).

We first consider a solution of (12.4.1) representing Small deviations around a steady equilibrium state Uo in the form

u = uo+u,cxp[i("-w()].

(12.4.2)

Subsututing (12.4.2) in (12.4.1) and retaining only the fint power of U, gives kP=L f re=t( K ~ o - E kH ~ o ) ] U , = O .

[(-;)I+&+(ik)"

(12.4.3)

when the zem subscript denotes rhe value when U = Uo.Thus equation (12.4.3) leads to the dispersion relation

For the case of long waves, that IS, for small k , equation (12.4.3) can be solved by the method of successive approrimallonr. The zeroth order approximation IE given by

(-;

I + A.) U,,,,= 0.

Obviously, the zeroth order dispersion relation is of order" in

(12.4.5)

.I;[

. ,

We a~sume

that 1, is a nondegcnerate cigenvalue of & with L, and 4 BE the corresponding I& and right eigcnvectors, respectively. To this approximation, we find

E=ao. u,, =%

orsomescalarmultipleof%.

(12.4.6)

510

Arympcotie Metho& ond Nonlinear Evolution Equations

Then, we obtain the next approximation by substituting I, for

?!

and

4 for

U,in the neglected terms. Consequently,

so that multiplication of this equation by

& yields

Proceeding in this manner gives a higher order approximation to

1 = I , + C , k-'

+ C , k2'v''+...,

(3 as

(12.4.9)

whne the coefticient C, is given by

and similarly for C,. In this system, it is necessary to assume that C, # 0. Neglecting the third term in the original equation (12.4.1). the characteristic curyes of the reduced quation can bc expressed in the fmm

dx= I , + & I ,+O(&').

(12.4.11)

I

where E is a small nonzero parameter which determines the effects of nonlincori@. A simple comparison of (12.4.9) with (12.4.11) leads to the fact that there can be eouplrng between nonlinearity and dispersion (ordissipation) .% theI order of &. provided that K-E'.

o=(p-l)-'.

(12.4.12)

This mcani that &" times wavelength is of order unity. so that we can define Ule moving coordinate 5 by

~=&-(x-I,r),

(12.4.13)

in which the w w e can be descnbed when nonlinearity is weak and wavelength IS long. Differentiating (12.4.13) with respect to x gives

Or, equivalently,

dS=L+O(,,, 11 = &.*IX. (12.4.15ab) d? 2, This allowvs us to introduce a siretched variable I l by (12.4.15b) so that %O(l). dv Similarly, differentiating (12.4.13) with respect 10 I gives

3 = E.*'[.I, *=&a+'

(12.4.16)

+O(E)].

This enables us to define another sncrchcd vnrioble

r by

*

(12.4.17)

suchthat g = O ( l ) .

dr

We thus oblain two sels of m l c h e d variables: (i) 5.7 and (ii) 5 . 7 . Using the first set defined by (12.4.13) and (12.4.15b) gives the characteristics m terms of these Stretched variables in the form

I=.a,+En,d5.

(12.4.18)

dq

I

The fmsl set of stretched variables can bc used

to

deal with an initial-value

problem. Similarly. the chmteristtcs assmiated with the second set

I = 1, +dS. dr

<,7

an given by

(12.4.19)

dr

The second set of stretched variables i n d u c e s the so called Gordner-Morikmuo rmnsfomtotion and can usually be applied to deal with boundary-value problems.

It IS clear from the above analysis that the scale of coordinate stretching 15 uniquely determined by the given governing equation (12.4.1). The reduction penurbation analysis has been develop3 by .Tamuli and Wei (1968) for the care of a homogeneous medium and, subsequently, modified by A ~ a n oand On0

511

Asymptotic Metbdr o d Nonlinear Evvoiurion Equorions

(1971) to B C E O U ~for ~ moderate nonhomogenity. When the medium is not homogeneous, as is the case with most physical systems. the interaction of the wave with the nonhomogenity also becomes imponant. To state the Stretched variable transformations associated with a nonhomogeneous medium,we briefly outline the basic ideas of Asano and Ono (1971) as follows: The steady state of the system of (12.4.1) with the term SV, on the Id-hand ride, denoted by the lem subrcnpt. ISdescribed by

This arises from a more general system of equations for weakly nonlinear waves in B nonhomogenaus medium. With the assumption that Uoand V a n slowly varying functions of x, the stretchedvariable 7 can be defined by (12.4.15b) with a = ( p - l ) - ' . We can also determine the order of & from the slow variations of U, and V with X. To account for the nonhomogenity of the medium. we introduce the stretched vanable 5 and r, redefined by the following relations:

<=&'[jI:

h - t ]

and ~ , = E ' * ' x .

(12.4.21ab)

where I, is a nondegenerate eigenvahte of the matrix A,, equal to thc velocity of the linear wave. In terms of stretched coordinates. equation (12.4.20) befomes

4u., + Bo V" = 0 ,

(12.4.22)

where the terms of the order &' are ignored. The reader is referred to Asano and On0 (1971) or Bhatnagar (1979) for a complete analysis. However, we simply state the final evolution equation for O(5.7) tntheform

@,+(a, $+a,)$,+p, o<, + Y , O + Y~ = o ,

(12.4.23)

where partial derivatives in the third term of (12.423) ~ c u r timer, p

u, = R. O(6. ?)+ W(7). a,= ( I : b %['Lo {(V. .A),

.%)%. a, = ( I : Lo 4)' b {(Vd), .W}%,

I2.4 Weakly Nonlinear Dzspersiw Syramr

513

Following Tanivti and Wei (1968). wc outline the fundamental ideas of the reductive perturbation method for B systematic nduction of a fairly general nonlinear system of equations (12.4.1) to B single, tractable, nonlinear evolution equation describing the far field behavior. We inmducc the Gardncr-MorikawaIranrformation

<=

( x - I I), r= &a*' I, n = ( p - I).',

E.

(12.4.24)

and assume expansions about a eonslant solution U"' in l m s of small parameter & of the form (I= (110)

+Eu'I)+ & a

A=x&'A,,

+....

(I")

(12.4.25)

HE=X&'Ht,, K,P=Xe'KE,. ,io

,=o

(12.4.26)

I-U

Substituting (12.4.25)-(12.4.26) in (12.4.1) and using (12.4.24) enables us to rewrite (12.4.1) in terms of derivatives with re~pectto 5 and r . Equating the cocffiicientsof like powers m & to zero yields

where, again, V v denotes the gradient operator with respect to (1. (Uo .V,)

represents the operator

2

u,

,=,

2, and A, is winen as Ju,

U @ (Vv -

I U") , (V,,

Inmdueing the right eigenvector Ro of

A)o.

4corresponding to the eigcnvalues

A . so that (4- I / ) & = O and integrating (12.4.27) with the boundary conditions U"' + 0 , as

-

5 + , gives (Ill'

= P ( 5 .r) & + v,( r ) ,

(12.429)

Asympronc Methods nnd Nonlinear Evolution Equrionr

514

whcn @"' is one of the components of (I"',and V , ( r ) is an mbiuary vectorvalued function of 7 to be determined from the given boundary condition for

U"'. Since U"'+O

To solve for

as

<+-,V,(r)=O.andthesolution(12.4.29)becomes U"' = @"I (5. T) 4 . (12.4.30)

2 U"'. a compatibility condition is required. To find this a!

condition, we muluply (12.4.28)on the left by the left eigenvcctor

b and use

thefactthat b ( - I f + % ) = O toobtain

Substituting (12.4.30) in this compatibility condition and dropping the supencript (I), leads to the nonlinear evolution equation for 9"'(<.z) of the form

(12.4.32)

When p = 3, equation (12.4.32)reduces to the KdV equation, whereas for p = 2 , (12.4.32)becomes the Burgers equation. Thus, the far field of the general nonlinear system (12.4.1)associated with the I characteristic field sausfies the nonlinear evolution equation (12.4.32)which admits the ~olutionto the order O ( E )in the form

U=u"'+s@(<,r) 4.

(12.4.35)

Taniuti and Wei (1968)have shown that the reduction penurbatmn method can be extended to the exceptional case in which the eigenspace of 4 comprises invariant subspaces. Su and Oardner (1969)derived both the KdV equation and the Burgers equation by using a similar pelturbation technique.

Arymptoric Mahods and Nonimmor Evolution Equorions

516

Taniuti and Yajima (1973). On the other hand, Noraki end Taniuti (1973). Oikawa and Yalima (1973, 1974) have considered the nonlinear interaction of three manochomatic waves and the interactions of solitary waves and nonlinear modulating waves. In addition. we may mention that similar reduction perturbation theory has been developed in other contexts, for example. in studying the behavior of the partial diffcrenusl equations in the neighborhood of a critical point and In describing the complete history of a pulse with a curved wavefront. Relevant references include Kulikovskii and Slobodkina (1967). Bhatnagar and Prasad (1971)andPrasad(1973, 1975). Wenextillurwatethemethod byeramples.

Exornplr 12.4.1 (Osciliorions of a Nonlinear Elastic String with Dispersive Effects). This example is taken from Taniuti and Nishihara (1983). me transverse displacement y(x.1) of such a string of infinite length is governed by the nonlinearequation

- d y - =o.

y,, - c ' y y

(12.4.40)

and a ir a real constant. Equation (12.4.40) can be

where c1 = ( l + y : r u r

and hence, it admits a plane wave solution y ( x , t ) = A exp[i(lu-wl)].providedthatthedisprrsionrelaUon

lineanzed when c - 1 ,

wr =k' -a2'k

(12.4.41)

is satisfied. For appropriate long wavm (large wavenumber or small wavelength k), w2 > 0 . and so, w ( k ) is real. which confirms that the waves are dispersive.

We now wis y, = u and y, = v, so that quiltion (12.4.40) can be replaced by a system of equations like (12.4.1) which has the form

U, + A U ,

+ KU-

=O.

ulhae U=[:],

and

A=[:l

K=[l

.I:-

(12.4.42)

The cigcnvalues of the matrix A are d") = f c , and hence. it follows that

,Xi=* 1, for oscillations about the equilibrium position in which u = O , whereas $ = ( f l , I ) Consequently,

end $ ) = ( I , i l ) ' =

pwd'*"] $)=O

[:I) .

so that Vu.%'*)=(O,O).

which confirms that each of the two

(EP'P'ZI)

518

Asymptotic Methods ond Nonlinear Evolulim Equorionr

-v(y-i)

K(Y-I)R-'

PIP'

0

-K(~-I)R-'

0

: i

Kl=[

,];

K:=[

0 v(r-l)

i),

1

0

0

0 01 101.

The eigenvaluesof matrix A are u, u f for the constant state

(1"'

= (1.0. y ' ) . The eigenvectors

& and

movingwiththeveloeity Io=l,aregiven hy & = ( O , I , l ) l h i r leads to the Burgen equation J'p"'

JP'"+qp"l @+sl Jt;

JI

R, for B wave

and &=(1.1.1).

(12.4.47) Jt;

where p"' is the first-order term of the expansion p = I + ~ p " ' + - . , the boundary condition U + U@',as x

+ce, has becn assumed,and

a,= y+l). I( and p, =--!

[ ( y R).' r(~ - 1 +)v ]~

Exampk 12.4.3 (Ion Acoustic Waves ond lhc KdV Equalion). The motion of cold ions in B hot electron gar with the Boltamann distribution law IS governedby equiuions Jr

(12.4.48)

Jx

[321

(12.4.49)

= eE .

(12.4.51) JX

519

12.4 Weokly Nonlinear Dirperme System

wherc the subscripts i and e refer to quanlilias related to ions and electrons, respectively, n , ( j = i , e ) isthe density, u , ( j - i . e ) is the flow velocity, T, is the product of the Boltzmann constant and the temperature. e is the charge of ions, --c is the charge of electrons,and E is the electric field. TO apply the reductive perturbation method to this System of ion-acoustic wave^, we eliminate n, and E fromequatiom (12.4.483-(12.4.50 to obtain (12.4.52)

This system of equations corresponds to syrlsm (12.4.1) with p = 3 and s = I . The associated matrices U.A. H, and K are given by

K2=(" 0

For

u"

=

['J

'1.

the ctgenvalues of

obmn

G=(l,l),

H,=O,

a n d K , = r0

0

4 are 1, = + I .

&=[:I,

'1. 0

For the ease of 1, = 1, we

and V u l o = ( O , l ) .

Thus, from (12.4.33) snd (12.4.34). it turns out that a,= I . and the dcnsity

0, =L, and

of electrons n:) to the first-order approximation satisfies the KdV

520

Asymptotic Uethals and Nonlinear Evolution Equationr

In thermodynamic equilibrium at a constant temperature, elecvons must satisfy the Boltzmann distribution law n, =no exp

9 , where

n, IS the

(1;) 0 IS the elecvostatie

numkr density of electrons in the undisturbed slate and

I::(

potential associated with the wave motions. Expanding exp - , we can obtain

$' = B"',

and hence. the rolilon solution of the KdV equation agrees

with !he solution for

0"'.

Exomplr 12.4.1 (Nonlinear Shnllow Water Waver on M Uneven Bonom). For long waves on a beach. Peregrine (1967) derived !he following system of coupled equations: *+(u.V)u+Vh+i

*+ t

6

H ' d V ( V . u ) - i H d V [ V . ( H z ) ] = O , (12.4.55) at 2 m

V . [ ( H + h)u]=0.

(12.4.56)

where u is the horizontal velocity averaged over the vertical direction, h i s the wave amplitude, and H is the depth of sull water which depends only on the haizontll coordinates. For the motion of water in the rdirection only, Pere~ne'sequations become h, +ah, + ( H + h)u. + u H* = 0,

(12.4.57)

u, +uux + ~ , - ~ H ( H u , ) ~ + ~ H =' Ou . _

(12.4.58)

It is noted here that only the first-order x- and 1-derivatives are involved in equation (12.4.57). whereas equation (12.4.58) contains the third-order derivatives in which the x-denvative IS repealed twice. but the I-derivative occurs only once. These comments an useful for chosing the mavices H t and Kt in !he original equation (12.4.1). In this problem. p = 3 and equations (12.4.57)(12.4.58) can be expressed in the form (12.4.1) with an extra term BV. on its left side by chwsing s = I and

12.5 Sfrongly Nonlinror Dispersive Systems

521

K:=l,

K: =O.

For the equilibrium state h = u = 0, we have the matrix

A left eigenveetor 4 and a right eigenvector 4 are given by

To Write the nonlinear evolution equation (12.4.23).we calculate its cocffieientr a,$,.and y, for I , = v% as

Conssqucnlly, the KdV quation far y = H"

y,,

+( 5

H-"') y y r

0 assumes the form

+(+a) yet< 0, =

(12.4.60)

where the coefficients of this equation are functions only of q .

12.5 Strongly Nonlinear Dispersive Systems and the Nonlinear Schrodinger Equation We consider a general, strongly nonlinear, dispersive system that can be described by l k equauon

U,+AU.+B+

[h, fi

:

H -+Kfd

U=O,

a)]

(12.5.1)

where (Iis B column vector with then components u,.u2....,u.,(n 2 2) A , H! and Kf

at^

n x n matrices, and B is a column vector of n components.all of

Arymproric Methods and Nonlinear Evolution Equations

521

which are functions of U . We make an imponanl msumption that the linear dispersion relation of (12.5.1) admits smng dispersion because of the term B. For simplicity, we consider the problem of nonlincar wave modulation based on a system of equations without the last term in (12.5.1), EO that (l2.5.1) gives the firSt-ordereq"ati0"

U, + A l l ,

+ B = 0.

(12.5.2)

Many physical problems exhibiting strong dispersion are described by this equation. A eonstant equilibrium solution U = Uoo f this equation is required to satisfy the algebme equation

B(U,)= 0 .

(12.5.3)

We assume, here, that a eon~tantsolution U, exists end that B plane wave solution of infinitesimal amplitude ir superimposed on the con~mnlstate, so that

U=U,+(SU,) cxp[i(lu-wr)]+e.c..

(12.5.4)

where C.C. stands for the complex conjugate of the preceding expression. Substituting (12.5.4) in (12.5.2) and linearizmg with respect to 6 U , gives the lincar dispersion nlation

IT t o / i i k A , + V , S, I=o.

(12.5.5)

when I is the unit mauir. 4and Bo am mamees evaluated a1 U = We,and V" is the gradient in U space. Relation (12.5.5) determiner w as a function of k. and hence, we msume that 0 is given by a single real root of (12.5.5). We then invoducc amabix Wmdefined by (12.5.6)

Wm=Fimwlf imk%+V,B,.

To consider the moddalion of a linear plane wave. we introduce the slow variables <=&(x-At)

and c = E * f ,

(12.5.7ab)

wh en A = ( z ) is the group velocity. The main objective h e n is to

investigate how such a plans wave is modulated by nonlinear effecu.We assume that the solution of (12.5.2) IS of the form U = U o + x ~ 'urn,U n= ~ U ~ ) ( < erp[im(h-wr)]. , c ) "4

up".

(12.5.8ab)

I-..

whne U" is asrvmd to bc red, so [ha<

ff?) is qua1

to

iu oompler eonlugales

12.5 Strongly Nonlinenor Dispersive Syslrm

523

We consider modulation of the plane carrier wave with wavenumber k and h e m y 0 ) . Wemay set U y ' = O except for m = f l . Substituting (12.5.8ab) into (12.5.2) and using (12.5.7ab) leads 10 the following perturbationequations q E ) :

w.uy,i"'=o,

(12.5.9)

(12.5.10)

where ( .)'"I

represents the coefficient of the mth harmonic and

z"'leip[ir(G-wt)]. Thus, h m (12.5.9). we obtain, Uf"'=@,({,r)R

for

where @, is a complex function of slow vanablcs

Iml=l,

(12.5.12)

5 and 7 to be determined and

R is a dght eigenvector of W, conespanding to a eigenvalue A , so that W,R=O,

whereas, for 1 m I+ I

(12.5.13)

12.5 Strongly Nonlzneaor Dispersive Syrnrns

525

and hence, we note that. for Irnl-1, the nonlinear terms in (12.5.11) do not contain 0‘”.If we multiply equation (12.5.11) for ( r n ( = l by L from lhe lcft and use equations (12.5.12>(12.5.14), and (12.5.19)-(12.5.21), we can eliminate

all terms eontaning Q”’ because the first and the second terns vanish by equations (12.5.17) and (12.5.16). Consequently, all terms involving

0‘”

vanish, and hence, we o h n the equation for 0‘‘’in the form

where the coefficientsp and q are given by

ax

(12.5.23)

526

Asymptotic Mahods md Nonlinear Evolution Equarionr

-iw'R+Zi(-LI+%)dR+W,~=O.

(12.5.28)

dk'

dk

This is multiplied by L fmm the left to obtain, by (12.5.17). -iw"(L.

R)+ZiL(-ll+%).

(%)=O. aR

(12.5.29)

w" from (12.5.23). We then derive thcgenrrolizcdnonlineor

which gives p='

Sckr&dhger cqunrion

io,+'w*q+<{

+q1++rm=o,

(12.5.30)

whne $ isafunctionof < = E ( x - w ' I ) B ~ ~ ~ = E ' Imdqisnotneecsrarily real. In particular, when q is real, (12.5.30) is called the nonlinear Sckradingsr equnlion. Asano (1974) obtained the most general form of this equation for the case of Ihre-dimensional mdulation of waves BS

where k, (i = 1.2.3) and w denote the components of the wavevector and the frequency of the p1qe Earner wwe. respectively. and

<, = E

last term i 7 $ is involved to incorporate wcak dissipation and/or unstable

effecu. Equation (12.5.30) can be solved exactly. The uansformations

{=X.

i w " t = T and

v=lLf# W"

are used to put it into the canonical form

idW+yrm+2alyrI1yr=0,

m

(12.5.32)

where a= sgngn(1). This canonical equation has been solved in Chaptcr 10 by

*."

\-

,

the inverse scattering method. For a=+1, the asymptotic solution of the initial-value problem for (12.5.32) represents a sequence of envelopesolitons

12.6 The Perturbofion Method o/ Ostmvsky ond Prlinarh

527

y ( X , T ) = Z q seeh { 2 q ( X + 4 v T - X 0 ) ) x exp[-2ivX-4i(v2

-q2)-i$0]

(12.5.33)

and radiation modes.where q and Y arc parameters to be determined from the initial conditions.

12.6 The perturbation Method of Ostrovsky and Pelinosky Whitham's theory of averaged Lagrangian principles has been a major conuibution to the field of nonlinear dispersive waves. This theory IS of great interest fmm the viewpoint of its mathematical generality and physical clarity. Bared on his averaged Lagrangian principles. Whitham derived his fundamental equations for describing slowly varying nonlinear dispersive waves. Thn method is directly applicable to a conscrv~uveSystem for which the Lagrangian is known. However, the determination of the Lagrangian is not always an easy problem. Furthermore, Whitham's theory cannot be applied readily to nonconservative dynamical systems. To extend Whitham's theory, Osmvsky and Pelmovrky (1971, 1972) have developed an asymptotic perturbation method for B general system of first-order nonlinear p m a l differential equations that can be applied to both conservative and nonconservative systems. They also considered a dynamieal system which can bc described by nonlinear p d a l differential equations ofthe Lagrangiantype tho1 include dtsspatmn. Oswvrky and Pclinovsky (1972) also showed that the equations of the first approximation can be deduced from the generalized Hamilton's variational principle. This section deals with the penurbation method due to OsUovrky and Pelinovsky. We consider a system of nonlinear panial diffcrcntial equations ~ ( S , c . , u , + , u , ) = EN(S,T,U,U,,U,).

(12.6.1)

where u(x,r) is an n-dimensional column vector function. operators M and N denote sets of nonlinear functions EX and c = & i , 6 and x m e thrcedimensional ~ e c t o r and ~ . the parameter E characterizes the deviation from the stationary and uniform state of the medium. We also assume that operators M and N are sufficiently smwth m terms of their arguments. For E = 0 and for constants 8 and r , the system of equations (12.6.1) is assumed to have stationary plane wave Solutions in the form

u = U ( O ) and O = l r . r - o r .

(12.6.2ab)

There solutionscan be determined from the ordinary differential equations

Asymplollc Mdhds Md Nonlimor Evolution Equnaons

518

M{<.T.

u(e),-mu,,

KU,)=O,

(12.6.3)

where 6 and I are assumed to be constants. It is noted that solutions of (12.6.3) depend on m ( S n ) arbitrary constanl~of integration 9. A=(Ax,A, .....Am) and on the parameters 6, r, I, and 0 . We also assume lhal the frequency w , the wavenumber vector I, and constantA satisfy the dispersion relation

w=w(~,A),

(12.6.4)

and ulat they will be chosen so that u = U(B) is a penndie function of 6 with period 2x. The nonlinear wave profile IE then determined from the basic quation (12.6.3) and may. m general,k different from a sinusoidal shape We now look for B solaion in the form of the asymptotic series

so that

&+*=o.

(12.6.9)

a: m

We also expand operators M and N in an asymptotic series to obmn M = M 1 o ' ( U , - ~ U ,K. U e )

- we,,,KU.)+

E N = E N"] (I/. w

where each term of the form

-

o(E'), (I represents

(12 6.11) a pmduct of a square mauix

[:'0']

and a column vector. We substitute (12.6.5)-(12.6.1I) in (12.6.1) and equate coefficients of similar powerr of E 10 obtain

(8,

&. &)

6,7,

Y"'

= Q"',

(12.6.12)

whne n=1,2,3,...,

and similar expressions for Q"',Q"',

To determine functions 0 andA to any desired approximation. we have to solve a system of nonhomogeneous. linear. panial differential equations with periodic cocfficicnts and nonhomogeneous terms, which BIC also periodic in 8 . Using the following results

d..d

JB

in equations (12.6.12).the solution pc'

and

d,-wd

(12.6.15ab)

at as d"'can be represcntcd in the form

= R I R - ' Ql"'

do',

(12.6.16)

where R is a matrix consisting of vectors of the System of solutions of the variational equations

(12.6.17)

and R-' is UX inverse of the matrix R. It should be painted particular vectors is found in terms of W a s follows:

out lhat the set of rn

Clearly, R, is a periodic function of 0, and R, can be expressed in the general form R, =

E +B, s c, JA>

d8

(12.6.19)

whne B, is an arbitrary constant to be determined from the dependence of

w onA. The remaining (n - m) vectm can be expressed as

Y

and

530

Asymprollc Metkals

R~= e v ( l , 8 )

and Nonlmor Evobrtion Equorioni

fm.

(12.6.20)

where f,(S) IS a periodic function with period 2a. We assume that the characteristic exponent A, has no p o t j v e real pan a d . if Re A, = 0 , rhcn,

Im A, #O. f l , .... Substituting R and R-' in (12.6.16) yields

The last term of (12.6.21) is bounded for all 8 due to the nature of A . In order that u'"' renuins bounded as a functionof 8 , it is necessary that it satisfies the orthogonality conditions

j U A ,Q'"'d8=0 and jU,Q"'dO=O,

(12.6.22ah)

f o r n = l , l ,... a n d r = 2 . 3 ,...,m. The equations of the first-orderapproximation can k obtained in a different form. We mrrltrplp cqrration (126.1) by U,. integrate o w 8 , and then, subtract equation (12.6.22a) h m the resulting integral expression to obtain

-

y M ( S . T , U , U , , U , ) UA,d 8 = E 7 N ( S , 7 , U , w U ~Y,U , ) UA,d o . (12.6.23) 1

Similarly. multiplying (12.6.1) by Uo,inlegrating over 8 , and subtracting equation (12.6.22b) from the integrated result gives

y M ( < , f , U , U , , U , ) Ue d o = € T N ( C . 7 , U , - m U # , YU,) U,dO. (12.6.24)

Three equations (12.6.9). (12.6.23). and (12.6.24) form a. closed set for the unknown functions Y , W , and A. It is necessary to solve the second-order approximaOon equations to determine the phase function 8 of wder

a€).

In their paper, Osvovsky and Pelinovsky (1972) have shown that equstlons of the first-order approximation can k derived fmm the generalized Hamilton variational principle ~n the averaged form. They examined the system of equations

&(3+ $3-2

= 0.

(12.6.25)

12.7 The Method of Multiple Scales

0 = 0'0' + &#"' + O(&'),

531

(12.6.26)

where X is the Lagrangian density. 4 is the density of the nonconscrvativc term, "(*,I) is an n-dimensional vector function, and & is B small parameter. We assume that X and 0 BX sufficiently smwth functions of u, y. and u, and alsoofslowvariables & = E X and r i & t . Equation (12.6.25) can be derived from the generalized Hamilton variational princ,ple I j ( S X + S W ) drdt=O,

(12.6.27)

where S W = 0 S u . We conclude the permrhation analysis ofOrmovsky and Pelinovsky by adding a couple comments. First, this analysis is valid for nonlinear systems with arbitrary, hut not necessarily, w e d nonlinearily. Second, the reader is referred to work of Osvovsky and Pelimwky (1971, 1972) and Gorsehkov ct al. (1974) for funhcr information.

12.7 The Method of Multiple Scales Multiple scales, or more precisely. scales of different orders arise in many manifpst themselves physical problems because different physical effects UEUBIIY over different lengih and time scales. Naturally, the concept of multiple ~calesis involved, explicitly or implicitly, in almost all asymptotic perturbation methods. Hence. it seems to have wide applicability to many problems that involve physical phenomena which mcur in relation to various scaler. Although the method of multiple scales covers a wide variety of perturbation procedures, the method will be formulated here in the context of nonlinear wave propagation. Historically. Sturrnk (1957) first introduced the method of multiple scales, or more precisely, the so called derivative expomion method for an investigation of nonlinear effects in electron plasmas. Nonlinear effects are analyzed by the above perturbation procedure to study the incoherent interaction which is responsible for specval decay, that is, for the redistribution of energy in wavcnumbcr space - the breakdown of organized large-scale motion into disorganized small-scale motion. Subsequently. Sand" (1963, 1965, 1967) used the method of multiple sealei that completely Separates the different time scales exhibited by the evolution o f a gas to provide the foundation of nonequilibrium stati~ticalmechanics. Several other authors including Stuart (1960), Watson (1960), Hasirnoto and Ono (1972). Fricrnan (1963). Nayfeh (1965ab. 1971. 1973). Nayfeh and Hassan (1971), and Kawahara (1973. 1975ab) successfully applied the derivative expansion method to solve different problems in fluid mechanics and plasma physics. There authors showed that a systematic application of the derivative expansion meUlod can effectively Lake into account

531

Asymptotic Methodr and N o d i n e m Evolution Equtionr

the full effect of amplitude modulation of a quasi-monochromatic wave. and thereby can lead to the nonlinear SchrMinger equation. It has also been shown that the derivative expansion can be applied successfully to problems with longwave approximation which leads to the Korteweg.de Vrics equation. We considerthe nonlinear parUaldifferential equation

4i.5 1

(: ad,

1 u ( x . t ) = N -, -,

where

1

p u*(x,I), xsR, 1>0.

(12.7.1)

L and N are differential operators and 1,p are fired parameters. W e

introduce the sets of independent variables defined by I.=&".%

X ~ . X , , X , , . . . , X ~ and

I~,I,,I~....,~~

(12.7.2ab)

and r . = & " f ,

where & is a small parameter characterizing the smallness of the associated terms. Consequently, the dependent variable U ( X , I ) can be regarded as a function of tkse new variables, so that u(x,I)= u ( x ~ . x.....~~,r~.r,,....fi). , Since the method is called the derivative expansion method, it is appropriate to i n d u c e expansions of the derivative operators

Using these expansions in (lZ.1.l) leads to the following results

+o(&~*'), (12.7.4b) We also assume that

U(X.I) has the asymptotic representation

"( s,.,,..... I. , .. I,."'.I~) = $&"

un(xo.x,,

x.. to,I, ,.".I.)+

( ( ( .

O(&-'). (12.7.5)

"4

In general, "(*.I) can be expanded in terns of another small parameter 6 which measures the degree of nonlinearity of the wave field, and it is assumed

534

Asymptotic UrB&

and Nonlimor Evolution i?qmdonr

Utilizing there operaton, equauon (12.7.9) reduces to B system of perturbation equatms I" the form: O(E): L u , = O .

(12.7.10)

O(&*): & n 2 + I , u, = N o u:,

(12.7 11)

O(E') :

O(2):

ku,+ I,u2 +& u, = N,[Zu,u,]+N,u:, r, u. +r, + r, u* +r,", = N, [u: + 2u,u,]+ N , [2u,u,]+ N , u:.

(12.7.12)

")

(12.7.13)

We consider a smngly dispersive wave system and it is necessary to take into account nonlinear mdulation of the wavevain, that is, amplitude and phase suffer fmm a slow variation over a localized region in space. Consequently. it IS reasonable to assume that the first-order perturbation solution of the first equation (12.7.10) has the form a, = A ( x , , x ,

,...,x , , r , , f ~ , , . . , fexp(i@)+e.c., ~)

(12.7.14)

where 0 = ( k x , - ofm), the amplitude A is a complex function of the slow variables, and C.C. represents Ule complex canjugate of the preceding expression. Moreover,k and UJ satisfy the dispersion relation (12.7.7). Invoking (12.7.14) in the second equation (12.7.11) of the perturbation system yields ,!,u

whne

---I

'-

[**-*"A ao Jt,

*

exp(i 9 )-2k'A* exp(2i @)+c.c.,

(12.7.15)

ak a x , ]

*

= -2(1+ L k 2 ) m = 2(c2 - 1 w')k. (12.7.16ab) JUJ ak To determine u, from (12.7.15). it is necessary to integrate it, so that the

result producer a so-called secular (unbounded) rerm. that is, the term gmws without bound, as time f + -, due to the form of and erp(i@)in the first term on the nght-hand side of (12.7.15). 'Ibis means that such B solution for u2 will not be uniformly valid as I + -, no matter however small E may be. To eliminate the nonunifomity rn the perturbation expansion, the coefficient of crp(i9) in (12.7.15) must vanish identically. This leads to the following condition Far non~eeulanty

r,

535

12.7 The Merhod of Mulnple Scaler

(12.7.17) When

(12.7.18) represents the linearized gmup velocity. Consequently, the uniformly valid solution for u2 can he obtained from (12.7.15) in the form

+~(~,.~,.-.,'.r,.r,...r.).

(12.7.19)

where F and G ace complex and real functions rc~pcctively.of higher order scales that can be found fmm higher order pcrmrbauons Substituting u, and u2 in equation (12.7.12) leads to another nonsecular (bounded) condition

, JD

JA ( J m Jt,

JD JA

1 J'D

J'A

Jk J X ~ ] 2 ( do'

Jr:

~

J'D J'A JkJw Jx,Jr,

~

J'D J'A Jk' J r : )

and its complex conjugate relation. The vmiation of G follows from the non~ecularcondition obtained from the fourth equation (12.7.13) of the perturbation system, and hence. it Nms out that

(12.7.21)

-

If A and G depend only on x, and 1, through 5 = x , C, r, , that is. if they are considered in a coordinate system moving with the group velocity $, equation (12.7.21)can be integrated with respect to to obtain

5

and, then, can he solved

C(<.x, ,...,x.. t2,r3,....rm)

= (Ci -c')'AA*

providedthat C 8 # e and a = a ( x , , x ,

,...,x-,r,,r, ,...,I - )

+a,

(12.7.22)

is aslowly varying

functions of higher order slow vanahles to be determined from the initial and

536

Asymptotic Methods ond Nonlinmar Evolurion Equations

boundary conditions.However. if C, = c , the resonant interaction between short and long wayes O C C U ~ .and this case can be handly separately. To obtain an equation for the amplitude A, it is newrsary to find F involved in (12.7.20). Since F is a ccefficient of a secular term involved in uI and PO, it is an unbounded term proportional to exp(i0). it follows that F is associated with lhe unbounded terms of the higher order solutions en. It is possible to show that these unbounded (resonant) terms can be transferred to the lower order solution u, and the transferred quantity then can be considered as a new amplitude A . Therefore, the last term in (12.7.20) may be dropped. Finally, using of (12.7.17) and (12.7.22) in the nonsecular condition (12.7.20) yields

where A = A ( g ( , x ,

,...,r, I,.I, ,...,I-),

a is independent of

5. and

hence.

(12.7.22) indicates that u can be treated as a constant with respect to 5 dinate. In a frame of reference moving wilh the group velocity C. and the cwrdinate wnsformations defined by

= 1 ( x 2 -Cn r,)=(il

-C, I,)= ‘(x-C,

I) and r = 1 2 = EI, = E l i . (12.7.24ab)

E

we can express equation (12.7.23) in the following canonical form of the

mnlinear Seh&inger e q w h

Thus. the upshot of this perturbation analysis is that the amplitude modulation A (6.7)for a strongly dispersive system satisfies the NLS equation. We close this section by adding some relevant comments on the asymptotic expansion in terms of an other small parameter 6 . First, the coordinate transformations defined by (12.7.24ab) are nothing but thc Gardner-Morikawa transformations that have been used in the reductive perturbation analvsis. Second. u(x.1) has a more general asymptotic expansion in the form u ( r , 1 ) = ~ 6 ” u ~ ( x , . x , . . - .ro.l, , = ~;.., , I-),

(12.7.26)

1-1

where 6 is a new small parameter characterizing the smallness of terms that measure the degree of nonlinearity. Several cases arise depending on the dative

12.7 Tke Method ofMultiple Scales

537

implfanee of S and E . The case S = E has already been discussed above. The case S = E' cornsponds to the dispersive w w e Systems dominated by linear propagation of envelopes. This case can be analyzed similarly to the case of S = E , and hence, the wave amplitude A satisfies the linear rguarion similar to (12.7.25) wirhourtk nonlinearterm I A r A . In the ease S=&. the nonlinear interaction dominates, and hence. the nonlinear term occurs in the first nonsecular condition. A similar analysis leads to the following equation:

where A and a are functions of slow scales X , . X , : - . X ~ , r,,.r2;...rm. The independent variables in (12.7.27) are the first-order slow variables x , and I,. The slowly varying function a can be found with the aid of Ule same equation (12.7.21) which was obtained fmm B higher order perturbation analysis. Thus. cx canbevcatedasafunctionofslowvariables x,,x,, ...,xm,r2.t,....,f to be determined from the prescribed boundary andlor initial conditions. Hence. the amplitude equation (12.7.27) in the nonsecular condition remains valid far the first-order slow variables. More slowervariables x2,t,,x,,r,, ... in (12.7.27) can be mated as slow parameters for this order of approximation. Inow and Matsumoto (1974) applied the method of multiple scales to a general system of equations of the form A(U)

z+ E+ dt ax B(U)

C(U)= 0.

(12.7.28)

where UisaeolumnvectorwithnEamponents u , , u , : . . , u m , A and B are n x n maulces. and C i s B column vector. According to the method of multiple scales. we introduce a new set of independent variables as x.

= E" x and I* = 2 I

(12.7.29ab)

where n=0,1,2. ...,rn and, then. the dependent variable U is expanded in the

fm

where 0, is condition

B

constant solution vector which satisfies the compatibility

C(UJ=O.

(12.7.31)

Invoking the asymptotic expansion of the derivatives into multiple scale independent variables. BS in (12.7.3ab). the perturbation analysis for the general

539

12.8 The Method 0fMultiplr Scaler

In this ease, the Boussinesq e q u a t m reduces to the following coupled system.

o(E‘):

4”,= 0.

(12.8.6)

O(E’).

Lp,=o.

(12.8.7)

O(e6):

Lp,+L,n,=N,u:.

(12.8.8)

Equation (12.8.6) yields the linear equation

This admits a solution for u, which is function of 5, = x , -cr, for waves pmpagaung 10 the dght 01 of 5, = x , +cr, for waves traveling to the left with constant velocily e. Using these results m (12.8.7) combined with terms of order o(E’)yields

ibis equation is satisfied If u, depends on x, and r2 through hence. equauon (12.8.8) reduces to the form

5, = x1 -cr8, and

Fmally, we assume that u2 depends on x , and I, through thatequation(12.8.11)with u, = q l takestheform

in which integration is carried out with respect to

5, = x, - er,

EO

4,.

In a f-e of reference moving with the phase veloeiry c of long WBWE, that is, by Setting 5, =x,-c13, r=r, and. then, replacing 5, by 5 . equation (12.8.12) reduces to the canonical form of the Korteweg.de Vdes equation in terms of the independent variables 5 and r

540

Asympmric Mahods and Nonbncor Evolution Eguorions

This revealsthe fact thaf when deriving the KdV equation, the frameof reference moves with the phase velocity but nof with the group veloeily of the wave. Case(n): s = & 3 . In this ease. !he asymptotic expansion enabler us to derive an equation similar to (12 8.12) without the nonlinear term. Case(iii): S = E . In this case, we obtain a nonlinear equation without a dispersive term in the form

dm+cl+' Jt,

Jx,

2r(

21

0-

=o.

(12.8.14)

Obviously. the independent variables are the lower order %ales x2 and equation (12.8.14).

t2

in

Answers and Hints to Selected Exercises 1.12 Exercises 1. (a) Hyperbolic, y - x = c , a n d y = ~ x + c , ,

a = <+ q.

0 = < - q.

uIQ=i(un-:):

urn -u4# =$(urn -up

-:).

(b) Hyperbolic, x + 2 y = 4 , x + y = c ? : u { . = q - < . (c) Hyperbolic for y < 0 and elliptic for y > 0. Parabolic for y = 0. Forthehyperbolic case,

< =x+2&.

~orththee~iptlccase, a=x,

q=x-Z&,

0 = 2 ~ ; um+auD=+uu. ;

(d) Hypbolic, < = y ’ - x ’ = e , ,

q=y-x=c,;

u c q + i u -0

+ .,)+

q u < + < a , = 0. (0Hyperbolic for x < 0 and elliptic for x > 0. (e)

5 = y1.

1) = 2 : 2
For x < O : < , S = ~ Y + ( - X ) ~ ,( y - c ) = + ~ ( - x ) ” ”{“=-

<-

R

(cubicparabolas)

(5-4 6(< - 11) I

Forx>O, a=(+)y.

D=-P;

um+uub+

-0

LsI 6 -

a and 0 satisfy the Bcltrami equations p, = -& a,,0,= 1a,.

&

(g) Hyperbolic in

I x 1 < 2 ( y l , elliptic in Ix1>21yl, and parabolic in

IxI= 21Y I. 01) Hyperbolic in the first and third quadrants, elliptic in the second and 0)

founh quadrants. Elliptic for Ix I c (I,parabolic far 1 x I= n, and hypeholic far I x I> a .

(i)

Elliptic for y > e.’

parabolic for y = c-’. and hyperbolic for y c e ’ .

3. For ,%=-a'.u ( x . y ) = ( A c o s a r + B s i n a r ) ( C c o s h q + D s i n h ay)

4. Hint:. X"'= C X . 7" =-A'&" X ( x ) = A c a s h k + B c o s k + C sinhk + D s i n k ~ ( r ) = ~ e m ( d * c t )sin(.X*ct), +~ B=-A.

D=-c,

-

X ( x ) = A(corh k - cm k) + C(sinhk sin k). From X ( I ) = 0 = X'(C), we obtain cosrll- eosU

sinhrll -sinrll

sinhlV +rinU

cmh At-cmrll

= O *eoslVemhU=l.

Nonhneaor Pond Diflerentiol Egunrions

545

45. Hint:. The solution of h e dual integral equation

I k I @ ) A(k)dk = uo. 0 5 r S a. Jk’ I&)

A(k)dk = 0, n < r c -,

is given by 46. Hint: See Debnath. 1994. pp 103-105.

49. Hint:

.-I[

(s2 + a’y

exp{-k(?

-

+a’,’]] = H(t k ) Jo( a m ) .

52. Hint: Use the joint Hankel and Laplaceumsformmethod. 54. Ux the Hankel transform and derive

x J ~ ( k a ) d a ] ~ ~ ~ { o ( t - r ] where d ~ ] , o*=gk. 56. Hint: Use the joint Laplace and Fourier uanrform.

j a e exp(*r)&,

~(x.t)=W

2 m -

w h e r e o ’ = g and m

a=(azka++d)+,

w’=x. m

58. Hmt. Set

59. Apply the joint Laplace and double Founer transform

546

A n m m ond Hints to Selectedfiercirer

60. Hint: G. -e'G,

+d'G= S(x)S(r). 7he joint Laplace and Fourier urnsfom giver

me inverse mnsforms give

2.7

Exercises

547

Nonlinear Panid Diffenntiol ffiuoaons

Exercises

3.6

(d) u = f ( y - t m P x ) ,

(0

(e)

(b) yp-xq=y'-x'. (b) u=f(bx-oy).

2. (a) z p - y q= x - y, 3. (8) .= f ( y ) ,

(c)

px+qy=q'. u=f(yi'),

. ) (7

(4 u = f '*-"

/(y,

(x-y)'-u')=o.

x1 + y* =C,, y(u- y)= C,.

Hint: ( u - y)dy+ yd(u- y) = O

(9) u=y+y-'f(x*+Y*).

(h)

u +logx = f(xy),

ti) /(x' +u'~ Y' +u') = 0 .

5. (a) u = s i n ( x- z y ) ,

(b) u=cxp(x'-y'),

(c)

u = q +uf= x(y l + Z)- (,$ ) ] ,

(e)

"=

1'

(d) u=sin(y-:x').

TY*+crp[-(r'-y')]

fCfX>Y

;x~+clp[-(y'-x~)]

for x c y

(f) Hmt: y = i x ' + C , .

u=C:x+C,,

1.

"=x(y-Lq+,(y-L~), *=~(y-~~z)]+~~p(~-:~l]. (g)

r=c,and "+'-C,.

(h) Hmt: x + y = C , ,

C,=l+',

c:

u=y+g-l, yfO.

dy=A, u'+c~SC,txp(-Zy). --u 2+c:

From the Cauchy data I + C: = C,, and hence,

(i) $-;=I.

$(;)=

which implies that x = C, cip

Nonlmaor Paniol DiffemnrzolEquoaons (C)

549

.r=L(r+s)~, y=u=r+s

13. Hint: I h e initial curve is a characterisuc.and hence, no solution exists.

14. (a) u = e x p *(+AY ) ,

IS.

(b) u=sin[(-y].

HmI.

3uy=u2+x2+y',

xdr+ydy+udu.

16. (a) x ( s , c ) = q y ( s , r ) = l f - + m s + ~ , u(s,r)=r+r?s

r = x,

I = (1 +ar)-'

(b) y = c + f ( v - z ) ,

(y

;x')a.

and hence,

Zy=u'+(u-x)',

u(O.y)=&.

(z+ y + u ) ( y - u)' = e, and (x + y + N ) ( U - x)' = c2

(b) Hint.

"=*. 1

Hence q = a .

-Y

x u ~ u ~ + o l -So, ~ .$ = v g i v i n g

x'=u'+Zou'+b

and, thus. x' - u' - Z n ' q = b . 18. Hint: d r = d y = d r a n d h e n c e f = e , X

Y

Z

z

l=d. z

c=(f)corO, d=(f)smO and z = b t a n - ' ( t a n @ ) = b Q .

Thus Ihe curves are x bQ = orcos@ and 19. F ( x + y + u , ( x - 2 y ) * + 3 u ' } = O . (1-2y)'

+3u* =(I+ y + u ) * .

Hint.

y b @= or sin 8 .

(dr-2dy)=i 9u

-3(x-2y)'

Answers Md

550

Hints

to SelectedEvrrisrr

20. F ( x ’ + y , y u ) = O .

(x*+y)‘=yu.

21. Hint: x - y + r = e , .

A = *+:+&), andhence, -(x+y+z)

8 2 +(I + y + 2 ) l = ct. F((X - Y + z). 82’ +(x + y + $ } c;

+e, = 20’. or (x- y + 2)’ +(x+ y + z)’

22. F(x’ +y2

+22.

(a) y* - 2 y r

=o

y2 - 2 y z - 2 )

-2

=0.

+ 8 2 = 20’.

= 0. two planes y = (I *&)r

(b) x’+Zyz+Z~’=O,aquadri~conewithveneiattheaigin. (c) r’-2yz+2y2=0,aquadric conswithucrtexatthearigin

4.6

Exercises

1. (a) 1 6 u = ( x + 4 y ) ’ ,

(b) u = a ( x + y ) .

2.

(b) Z m = x + o y + b .

(a) & l o g u = x + o y + b ,

(d) u = o y + f ( x + b ) ’ .

(9) 1(=.logx+(4-a’)ilogy+b,

+

(h) u = a(l+ XI);+;(q)’b

3 . (a) l o g u = x + y - I . (c) u ( x , y ) = x + y ,

(e) u = * y whichissingularat y = 2 , 2(Y - 2 )

(0

4u=(x+y+2)’.

4 . Hint: At s = O , p ( r , x ) = q ( t . s ) = l . then, p(t,s)=q(f,s)=l x(1,s) =(1-2)e’ +2. y(1.s) = I+e’, u(r,s)

= (t - I)c’ +2, u ( x , y ) = x + y - I.

8. Hint: x ( O , s ) = O , y ( O , s ) = s , u(O,s)=-s. x ( l , s ) =-sinhr, y(r.s)= aeoshr, “(1,s) = -L(l+cosht),

u(x, y ) = - y ( x ’

+ I);.

10. Hint: Characteristicsme x -2cr = s where s is a parameter. The points on the chmcteri~ficlines navel with speed 2 e , whenas p i n t s on the wave profile u = c(x -n) move with speed c. The wave profile u is not constsnt on the charactensticlines.

function, such that f ( l ) = O .

Nonlinear Poniol DrbYemnrial Eqmlzons

551

(b) Characteristicsare xy = constanl (a family of hyperbolas). u=nlogy+f(xy), (c)

u = t

Att=O.p=O. q=L:

.g($)+2(~j.

sp=tanh((5).

.(,,,)=-2,’.”(5)sech(5). u(s.r)=Zrrech(f), 12. Hint:

I dp

sq=sech((5).

y(s,l)=Zs’sech’ (:)-s’. (“’-2yY =4(x*+y’).

Wrile the characteristic strip equations

I= F,. ...

Since

I dq

_ _dI= _ _ d I , p and q are proportional to each other. The initial data

p&) and qa(s) are !he solutions of p o ( s ) = O and ( p i + q i ) i =tan 8 so thal p,(s)=Oandq,(s)=tan8. Since p , ( s ) = O , the constant of proportionality must be zero, and so p ( l , s ) = O . We then find x(I.s)=s

and

r(t,s)=cas(8-r).

q ( f , s ) = Lm(8 - 1)

Then

~=-[q2+qcos(8-r)]

y(1.8) = - a 4 8 - l ) + Zsin 8

.

gives

or (y-Zsin8)*+z2 = I whichFepressntsaeylinder. 14. Hint:

($)=d(s)’,

d ( l - r 2 ) t = * d ( ~ .)%=+fId ( r . o s 8 + y r i n 8 ) .

5.5 1.

Exercises

(a) u(x.r)=sin<,

<=x-tsin<.

(b) u ( x . r ) = l , u + -

as 1-1.

1-1

2. Hint: (u), +(id) = O . Multiplying 11 by n u”.’ gives the first result. 2 , 4.

<=r-lcrp(-~)

forx>0:

.(..l)=..p(-~)

A m e n and Hints 10 Selected fieroses

552

7.

u = I + A . x = .I-"+ B whcreA andB an constants

8. Hint: Shock r = Z ( x - l )

from (1.0) to (2.2). and shock x = h I from

(2,2). u = O f o r x < O , and. in x > O , u = ~ to the left of x=

9.

h I.u = 0

to

I=X

and

the right of shocks and u = I in a triangular region

boundedby t = O , r = x , and t=Z(x-I). u = - l totheleft oftheline r + x = O . a n d u = l tothe r ightof the line ~ - r - l . B ~ t ~ ~ e n t h ~ ~~~- (I2m ~ -c1s) /, ( 1 + 2 r ) .

10. Characteristics from O I X S I . 1 = 0 intersect at

(a,;).

Characteristics

from x SO,x 2 1 also intersect in the region I 2 1 and hence, no singlevalued solution. Shock parallel to the +,I)

=(Zx-l)/(1-21)

r-axis from

(+,+).

m the triangular region x - I =0, X + I = I, r = O .

},

u ( x , I ) = { ~ l , totheleftofshock 1, to thenghtofshack 1 1 . When o 0. - a to the left of the line x =-
I = 201.

Between these lines. u =

I

200.

The graph of p(x.0) is linearly decreasingin x from

( -). *).

25 3+x-151

'

1

I<-(M)I+&)

--E

-1+301Lx<151'

1-151

p(x.1) =, 25(,-

I51 S x S 1+301 ,

x+15I

50,

Ix- 30t lt I

to

+&.

Anrwcrr md Hints to Selected Erercrsrr

554

12. Hint: The equationsm the mahix form IS

The eigenvalucs are thc rwu of the equation u-1

e'p

0

13. Hint d u + A , , d v = O alongthecharacteristics C,md C,,resptivelY. a(du)'+Zb dudu+c(du)' = O whlch corresponds to the characteristics

a(&)' - Zbdxdr+ e(dt)* = 0 .

dx = 1dr 14. Hint: Along

the characteristics

%+Lau+=O dr 2

q=x-cr.

C, : q=x-cr=constant,

Similarly, C. :<=x+Cr=constmt,

15. Hint: p = p ( x ( r ) , r),

and

with the solution u + ( r ) = ~ , ( O ) c r p ( - ~ n t ) with ++inu.=~

*=*+&.*, and then. use equations (6.3.1) dr 21 dr JX

and (6.3.2).

16. Hmt: d,, =ufe,

(2

+1 b

I,2 =(I, f 2 ) . (n+Ze)=H,

along C, :d"=n+c.

(u-2c)=/f2

along C.

'Jx)

dr

:&=U-C.

dr

18. Hint:SeeJeffreymdTaniuti (19M),pages72-73.

19. Hint: Equate x,, and x,, to derive ulc Euler-Poisson-Darbour equation. 22. Hint. The fimt equation can bt written as * + d i v ( p . p u , p w ) = O

Nonlinear Parlid DcflennnOl4~rions

555

The f i t and second equations can be combinedto find the result

puw

pvw

p+pwl

The above two resulll lead to the desired conservation form.

1] , 2

2

end. for both cases (i) and (ii),

26. Characteristicsdrawn fmm the negative x-axis are vertlcal lines Pconstmt. and,alongthesccharaetcnsties, p=po. and p(x,f)=p, for x c O . t > O . Characteristics drawn from points x>n satisfy i(t)=.(p(.,o).o)=,(o)=p~. I" x > n + p m r . r>o, p(x,t)=o. i n o<x
556

A m w m and Hmfr to Lleocd Eurczser

8.14 Exercises 1 . (a) a = p = l a n d y = Z , (b) P = 2 a (c) y = a - P . (dl p = 2 a . 2. In D,andfor c > Z , u ( S ) = A e x p ( m , S ) + R e r p ( ~ 5 ) , where m,. m, = "-c,

+GI.

In D2, "(5)=l+A,e~p(',5)+R,ex~(n,5),

To determine A, R, A, and 8,. we use (i) "(0)= (ii) I values ,of "'(0) in

D,and D2arcequal,and(iii) u(--)=landu(-)=O. In D,for 5 > 0

.(5)='-

4G

[{i'+2c-Je'}erp(m,i;)

+{G - 2 e +fi}erp(m,<)]. In D2for 5 < 0 , u ( 5 ) = I - i e x p ( n , < ) . When c = 2, we find solutions in D,end m D2

in

for

<
8. Hint: @=o'*'yr. and sets = 0 to find the similarity solutions. 9. Hint: The boundary conditions arc u(q=O)=Oandu(q+-)=l u('I=0)=v,. 11. """+f(q'""+S7"'+3V)=0

Bibliography

The following bibliography is not, by any means,

a complete one for the subject. For the most p m . it consists of books and papers to which referenceis made in the text. Many other selected books and papers related to matend of the sublect have been included so that !key may serve to stimulate new interest in future SNdy and research.

Abdelkadcr. M.A. (1982). Travelling wave solutions for B generalized Fisher equation. I Moth. Anal. Md Appl. 85, 287-290. Ablowitz, M.J. and Clarkson. P A (1991). Solitons, Nonlinear Evolution Equnrionr ond hverrs Sconering. Cambridge University Pnss. Cambridge. Ablowitz. M.J. and Habeman. R. (1975b). Nonlinear evolution equations - two and three dimensions, Phys. Rev. Len. 35, 1185-1188. Ablowiu, M.J. and Scgur. H. (1981). Solitons and the Inverse Scattering Tronrfomt. SIAM Publications. Philadelphia. Ablowitz, M.J. and Zcppetella. A. (1979). Explicit solutions of Fisher's equation for a special wave speed. Bull. Moth. B i d . 41, 835-840. Ablowitc. M.J., &up, D.J., Newell, A.C., and Segur, H. (1974). The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Moth. 53. 249-315. Ablowrtz, M.J. and Segur, H. (1979). On the evolution of packets of water waves, J. Fluid Mech. 92.691-715. Abramowitz. M. and Stegun, I A. (1964). Handbook of Mothemorieol Functions. NBS. Appl. Math. Series 55. Washington. D.C. AkylaJ, T R . (1994). Three-Dimensional long water-wave phenomena. Ann. Rev. Fluid Mech. 26, 191-210.

560

Bibliography

Ames, W.F. (1965). Nonlinear P a n i d Differential Equations in Engineering, Vol. I, Academic Press,New York. Ames, W.F. (1912). Nonlinear Poniol Differential Eqmtiom in Engineering. Vol. 11, Academic Press. New York. Ammeman, A.J. and Cavalli-Sforza, L.L. (1911). Measuring the rate of spread of early farming, Man 6,614-688. Ammeman, A.J. and Cavalli-Sfana, L.L. (1983). The Neolithic Transition and the Genetics of in Eurooe. , Pooulorions , . . Princeton Universitv Press. Princeton. Aaki, K. (1987). Gene-culture waves of advance. 1. Moth. Bid. 25.453-464. Arecchi, F.T.. Masaerini, G.L., end Schwendimann, P. (1969). Electromagnetic propagation in a resonant medium, Riv. Nuovo Cimcnto. 1. 181-192. Ark, R. (1975). The Mothemoricol Theory of Diffusion and Reaction in Permeable Caralyst. Oxford University Press. Oxford. h o l d . R., Showalter, K., and Tyson. 1.1. (1987). Propagation of chemical reactions in space, J. Chcm. Educ. 64,140-142. Aronron, D.G. (1986). The pomus medium equation. in Nonlinear Diffusion Problems, Lecture Notes in Mathematics, No. 1224, Springer-Verlag.New York. 1-46. Amnson. D.G. (1980). Density-Dependent interaetion-diffusion systems in Dynamics ond Modelling of Reactive Systems (ed. Stewart W.E., Ray. W.H. and Conlcy. C.C.). Academic Press. Boston. 161-116. Assno. N. (1910). Reductive perturbation method for nonlinear wave propagation in inhomogeneour media Ill. J. Phys. Soc. Jopnn. 29, 220224.

Arano, N. (1914). Modulation for nonlinear w w e in dissipative or unstable media. J. Phys. Soc. Japan, 36, 861-868. Asano, N. and Om, H. (1911). Nonlinear dispcrsivc or dissipative waves. 1. Phys. Soc. Japan, 31, 1830-1836. Asano. N. and Taniuti, T.(1970). Reductive penurbation method for nonlinear wave propagation in inhomogeneous media 11. 1. Phys. Soc. Japan. 29.

209-214.

Asano. N., Taniuti, T. and Yajima. N. (1969). A pcnurbation method for a nonlinear wave modulation 11.1. Math. Phys. 10, 2020-2024. Atiyah. M.. Gibbon, I.D. and Wilson. G. (1985). New developments in the theory and applications of solitons, Phil. Trans. Roy. Soe. London, 315.

___

797.469

Balmforth, N.I. (1995). Solitary waves and homoclinic orbits, Ann. Rev. Fluid Mech. 27, 335-373. Barenhlatt, 0.1. (1919). Similarity. Self-similarity and Intcrmsdiotc A~ymptotles.Consultants Bureau, New York. Barcnhlatt, G.I. and Zcl'dovieh. Y.B. (1912). Self-Similar S O I Y I I ~ ~ as S intermediate asymptotics, Ann. Rev. Fluid Mcch. 4,285-312. Barone, A,. E~posito.F., Magee, C.J., and Scott. A.C. (1911). Theory and applications of sine-Gordon equotion, Rivista del Nuovo Cimento, 1, (2). 221-261.

561

Nonlinear Ponioi D~,qerenriolEquorions

Beberneo. 1. and Eberly. D. (1989). Mothemoticol Problems from Combuslion Theory, Spnnger-\ierlag. New York. Benjamin,T.B.and Lighthill. M.I. (1954). On cnoidal waves and bores. Pror Roy. Soe London. A224, 448-460. Benjamin. T.B.(1967). Instability of periodic wave m n s m nonlinear dispersive wtems. Sor London. ~,~ ~,P m r Rnv. ~, . A299.. 59-75. Benjamin, T.B. (1992). A new kind of solitary wave, 3. Fluid Mech. 245. 401~

~

~~

~~~

~

~~

411.

Benjamin,T.B.and Fea. I.E. (1967). The disintegration of wavetrains on deep water, Pan I , Theory,3. Fluid Mech. 27, 417-430. Benjamin. T.B..Bana, I.L., and Mahony, 1.1. (1972). Model equations for long waves in nonlinear dispersive systems. Phil. Trans. Roy. Soc. A272, 4778. Benncy, D.J. (1966). Long waves on liquid films, I Moth. Phys. 45. 150-155. Benney, D.J. and Rosbs. G. (1969). Wave instabilities. Studies Appl. Math. 48, 377-385. Ben-Jacob. E., Brand, H., Dee 0.. Kramer, L.. and Langer, I S . (1985). Pattern propagation m nonlinear dissipative systems, Phyrica D , 14, 348-364. Benton, E.R. and Platzman, G.W. (1972). A table of S O ~ U ~ ~ofO ~oneS dimensional Burgers equation, Pun.Appl. Math. M,195-212. Bcrezm, Y.A. and Kgpman, V.I. (1966). Nonlinear evolution of disturbances in plasmas and other dispersive media, Soviet Phys. JETP. 24. 1049-1056. Bespalov, V.I. and Talanov. V.I. (1966). Filamentary smcture of light beams in nonlinear liquids, J E T P k m 3. 307-310. Bhatoagar, P.L. (1979). Nonlineor Wows in One-Dimensional Disperrivc Systems. Oxford University Press,Oxford. Bhatnagar, P.L and Prasad, P (1970). Study of the self-similar and steady flows near singularities. Pan I, Proc. Roy Soc. London. A315; (1971). Part 11. Proe. Roy. Soe. London. (1971) A322, 45-62. Birkhoff, G. (1950). Hydrodymmrcs. Rinccton University Press,Princeton. Birkhoff, G.D. (1927). Dynomicol Syslem, American Mathematical Society,

Providence. Blackstock. D.T. (1964). Themoviscous attenuation of plane, periodic finiteamplitude sound waves. 1 Acoust. Soc. Amcr. 36,534-542. Bluman, G.W. and Kumci. S. (1989). Symmetries and Diferentiol Equations, Spnnger-Verlag. New York. Born, M. and Infeld. L. (1934). Foundations of a new field theory. Proc. Roy. Soc. London, A144, 425-451. Boussinesq. J. (1871a). Theorie de I'intumesccncc liquide appelb onde solitaire ou de translation se propageant dans un canal rcctangulaire. Compres Rendus, 72, 755-759. Boussinesq, M J. (1871b). Th6one gcneralc dcs mowenmenti qui smt propages dam un canal nctangulaire horizontal. Comptes Rendus, 72.256-260. Boussinerq. M.I. (1872) Thhdorie des ondes et des ...fond., 3. Moth. Pure Appl. Ser. 12) . , 17.. 55-108. Britton. N.F. (1986). Reaction-Diffusion Equations and Their Applications to Biology, Academic Press,New Yark.

561

Bibliography

Burgers, I.M. (1948). A mathematical model illustratine. the theory of turbulence, Ad”. Appl. Mech. 1, 171-199. Burgers, I.M. (1974). The NonlinearDi@ion Equnrion. Dordrecht: Reidel. Bunon, C.V. (1893). On plane and spherical sound waves of finite amplitude, Phil.. Mar 35. 317-333 Calogero, F. and Degasperis, A. (1978). Solution by the spectral-transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation, Lett Nuovo Cimento. 19, 150-154. Campbell, G.A. and Foster, R.M. (1948). Fourier Inrexrols .~for Procticol Applicationr, Van Nostrand. New Yak. Cano~a,1. (1969). Diffusion in nonlinenr multiplicative media, J. Math. Phyr. 10. IR62-IR6R Csnosa, I. (1973). On a nonlinear diffusion equation describing population growth, IBM 3. Rcs. Develop. 17, 307-313. Case, K.M. and Chiu, C.S. (1969). Burgers’ turbulence models, Phys. Fluids ~

~~~

~

~~~~

~

1

.

~

.

~~~~

12, 1799-1808.

Chen, Y. and Liu. P.L.-F. (1994). A pseudospcctral appmach for scattering of water waves. Proc. Roy. Soc. London, A445 619-636. Chen, Y.and Liu, P.L.-F. (1995). The unified Kadomsev-Petviashvili equation for interfacial waves, 1. Fiuid Me& 288, 383-408. Chen, Y. and Liu. P.L.-F. (1995). Modified Boussincsq equations and associated parabolic mcdels for water wave propagation. J. Fluid Mech. 228, 351-381. Chen, Y. and Wen, S. (1987). Traveling wave solutions to the two-dimensional Korteweg.de Vries equation, J. Math. Anal. Appi. 127,226-236. Choudhury, S.R. and Dehnath, L. (1994). Painleve analysis. similarity reductions. and s p i a l solutions of nonlinear evolution equations, In Ocean Wovcs Engineering (ed.M. Rahman), 2. 115.133. Chu, V.H. and Mei, C.C. (1970). On slowly-varying Stokes waves, J. Fluid Mech. 41, 873-887.

Chu. V.H. and Mei. C.C. (1971). lle nonlinear evolution of Stokes waves in deep water, J. Fluid Mech. 47, 337-351. Cole, J.D. (1951). On a quasilinear parabolic equation occurring in Berodynamics, Qunn AppL Moth. 9.225-236. Cornish, V. (1934). Ocean Waves and Kindred Geophysical Phenomena. Cambridge University Press, Cambridge. Courant. R. and Wedrichs, K.O. (1948). Supersonic Flow and Shock Waves, Interscience,New Yo& and London. Courant. R. and Hilben. D. 11953). . , Methods ofMothemotico1 Phvsicr. , . Vol. 1.. Wiley-Interscience.New York. Courant, R. and Hilbert, D. (1962). Mcthodr of Morhcmnricnl Physics. Vol. 2 , Wiley-Interscience. New York. Cramer, M.S. and Sen, R. (1992). A general scheme for the derivation of evolution equations describing mixed nonlinearity. Wove Mofion. 15.333355.

Crandall. R.E. (1991). Marhemaricaforthe Sciences, Addison-Wesley. Redwood City, CA.

Nonlinaor Partial Dig.mnriol Qmrions

563

Crank, 1. (1975). The Mathematics of Diffusion (Second Edition), Oxford Univmity Rcsr, Oxford. Clighton, D.G. (1979). Model equations of nonlinear acoustisr. Ann Rev. Fluid t , 11-21 .Mm-h ...... I.. .. . ..

Crighton, D.G. and Scott, J.P. (1979). Asymptotic solution of model equations in nonlinear acoustm, Phil. Trons. Roy. Soc. London, A292. 101-134. ~m~agation Dave". A. (1912). The . . _ of B weak nonlinear wave. J. FIuLl Mcch. 53, 769-781. Davey. A. and Stewareton. K. (1914). On three-dimensional packets of surface waves, Proc. Roy. Soc. London, A338, 101-110. Davis, H.T.(1962). Intmduction to Nonlinear Differential and Integml Equnrions. Dover Publications, New York. Davis. R.E. and Acrivos. A. (1967). Solitary internal waves in deep water. 1. Fluid Mech. 29. 593-607. DeLeonardis, R.M. end Trullinger. S.E.(1980). Theory of boundary effects on sme-Gordon solilons, J. Appl. Phys. 51 1211-1226. Debnath. L. (1983a). Nonlrneor Waves, Cambridge University Press, Cambridge.

Debnath, L. (1983b). Nonlinear dynamics of water waves and the nonlinear Schredinger equation, Diamond Jubilee Volume, Calcutta Mathematical Saeiety. pp. 63-93. Debnath. L. (1984). Advances in Nonlinear Wrmcs.Vol. 1. Pitman Publishing Company. London. Debnath, L. (1985a). Advances in Nonlinear Waves, Vol. 11, Pitman Publishing Company. London. Debnath, L. (1985b). Nonlinear dispersive waves and variational pnneipler. In Differemiol G ~ O W I V ,Calculus of Variations and Their Applications. (d. G.M Rassias and T M. Rassias) Marcel Dekkar, pp. 149-176. Debnath, L. (1987). Bifurcation and nonlinear instability in applied mathematics, In Nonlinear Analysis, (ed. Th.M. Rassias), pp. 161-266, World Scientific Publishing. Singapon. Dcbnath, L. (1989). The linear and nonlinear Cauchy-Poisson wave problems for an inviseid or viscoui liquid, In Topics in Mothcmaticol Anolysis (ed.T.M. Rssnas), pp. 123-155, World Scientific Publishing, Singapore. Debnath, L. (1992a). Nonlinear Dispersive Wove Systems. World Scientific Publishing, Singapore. Dcbnath. L. (1992b). Solitons and the inverse scattering uansfonn, Bull. Cal. Moth. Soc. 84, 475-504. Debnath, L. (1994). Nonlinear Water Waves, Academic Ress, Boston. Debnath, L. (1995). Inrenrol Tmnsfons and Their Applications, CRC Press, Baea Raton. Flonda. Dcbnath, L. and Choudhury, S.R. (1997). Nonlinear Instability Analysis. Volume 1. Computational Mechanics Publications. Southhampton, Eneland. Deb&, L. and Mikuanrki. P. (1990). lnrmducrion to Nilben Spaces with Applications.Academic Press. Boston.

564

Bibliography

Debnath. L. and Rehman, M. (1981). A theory of "onlinear wave loading on offshore structures. Inrcmar. 1.Morh. nndMarh. Sei. 4.589-613. Debnalh. L. and Shivamoggi. B.K. (1985). Some nonlinear model equations in dispersive systems and !heir solitary wave solutions in fluids and plasmas, In Advances in Nonlinear Waves 11, (ed. L. Debnath), Pitman Advanced Publishing Program, Boston, pp. 309-347. Debnath, L. and Shivamoggi. B K. (1986). Three-dimensional nonlinear Schrbdinger equation for finite-amplitude gravity waves in a fluid. Nuovo C m m t o , 94B, 140-148:(1987)998,247. Dcbnath, L. and Shivamoggi, B.K. (1990). Pseudo-variational pnnsiples and nonlinear surface waves in a dissipaltve fluid, Inrcmr. 1. Nonlinear Meek 25, 61-65. Dodd, R K. and Gibbon, I.D. (1977). The prolongation ~tmcturesof a higher order Konewcg.de Vrics equation, Proc. Roy. Soc. London. A358,287296. Dodd, R.K. and Fordy, A.P. (1983). me prolongation ~ t r u ~ t ~ rofe squasipolynomial flows. Pmc. Roy. Soc. London. A385 389-429. Dodd, R.K. and Fordy, A.P. (1984). Prolongation smetures of complex quasipolynomial evolution equations, 1. Phys. A,: Mark Gen. 17, 3249-3266. Drain. P.G. (1983).Solifom, Cambridge University Press, Cambridge. Drazin, P.G. and Johnson. R.S. (1989).Solirons: An Inrroduerion. Cambridge University Prerr, Cambridge. Dresner, L. (1983). Similarity Solurions of Nonlinear Ponial Differenrid Equtiom, Piunan Books, London. Drersler. R.F. (1949). Mathematical solution of problem of roll waves in inclined open channels, C o r n , Pure Appl. Mah. 2. 149-194. Dunbar, S.R. (1983). Travelling wave solutions of diffusive Lotka-Voltena equations. 1 Morh. Bio. 17, 11-32. Dutta, M. and Debnath. L. (1965). Elemenrs of rhr Theory of Nlipric and Assockred Funcrionr with Applications. World Press Pub. Ltd.. Calcutta. El-Sherbiny. H.M. and Debnath, L., (1997). wuivalence between Bkklund transformation and inverse scattering methods for a class of nonlinear differential equations, Inremot. Jour. Nonlinear Mech. 31.51-58. Engelbrecht, J. (1981). On theory of pulse mansmission in a nerve fibre, Proc. Roy. Soc. London, A375, 195-209. Engelbrecht. 1.K.. Fridman, V.E.. and Pelinovski, E.N. (1988). Nonlinear Evolurion Equ(lriom,Longman. London. Engelbrecht, I. and Peipman, T.(1992).Nonlinear WBVCE m a layer with energy mflux. Wove Morion 16. 173-181. Enz, U. (1963).Discrete mass.elementary length. and a topological invariant as B eonsequenes of a relativistic invariant principle. Phys. Rev. 131,13921394. Estrada, R. and Kanwal, R.P. (1994).Asymproric Analysis. Birkhauser, Bolton. Fermi. E., P a m , I., and Ulsm. S.(1955). Studies of nonlinear problems I, Los Alamos Report LA 1940, Leetuns in Applied Mathematics (ed. A.C. Newell). Amcr. Moth Soc. (1974)15, 143-156.

Nonlmor Pam01 Diffrrcnml Eyvrlions

565

Fickett. W. and Davies, W.C. (1979). Deronnrion, University of California Press, Berkeley. Fife. P.C. (1979). Mothrmticol Aspects of Reacting and Diffusing Sysrcms, Lecture Notes in Biomathematm. 28, Springer-Verlag. Berlin. Fisher. R.A. (1936). The wave of advance of advantageous genes, Ann. Eugenics. 7, 335-369. FmHugh. R. (1961). Impulses and physiological ~tatcsin theoretical models of nerve membrane. Btophys. 3. 1,445-466. Fobs, A.S. and Ablowitr. M.J. (1983). The inverse seattenng transform for the Benjamin-Om equation - a pivot to multidimensional equations. Stud. Appl. Math. 68, 1-10, Fordy. A.P. (1990). Prolongation structures of nonlinear evolution equations, Solifon Theory: A Survey of Results (ed. A.P. Fordy), Manchester University Press. Manchester, pp. 403-4W. Fomberg. B. and Whitham, G.B.(1978). A numerical and theoretical study of certain nonlinear wave phenomena, Phrlor. Trms. Roy. Soc. London, A289, 373-404. Freeman, N.C. (1980). Soliton interactions to two dimensions. Ad". Appl. Mech. 20 1-31 Frenkel. I. and Kontorova, T. (1939). On the theory of plastic deformation and twinning. Fiz. Zh. 1, 137-139. Fieman. E.A. (1963). On a new method in the theory of imversible processes. 3. Math Phys. 4, 410-418. Fulton, T.A. (1972). Aspects of vortices in long Josephson junctions, Bull. Amer. Phys. Soc. 17, 46-49. Ombedsan. P.R. (1986). Partial Diflerentiol Equations. 2nd Edition. Chelsea, New York. Gsrdner, C.S., Gnenc, J.M., Kruskal. M.D., and Miura, R.M. (1967). Methad for rnlving the KdV equation, Phys. Rev. Lert. 19, 1095-1097. Gardncr. C.S.. Greene, J.M., Kruskal, M.D., and M w a , R.M. (1914). Korteweg-dc Vries equation and generalnations. VI. Methods for exact solution, Comm Pure Appl. Moth. 27. 97-133. Gardner, C.S. and Morikawa, G.K. (1960). Similarity I" the asymptotic behavior of ollirim-free hydromagnetic wave$ and water waves. Courant Inst. Morh. Sci. Rep. NYO-9082, pp. 1-30. Gazdag. 1. and Canosa, 1. (1914). Nummcal solution of Fisher's equation, 1. .Ann! . r r . Pmh ..I.11 44<.4<7 Gms. D C (1971) TrofirSnmrr. John Wilcy, Nru Yuck. ( i c l f d . I M and Lcbitan. B M (1951) On thc dclennmation of a diffcwnual cauitlitin from IIS soccnal funruon. Am Moth Tnrnrl r2r I 251-304 Ghez,k. (19W.A Prikerof Di@sion'Pmbl& Ghn k ley:New York. Ghosh, K.K. and Debnath, L. (1997). Some exact solutions of nonlinear shallow water equations. Internal. 1.Nonlinear Mech. 31, 104-106. Oimfon. ID.(2985). A suwsy of rhe origins and physical importance ofsoliton equations, Phil. Trans. Roy. Soc. London,A315. 335-365. Gibbon, J.D., James, I.N.. and Moroz, I.M. (1979). An example of soliton behaviour in a rotating fluid, Proc. Roy. Soc. London, A367, 185-219.

...

566

Btbliogrophy

Gibbons, I., Thornhill. S.G., Wardrop, M.I., and ter Hmr, D. (1977). On the theory of Langmuir solitons. J. Plasma Phys. 11. 153-170. Gillbarg, D. and Trudmger, N.S.(1983). Elliptic Partial Differential Equotionr of Second Ordrr. 2nd Edition. Spnnpe~Vcrlag.Berlin Goldstem. H 1 IY65r. C l u r ~ aUl s r h m i r r . Addison-Wolc),Camhndgr. Goldstcon. S 11013, On the marhematics of crchange proecrrcr tn fmed columns. Pan\ I and II.Proc Rur So< London. ,1219, 151-185 Gold-tein. S and Munay. I D t I J S Y j On the mathemaiic~of exchange pn,.'cwc* ~n Oxcd c h m n . , 1'3rh 111. IV. V. Pmr R o y . Soc London. AZSZ. 334-375. Gursrhkov. K A,. O\truv.ky. L A. and Pclmovrky. E N . (1914, Sumc prohlcmr of esymplolc theory of nonllncar WBVCS. Proc IFEE. 6 2 , 151 I. 1517 t i ~ e n k r gH . (1959) An analyrii of traflic llo-, O p u Rer 1.79-85 Crm3haw. R. (1985) Evulutiun equatmns fur wcakly nonlinear. long internal

umes in a rotating I l J d Stud Appl Uoth 7 3 , 1-33 Gnnmd. P (1991 Arrrmrund Wowr. Oxlord Univeniiy Rc$r.Oxford Gurney. W S C and Nnhct. K M (1915).The r:pulmun of tnhomogznzuus population\. J Theor Bio 52. 441.457 Curtin. M E . and MirCamy. R C 119171. On the dilfuimn of biological popula!ion?, Umh Rmrrtmrrr. 33. 35-49 Habcrmm. K ( 19XR) Thc muduldted phuc FhiR for weakly dtwpated nonlmar O I C I I I I I O ~ u d $ c b of !he Koncwcg.dc Vncs t)pc. Stud Appl U d h 78. 71911

Hagslrom. T. and Kcllcr. H B (1986) Thc numerical caIc"Iat!om of mvclling w m e iolutions of nonlin~arparabolic equations (Rcpnnll Hanghi. F A (19631 M o t h e r n a r d Theonrr of T m f i c F b w . Academic Prerr. New York Hammack. I L.and Scgur. H . (1974) 7he Koncweg.de Vnercqualmn and walcr U I V C P , Pan 2, Cornpariron with cxpcrtmcnu. I I.?ud Mcrh. 65. 289-314 Hanrcn. A G (1964, P m h n n Ano1)rer of B o u n d q V d u e Problrml in Enginernng. I'rcntue Ilall. Engleuud Cliffs. New l e r ~ c y H a q a u r . A (1990, Optwul Solmnr m h b e r ) . (Liccond Edition). Spnngo \'erlag, Rerll" Harcpaua. A and Tappcn, b (1973). Tranmhslon o f stationary nonlinsar opw31 pul~c.tn di~pcrsivcdielccmc fibres. 1 Anomalous dmpcrrion. Appl Ph>r L a 1 23. 142-144 Hawnciio. tI (1972). A sol~tooon a Y D ~ C Xfilament. J Fluid Uerh 51. 417485 Hawnolo. H and On". H. (1912) Nonlinear modulation of graviry umcs. I Phys. Soe. Japan, 33. 805-811. Hayes, W.D. (1973). Group velocity and nonlinear dispersive wave propagation, Pror Rov .%r Iflnndon. . . .., 6332. 199-221 . Herman, R., Montroll. E.W.. Polls. R.B., and Rothcry, R.W. (1959). Traffic dynamics: Analysis of Stability i n car following, Opcr. Res. 7, 66.106. ~~~

Nonlrnrar Ponial Diffrrrnriol Eqquorions

567

Hernandez. J. (1986). Qualitative methods for nonlinear differential equations. in Nonlinear Difision Problems. Lecture Notes in Mathematics. No. 1224, Springer-Verlag. New York. pp. 47-1 18. Himta, R. (1971). Exact solution of the Korteweg-deVrics equation for multiple collisions of solitons, Phys. Rev. Lett. 27, 1192-1194. Himta, R. (19739. Exact envelope-soliton solutions of a nonlinear wave equation, J. Moth. Phys 14. 805-809. Himta, R. (1973b). Exact N-solutions of the wave equation of long waves in shallow water and in nonlinear lattices. J. Math. Phyr. 14, 810-814. Himta. R. and Suzuulu, K. (1970). Studies on lattice ~ o l i t ~ nbys using clcemsal networks, J. Phys. Soc. Japan. 28, 1366-1367. Himta. R. and Sumki, K.(1973). 'Illheoretical and experimental studies of lattice solitons in nonlinear lumped networks,Proc. IEEE, 61, 1483-1491. Hodgkin. A.L. and Huxley. A.F. (1952). A qualitative description of membrane current and its applications to conduction and excitation in nerve. J. Phyriol. (London) 117. 500-544. Holliday, D. (1973). Nonlinear gravity-capillary surface waves in a slowly varying current, J. Fluid Mech. 57,797-802. Hopf, E. (1950). The pmial drfferential equation u, +xu. =pun. Comm. Pure Appl. Moth. 3, 201-230. Hoppensteadt. F.C. (1975). Mothcmotrcol T h e o r r r r of Populations: Demographics, Genetics and Epidemic. CBMS Lectures Vol. 20. SlAM Publications, Philadelphia. Ichikawa. Y.H. (1979). Topics on solitons in plasmas. Physico Scripta. 20, 296-305. Ichikawa. V.H., Imamura, T., and Taniuti. T. (1972). Nonlinear wave modulation in collisionless plasma, 1. Phys. Soc.Japan. 33. 189-197. Iizuka, T. and Wadati, M. (1992). Shallow water waves over an uneven bonom and an inhomogeneous KP equation, Chaos, Solitons. and Fraetolr, 2,575582.

Infeld, E. (1980) On three-dimensional generalizations of the Bousiinesq and Koewcg.de V t k equations. Quon Appl. M o t h , Vol. XXXVIII, 277-287. Infeld. E. and Rowlands. G. (1990). Nonlinror Waves, Solitons and Choos, Cambndge University Press. Cambridge. Infeld, E., Ziemkiewicr, I., and Rowlands. 0. (1987). The two surface dimensional stabililv of Deriodic and solitar~w a v m a i n ~on a wafer surface over arbitrary depth: Ph;s. Fluids, 30,23362338. Inouc, Y. and MafSumolo. Y. (1974). Nonlinear wave modulation in dispersive media. J. Phys. Soc. Japan, 36. 1446.1455. J e t k y . A (1976) Quadlinear Hyperbolic System and Waves. Research Notes m Mathematics 5, Pitman Publishing Company, London. Jeffrey, A. and Engelbrecht, J. (1994). Nonlinear Waves in Solids, SpringerVerlag, New York. Jeffrey. A. and Kakutani, T. (1972). Weak nonlinear dispersive waves: A discussion centered around the KdV equation, SIAM Rev. 14,582-643. Jeffrey, A. and Kawahara. T. (1982). Asymptotic Methods in Nonlinear Wove Theory,Piman Advanced Publishing Program, Boston. ~~~

568

Biblrogrophy

Jeffrey. A. and Taniuti. T. (1964).Nonlinear Wave Propagarion, Academic Press. New York. John, F. (1982). Paniol Differential Equnlions, 4th Fdition, Springer-Vcrlag, New York. Johnson, R.S. (1970). A nonlinear equations incorporating damping and dispersion, 1. Fluid Mech. 42.49-M). Johnson, R.S. (1980).Water waves and Korteweg-de Vries equations, J. Fluid Mech. 97,701-719. Johnson, R.S. (1983). The KdV equation end related problems in water wave theory, In Nonlinear Waves (ed. L Debnath), DD. 25-43. Cambridge University Press,Cambridge. Jones. D.S. (1966).Generalized Funefiona, Academic Ress, New Ynk. Jones. D.S. and Slecman. B.D. (1983). Differential EqUnrions Md Mothcmnficol Bioloev. ”, . Allen and Unwin. London. Josephson. B.D. (1965). Supercumnts through barriers. Adv. Phyr. 14. 419451. Kadomuev. B.B. and Pctviashivili, V.I., (1970).On stability of solitary waves ~n weakly dispcrrivc media, Dokl. A k d . Nouk SSSR 192.753-756So”. Phys. Dok. 15, 539-541. Kako. F. and Yajima, N. (1980). Interaction of ion-acoustic solitons in multidimensional space, J. Phys. Soc. Japm, 49.2063-2071. Kakutani, T. (1971a).Effect of an uneven bottom on gravity waves, 3. Phys. Soc. Japan. 30, 272-276. Kakutani. T. (1971b). Weak nonlinear magneto-acoustic waves in an inhomogeneous plasma, 1. Phys. Soc. Japan. 31, 1246-1248. KabIani. T. and Ono. H (1969). Weak nonlinear hydromagnetic waves in a cold collision-free plasma, J. Phys. Soc. Japan. 26, 1305-1318. Kakutani, T.. O m , H.. Taniuti, T. and Wen, C.C. (1968). Reductive perturbation method in nonlinear wave propagation 11. Application to hydromagnetlc waves in cold plasma. 3. Phys. Soc. Japan, 24, 1159-1166. Kaliappan, P. (1984). An exact solntion for travelling WWCS. Physien,D11, 368-374. Kametaka, Y. (1976). On the nonlinear diffusion equation of KolmagorovPeuowki-Piskunov ‘ype. Osnk J. Molh. 13, 11-66. Karpman. V.I. (1967). An asymptotic solution of the Kortcweg-dcVries equations. Phys. Len. 2% 708-709. Karpman, V.I. (1971). High frequency clecvomagnetie field in plasma with negative dielectric consUnt, PIosmn Physics. 13.477-490. Kamman. V.I. (.1975.a).. Nonlinear Wows in Disocrsive Media. Pereamon Rcss. London. Karpman,V.I. (1975b).On the dynamics of sonic-Langmuir solitons, Phys. Scnpfo, 11, 263-265. Karpman. V.I. and Krushkal, E.M.(1969). Modulated waves in nonlinear dispersive media. So”. Phys. JETP, 28. 277-281. Kauis, C. and Akylas. T.R. (1987). Solitary intcmal waves in a rotating channel: A numerical study. Phys. Fluids. 30.297-301.

Nonlinear P m h l DzfermmI

569

Qmfions

Kaup. D.J. (1980). The Estabrmk-Wahlquist method with examples of applications, Physica, ID. 391-411. Kaup, D.J. and Newell. A.C. (1978). Solitons as particles. oscillators, and in slowly changing media: B singular perturbation theory. Proc. Roy. Soc. London. A361. 413-446. Kawahara. T. (1973). The derivative-expansion method and nonlinear dirpenivc waves. 1. Phvs. Soc. Jmm. 35. 1537-1544. Kawahara.'T. (1975a). Noniinear self-moddauon of capillary-gravity waves on liquid layer, 1. Phys. Soc. Japan, 38,265-270. Kawahara, T. (1975b). Derivative-expansion method for nonlinear waves on a liquid layer of slowly varying depth, J. Phys. Sm.Japan. 38. 1200-1206. Kawahara, T. and Toha. S. (1987). Pulse interactions in an unstable dirsipativedispersive nonlinear syslem. Phys. Fluids. 31.2103-21 11. Kevorkan. J.Il990). Partial Dtff~08tmlGuntiona. BmLslCole, Pacific Gmvc. califomii Kevorkian, J.K. and Cole, I.D. (1996). Multiple Scale and Stngulnr Pemrrbofion Mefhods, Springer Vedag. New York. Kivrhar, Y.S.and Malomcd. B.A. (1989). Dynamics of solitons in nearly inteerable S,~ V S ~ C ~.~ Rev P. Mod Phvs 61. 763-916. Kalmogorov, A,, Petmvsky. 1.. and Piscunov, N. (1937). A study of the equation of diffusion with increase in the quantity of matter and its application to B biological problem. Bull. Llniv. Moscow. Scr. I n a m f . See A. 1. 1-25. Konno, K. and Wadati. M. (1975). Simple derivation of the Bicklund uansformation from the Riceati form of the invmc method. Prog. Theor. Phys. 53, 1652-1656. Kopell, N. and Howard. L.N. (1973). Plane wave solutionsto reaction-diffusion equations, Stud. Appl. Moth 42. 291-328. Korteweg. D.J. and de Vrics G. (1895). On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary L1

~

I

.

Kulikovskii. A.G. and Slobodkina, F.A. (1967). Equilibrium of arbitrary steady flows at the hansonic pints, PrikI. Mar. Mekh 31, 623-630. Kundu, A. (1984). Landau-Lifschitz and higher order nonlinear system0 gauge generated fmm nonlinear SchrEdinger type equations, J. Math. Phys. 25, 3433-3438.

Kundu, A. (1987). Exact solutions to higher order nonlinear equations through gauge mnsformations. Physico, 25D,399-406. Kynch, G.F. (1952). A theory of sedimentation, Trans. F a r d a y Soe.48, 166176.

Lagersuom. P.A., Cole, J.D., and Trilling, L. (1949). Problem in fhemy of YISCOUS comprcsriblcfluidr. Calif. Inst. Tech. Monograph, Pasedcna. Lake, B.M., Yuen, H.C., Rundgaldier, H.,and Ferguron, W.E. (1977). Nonlmcer dmpwater waves. Theory and crpnimcnt. Pan 2, Evolution of a continuous wave train, 1. Fluid Mech. 83.49-74.

570

Bibliography

Lamb, G.L. (1967). Propagation of ultrashort optical pulses, Phys. Lclr. 25A. 181-182.

Lamb. G.L. (1971). Analytical descriptions of ultrashort optical pulse propagation in a nsonant medium. Rev. Mod. Phys. 49.99-124. Lamb, G.L. (1973). Phase variation in coherent-optical-pulse propagation. Phys. Rev. Lctr. 31, 196-199.

Lamb. G.L. (1980). Elements of Soliron Theory, John Wdey. New York. Lamb. H. (1932). Hydrodynamics. 6th Edition, Cambridge University Press. Cambridge. Landau,L.D. and Lifshitz, E.M. (1959). FIurd Mechanics. Pergamon Press. NewYork. Lardncr. RW. (1976). The development of shock waves in nonlinear viscoclastic media, Prm. Roy. Soc. London,A347, 329-344. Lardner. R.W. (1986). Third order solutms of Burgers equation. Q u ~ RAppl. . Moth. 44, 293-302. Lardner. R.W. and Arya, I.C. (1980). Two generalisations of Burgers equation, Acro Meeh. 37. 179-190. Larson. D.A. (1978). Transient bounds and time mymplotie behaviour of S O I U ~ ~ O ~ SSlAM . J. Appl. Moth. 34, 93-103. Lax. P.D. (1954a). Weak ~ o l ~ t i oofnnonlinear ~ hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7, 159-193. Lax. P. (1954b). The initial value problem for nonlinear hyperbolic equations in two independent vanables, Ann. Morh. Stud. (Princeton) 33.21 1-229. Lax. P.D. (1957). Hyperbolic Systems of ~on~ervalion law. 11, Comm. Pure Appl. Marh. 10. 537-566. Lax, P.D. (1964). Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mark Phys. 5.61 1-613. Lax. P.D., (1968). Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Morh 21,467-490. Lax, P D . (1972). The formation and decay of shoek waves, Amer. Morh. Monrhly. 79, 227-241. Lax. P. (1973). Hypcrbolic Sysrems of Conscrvorion Lows and the Mahemorieol Themy of Shock Waves, Society for lndusmal end Applied Mathematics, Philadelphia Lax, P.D. (1975). Periodic solutions of the Koaweg-de Vries equation, Comm. Pure Appl. Math. 28, 141-188. Lax, P.D. (1978). A Hamiltmian approach to the KdV and other equations, I n Nonlinear Evolution Equaltons, (ed. M. Crandall), Academic Press, New York, pp. 207-224. Leibovich, S. (1970). Weakly nonlinear waves in rotating fluids, J. Fluid Mcck 42, 803-822. Leiboweh, S. and Seebass. A.R. (1972). Nonlinear Waves, Chapter N ,Cornell University Press, Ilhaca. Lc Veque, R.I. (1990). Numerical Merhodrfor Conscmarion Lows, Birkhluser Verlag, Boston.

Nonlinear Pam'd Differenrial Equations

571

Lighthill, M.I. (1956). Viscosity effects in sound waves of finite amplitude. In Surveys in Mechanic$, (ed. G.K.Batchelor and R.M. Davies), Cambridge University Ress, Cambridge. pp. 250-351. Lighthill. M.J. (1957). River waves, Naval hydrodynamics publication 515 Nauonal Academy of Sciences-NationalResearch Councd. Lighthill, M.I. (1958). An Introduction to Fourier Analysis and Generalized F U ~ C I ~ OCambridge W. University Press,Cambndge. Lighth~ll.M.J. (1965). Group velocity. 3. InsI Math Appl. 1. 1-28. Lighthdl. M.J.(1967). Some special cases treated by the Whitham theory, Proc. Roy. Soc. London, A299. 38-53.

Lighthill, M.J. (1978). Waves m Fluids. Cambridge University Press, Cambridge. Lighthill, M.I. and Whitham, G.B. (1955). On kinematic waves: I. Flwd movement in long rivrs; 11. Theory of traffic flow on long crowded mads, Proc. Roy. Soc. London, A229, 281.345.

Logan. I.D. (1994). An Introduction to Nonlinear Paniol Diferentiol Equntions, Wiley-lntemienee. New Yo&. Logan, J.D. and Dunbar, S.R. (1992). Travelling waves in model reacting flows with reversible kinetics. IMA J. Appl. Math 49, 103-121. Logan, J.D. and Shores. T.S. (1993). Steady state soIulions in a model reacting flow problem. Appl. A n d . 48, 273-286. Logan. J.D. and Shores, T.S.(1993). On a system of nonlinear hyperbolic e ~ n ~ e r v a t mlaws n with sources. Math. Models Meth. Appl. Sci. 3, 341358.

LonguebHiggins. M.S. (1978a). The instabilities of gravity waves of finite amplitude in deep water I, Superharmonics, Proc. Roy. Soc. London, A360, 471-488. Longuel-Higginr. M S. (1978b). The instabilities of gravity waves of finite amplitude in deep water 11, Subhmonics, Proc. Roy. Sm.London,A360, 489-505.

Luke, I.C. (1966). A perturbation method for nonlinear dispersive wave problems. Proc. Roy. Soc. London, A292, 403-412. Luke, J.C. (1967). A variational principle for a fluid with a h e surface. 3. Fluid Mech. 27, 395-397.

Ma.Y.-C. (1979). The perturbed plane-wave solution of the cubic SchrMinger equation, Stud. Appl. Moth. 60, 43-58. Madsen, P.A., Murray. R., and Sorensen. O.R. (1991). A new form of the Boussinesq equations with improved linear dispersion characteristics, Coastal Eng. 15, 371-388. Madsen, P.A. and Sorensen, O.R. (1992). A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowlyvarying bathymelry, Coosrnl Eng. 18, 183.204. Madsen. P.A. and Sorensen, O.R. (1993). Bound waves and mad interactions in shallow water, Ocean Eng. 20, 359-388 Makhankov, V.G. (1978). Dynamics of classical solitons, Phys. Rep. 35. 1128.

572

Bibliography

Manoranjan, V.S. and Mitchell, A.R. (1983). A numerical Study of the Belourov-Zhabotinskii reaction using Galcrkin finite element methods, 1. Math. B i d 16, 251-260. McCall, S.L. and Hahn. E.L. (1967). Self-induced mnsparency by pulsed coherent light, Phys. Rev. Lei:. 18. 908.911. McCall. S.L and Hahn. E.L. (1969). Self-induced transparency. Phys. Rev. 183,457-485. McKean. H.P. (1970).Nagumo's equation, Ad". Moth.. 4,209-223. McKean, H.P. (1975). Application of Brownian motion to the equations of Kolmogorov-Peuovskii-Piskunov.Co r n , Pure Appl. Math. 28, 323-331. Mei, C.C. (1989). The Applied Dynamics of Ocean Suflance Waves, World Scientific Publishing. Singapore. Miles, I.W. (1980).Solitary waves, Ann. Rev. FIuidMech. 12. 11-43. Miles, J.W. (1981).The Korteweg-de V r i e ~equation: a historical essay. 1. Fluid Mcch. 106, 131-147. Miura, R.M., Gardner. C.S.. and Kruskal, M.D. (1968). KorteWeg.de Vrics equations and generalizations, 11: Existence of conservation laws and eon~tansof motion, 1. Moth. Phys. 9. 1204-1209. Mollenauer, L.F.. Gordon, I.P., and Islam. M N. (1986). Soliton propagation m long fibres wilb penodieally eompensatcd loss. IEEE J. Quantum Electron 22. 1284-1286. Montmll. E.W. and West. B.J. (1973). Models of population gmwlh, diffusion, competition and narrangemcnt. Synergetic (ed. H. Haken), B.G. Teubner, Stungan. pp. 143-156. Munrer. A,, Burgen, I.R., Gutierrer, 1.. Fijalkow, E., and Feir, M.R. (1981). Group vansfomations and the nonlinear heat diffusion equation. SIAM J. Appi. Moth. 40, 191-207. Murray, J.D. (1968). Singular perturnations of a class of nonlinear hyperbolic and parabolic equations, 1. Math. Phys. 4?. 111-133. Murray, I.D. (1970a).Perturbation effects on the decay of discontinuous solutions of nonlinear first order wave equations, SIAM 1. AppL Moth. 19, 273-298. Munay, J.D. (1970b). On the Gum-effect and other physical examples of permrbed C O O S W ~ Oequations, ~ I. Fluid Mech. 44,315-346. Murray, J.D. (1973). On Burgers' model equation for turbulence, 1. Fluid Meek 59, 263-279. Munay, I.D. (1977).N o n i i n ~ ~ o r - D i f f e r c n r i ~ lModels E ~ ~ t i ~in"Bioloav,Oxford University h s s , Oxford. .. Murray. J.D. (1993). Morhcmoricnl B i o i q y (Second Corrected Edition), Springer-Verlag, Berlin. Mymt-U, T. and Debnalb. L. (1987). Pam'oi Differcnnhi Equntionr for Scientists ondEnrinerrs (Third " . Edition). Rentice Hall. Emlewood Cliffs. N.I. Nagumo, I S . , Arimoto, S.. and Yoshizawa, S. (1962). An active pulse transmission line simulating W N axon, ~ Proc. INL Radio Eng. 50. 2061207I. Nangolnykh. K.A. and Ostmvsky, L.A. (1990). Nonlinear Wave Pmeesscs in Acoustics. Nauka. Moscow.

Nodimor Panid D!TemntmI Epuotionr

513

Nayfeh. A.H. (196%). A perturbation method for treating nonlinear oscillation problems, 1. Moth. Phys. 44, 368-374. Nayfeh. A.H. (1965b). Nonlinear oscillations in a hot eleetmn plasma. Phys. Fiuids. 8. 1896-1898. Nayfeh. A.H. (1971). Thirdharmonic resonance in the interaction of capillary and gravity WBVCS. J. FluidMcch. 48, 385-395. Nayfeh, A.H. (1973). PerturbationMethods,John Wiley. New York. Nayfeh, A.H. and Harsen, S.D. (1971). The method of multiple scales and nonlinear dispersive waves. J. Fluid Mech. 48. 463-475. Nayfeh, A.H. and Saric, W.H. (1971). Nonlinear Kelvin-Helmholtz instability. J. Fluid Mech. 55, 311-327. Newell, A.C. (1985). Solitons in Mothernotics and Physics. SIAM Publications,Philadelphia. Newell. A.C and Kmv. D.I. (1978). Solitons as varticlcs, oscillators, and in slowly changing media: A singular perturbatjon theory. Pmc. Roy, Soc. London, A361, 413-446. Newell, A.C. and Moloney, J.V. (1992). Nonlinear Optics, Addison-Wesley, New Ymk Ncwman. W.I. (1980). Some exact solutions to a nonlinear diffusion problem in population genetics and combustion. J. meor. Bio. 85. 325-334. Newman, W.I. (1983). Nonlinsar diffusion: self-simdarity and travelling waves, Pure Appl. Ceophys. 121, 417-441. NozaLi, K. and Taniuti, T. (1973). Propagation of solitary pulses in intsractioni of plasma waves, 1. Phys. Soc. Jopnn, 34, 796-800. Nuneiato, I W., Walsh, E.K.. Schuler. K.W.. and Barker. L.M. (1974). Wave propagation in nonlinear vissoelastir: solids, (ed. by S. Flflggc) In Encyclopediaof Physics. Vld4. Springer-Verlag.Berlin. Nwogu. 0. (1993). Alternative form of Boussinesq equations for n e a r s h a wave propagation, J. Wafenroys, Pon. Cwsml, Ocean Eng, ASCE. 119,618638. Nye, I.F. (1960). I h e response of glaciers and ice-sheets to seasonal and climatic changes, Pmc. Roy. Soc. London. A 2 5 6 559-584. Nye, J.F. (1963).The response of a glacier to changer I" Ihe rate of nourishment and wastage. Pme Roy Sm.London. A275, 87-1 12. Oikawa, M. and Yajima. N. (1973). Interactions of solitary waves - A perturbation approach to nonlinear systems, J. Phys. Soc.Jopon. 34, 10931099. Otkawa, M. and Yajima, N. (1974a). A class of exactly solvable nonlinear evolution equations. Pmg. Theor. Phys. Suppl. 54, 1576-1577. Oikawa, M. and Yajimia. N. (1974b). A perturbation approach to nonlinear systems 11. Interaction of nonlinear modulated waves, 1. Phys. Soc. Japan, 37, 486.496 . .. ... Okubo, A. (1980). Diffusionond Ecological Pmblcms: Marhcmnticol Models, Sonnee~Verlas.Berlin. Ono, H. i1975) Llgcbraic solitary WBYCS rn swatifled fluids, 1. Pkys. Soc. Jnpon,39, 1082-1091. ~

~~

516

Bibliography

Scott, A.C., Chu. F.Y.F.. and McLaughlin, D.W. (1973). The soliton - A new concept in applied science, Pmc. IEEE.61. 1443.1483, Sddon. J.A. (1900). River hydraulics, Trans. Amrr. Sot. Civ. Engrs. 43, 179243. Scgur, H. (1973).The Konewcg-dc Vries equation and water waves. 1. Solutions of the equation, 1. Fiuid Mech 59.721-736. Sen. A.R. (1963). Surface waves due to blasts on and above liquids. J. Fluid Mech. 16, 65-81. Shigerada, N. (1980). Spatial distribution of dispersing animals. J. Moth. Bio. 9, 85-96. Shivamoggi, B.K. and Debnath. L. (1983). Modulational stability of Konewegde Vries and Boursinesq wavevains, I n r c m r . J. Moth. ond Math. Sci. 6, 811-817. Short, M. (1995). The initiation of detonation from general non-uniformly distributed initial conditions, Phil. Trans. Roy. Soc. London, A353, 173lnl _",.

Showalter, K. and Tyson, 1.1. (1987). Luther's 1906 discovery and analysis of chemical waves, J. Chcm. Educ. 64,742-744. Skyrme, T.H.R. (1958). A nonlinear theory of strong interactions, Proc. Roy. Soc. London. A247. 260-278. Skyrme, T.H R. (1961). Particle states of B quantized meson field, Pme. Roy. Soc. London.A262, 237-245. Sleeman. B.D. (1982). Small amplitude periodic waves for the FitzHHughNagumo equation. J. Math. Bio. 14, 309-325. Smollcr, 1. (1983). Shock Wows and Reaction-Diffusion Epotl~ns,SpringerVerlae. New Y a k . Sneddon. LN.(1951). Fourier Trm.sf'omu. MeGraw-Hill, New Y o t . Soewono. E. and Debnath, L. (1990). Asymptouc stability of solutions of the generalized Burgers equation. J. Mah. A n d Appl. 153. 179-189. Soewono, E. and Debnath, L. (1594). Classification of self-similar solutions to a generalized Burgers eq~auon,J. Math. Anof. Appl. 184.389-398. Sparrow, E.M., Quack, H.. and Boerner. C.J..(1970). Local "onsimilarity baundary-layer solutlons. AIAA J o u m l , 8. 1936-1942. Stakgold, I (1971) Branching of solutions on nonlinear equations, SIAM Rev. 13, 289-332. Stakgold, 1. (1979). Green's F u n d o m and Boundory Value Problem, WileyInterscience. New York. Stakgold, 1. and Payne. L.E. (1973). Nonlinear problems in nuclear reactor analysis. in Nonlinear Problems in the Physical Sciences and Biology, Lecture Notes in Mathematics. No. 322. Springer-Verlag. New York. pp. 298-307. Stewanson. K. and Stuan, J.T. (1971).A nonlinear instability theory for a wave system in plane Poiseuille flow, 1. Fluid Mcch 48, 529-545. Stoker, 1.1. (1957). Water Wnves, Interscience, New York. SNart, I. T.(1960). On the nonlinear mechanics of wave disturbances I" stable and unstable parallel flaws, Pan 1. The hasic behaviour in plane Poiseuille flow. J. Fluid Mech 9, 353-370.

Stuart, J.T. and DiPrima. R.C. (1978). The Eckhaus and Benjamin-Feir nsonance mechanisms. Proc. Roy. Soc. London, ,4362,27-41. Sturroek, P.A. (1957) Nonlinear effecls tn elecvon plasma. Proe. Roy. Soc. London. A242. 277-299. Su, C.S. and Gardner. C.S. (1969). The Korteweg.de Vries equation and

generalizations Ill. Derivation of the Korteweg-de Vries equation and Burgers' equation. J. Math. Phys. 10,536-539. Talanov, V.1. (1965).Self-foeusing of wave beams in nonlinear media. JETP Lett. 2. 138-141. Taniuti, T. (1974). Reductive perturbation method and far fields of wave equations, Prog. Than Phys. Suppl. 55. 1-35. Taniuti, T. and Nishihara. K. (1983).Nodineor Waves. Piunan Advanced Publishing Program. Boston. Taniuti, T. and Warhimi, H. (1968). Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma, Phyr Rev. Left. 21. 209-212. Taniuti. T. and Wei. C.C. (1968). Reductive perturbation method in nonlinear wave propagation-I, J. Phys. Soc. Japan.24,941-946. Tsniuti. T. and Yajima, N. (1969). Perturbation method for a nonlinear wave modulation I, J. Math. Phy$. 10. 1369-1372. Taniuti, T. and Yajima, N. (1973). Perturbation method for a nonlinear wave modulation 111, J. Moth. Phys. 14, 1389-1397. Tappen, F. and V m a . C M.(1970). Asymptotic theory of self-trapping of heat pulses in solids, Phys. Rev. Iatt. 25. 1108-1111. Thyagaraja. A. (1979). Rceurnnt motions in certain continuum dynamieal system, Phys. Fluids, 22. 2093-2096. Ihyagaraja, A. (1981). Recurrence. dimensionality. and Lagrange stlbility of solutions of the nonlinear SehrWinger equation, Phys. Fluids. 24, 19731975. Thyagaraja. A. (1983). Recumncc phenomcna and the number of effective degrees of freedom in nonlinear wave motions. In Nonlinear Waves (ed. L. Dcbnath), Cambridge University Press. Cambridge, pp. 308-325. Toda, M. (1967a). Vibration of a chain with nonlinear interaction. J. Phys. Soc. Japan.22.431-436. Toda. M. (1967b). Wave . ~rooa~ation . _ in anharmonic lattices. J. Phw. Soe. Japon,'23.501-506. Toda. M. and Wadati, M. (1973). A soliton and two solitons in an exponential lattice and related equations, J. Phys. Soc. Japan.34,18-25. Turin, A.M. (1952). The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. London. BZ37, 37-72. Valemi.. A. (19881. , , The asvmototic analvsis of nonlinear waves in mametothermo-elastic solids with thermal relaxation. ZAMP, 39,299-312 van Saarloor, W. (1989). Front propagation into unstable states: marginal stability as a dynamical mechanism for velocity selection, Phys. Rev. A

,.

. -..

.

97, 711.770 " __,

van W i j n g d e n , L. (1968).On the equation of motion for mixtures of liquid and gas bubble, J. FluidMceh. 33,465-474.

Bibliography

578

Wadati, M. (1972). The exact solution of the modified Kortsweg.de Vries equation, J. Phys. Soe Japm.32, 1681-1687. Wadati, M. (1973).The modified Korteweg.de Vries equation, 1. Phys. Soe. J a p m , 34, 1289-1296. Wadati. M.and T d a . M. (1972).The exact N-soliton solution of the Kortewegde Vrks equation, J. Phys. Sm.Japan, 32, 1403-1411. Wahlquist. H.D. and Estabmok. F.B. (1973). Biieklund transformations for solutions of the Korteweg.de V i e s equation. Phys. Rev. Lctt. 23. 1386I1PO ._Y,.

Wahlquist. H.D. end Estabrook, F.B. (1975). Prolongation structures and nonlinear evolution equations, J. Mark Phys. 16, 1-17. Wahlquist. H.D. and Estabrook. F.B. (1976). Prolongation SITUCIUI~S and nonlinear evolution equations. J. Moth. Phys. 17. 1403-l411. Walsh. R.A. (1969).Initial value problems associated with u,(x,t) = 6un(x.t) n(x,r)u.(x,r), 1. Moth. A w l . Appl. 26. 235-247. Wang. X.Y.(1985). Nerve propagation and wall in liquid crystals, Phy. Lett.

-

112A. 402-406. Wang, X.Y.(1986). Brochard-Leger wall in liquid crystals, Phys. REV. A34, 5179-5182. Wang. X.Y.,Zhu, Z.S., and Lu. Y.K.(1990). Solitary wave solutions of the generalized Burgem-Hurley equation. 1. Phys., Morh. Gen 23, 271-274. Ward. R.S. (1984).The Painlev6 property for the self-dual gauge-field equations. Phys. Lctt. IOZA, 279-282. Ward, R.S. (1985).Integrable and solvable systems and relations among them. Phil. Trans. Roy. Soc. London, A315. 451-457. Ward, R.S. (1986). Multi-dimensional integrable systems in Field Theory, Quantum Gravity, and Strings, 11 proceedings, Meudon and Paris. VI, Frnnce 1985/86 (eds. H.J. de Vcgn and N. Sonchcz). Lccr Norer Phys. 280, 106-110, Springer-Verlag. Berlin. Washimi, H. and Taniuti. T. (1966).Propagation of ion-acoustic solitary waves of small amplitude. Phys. Rev. Lcrt. 17,996-998. Watson, I. (1960).On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows, Pan 2. The development of B solution for plane Poiseuille flow and f a plane Couette flow, J. Fluid Mech, 9,371-389. Wehausen. 1.V. and Laitone. E.V. (1960). Surface waves, In Hondbuch der Physik. (ed. W. ROgge) 9. Springer-Verlag. Berlin. pp. 446-778. Weidman. P.D. and Manwonhy. T. (1978).Experiments on strong interactions between solitary waves, 1. FluidMech. 85, 417-431. Weiland, 1. and Wilhelmsson. H. (1977). Coherent Nonlinear feteroction of Waves in Plarmns. Pergamon F'ress, Oxford. Weiland, 1. Ichikawa, Y.H.. and Wilhelmsson, H. (1978). A perturbation expansion for the NLS equation with application to the influence of nonlinear Landau damping, Physic0 Scripla, 17,517-522. Whitham, G.B. (1956). On the propagation of weak shock waves. J. Fluid Mcck.. 1, 290-318.

Bibliography

580

W m v . V.E. and Shabat, A.B. (1973). Interaction between solitons in a stable medium, Sov. Phys. JETP, 37, 823-828. Zakharov, V.E. and Shabat, A.B. (1974). A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, Funel. Anal. Appl. 8. 226-235. Zakbarov, V.E. and Schulman, E.I. (1982). To the integrability of the system of two coupled nonlinear SchrSdinger equations. Phyrica, 4D,27C-274. Zauderer, E. (1989). Pmiiol DifferentidEqvatiomofApplied Morhemtier, 2nd

EdiUon, Why-Interscience. New York. Zcldovieh, Ya. B. (1980). Regime classification of an exothermic reaction with "on-uniform initial conditions, Combust. Flonrr, 39. 21 1-214. Zcldovich, Ya. B., Barenblat, 0.1.. Librovieh. V.B., and Makhviladze, G.M. (1985). The Mathematical Theory o j Combustion and Explosion. Consullanu Bureau, Division of Plenum h s s , New York. Zcldoweh. Ya. B.. Librovich, V.B.. Makhviladre, GM.,and Sivashimky, G.I. (1970). Dcvelooment of detonation m a non-uniformlv onhealed ~. *as_As*. A m . is. m & i . Zcldovich. Ya. B. and Raizcr, Yu. P.(1966). Physics ofshock W w c s and High Tenperarum HydmdyMmic Phenomena. Academic Ress. New York. Zcldovich. Ya. B. and Razier. Yu. P. (1968). Elements o j G u DyMmics and rhe Clrrrsical Theow o f b o c k Wwcr. Academic Press, New York.

_.

A AMelkader, 325 Ablowitl. 96.99.325.400. 413 absorption isotherm, 205 acoustic radiation problem, 62-64 Acrivos, 97 adiabatic invariant. 463

Airy. 332 Airy differential equation, 442 Airy function, 36. 358 Airy integral. 36 AKNS method. 4 m 4 0 1 Alfv6n wave velocity, 266 Ammeman. 95.302 antikink, 336,474,477 antisoliton. 473,474,477. 483 Aoki. 95,301 h c h i , 95

Aris, 95 h o l d . 95.301

Aronson,323 Asam, 97,50251 1-512.515.526 WOeiated Legendreequatlon,376 assaeiated Legendm function, 376 asymptotically stable, 306 average conservationequariOn,464 average Lagrangian.468,470 average variational principle. 469 axisymmeoic acoustic radiation problem, 62 arisymmeuic diffusion equaum, 61 axisymmeoic wave equation, 60

B Backlund msfo-tion.

403,450,484 Balmforth. 98 Bannblan, 324-326

386-388,

Barone. 95 Ben Jacob. 98 Benjamin, 97,98,280. 418 Benjamin-Born-Mahony@BM) equation. 97.265. 401

Benjamin-Feir (modulational) mstabihty. 424,426 Benjamin-Ono(BO)equation, 97 Benney. 96.422 Benton. 324 Bcrezin. 351.352 Bernoulli-Eula equation,93 Bespalov. 96 Bessel function, 59,62 Bhatnagar, 516 Bianchi theorem, 389 bihmonic equation, 9.64, 86 Birkhoff. 309 Blrkhoiitheonm. 418 Blackslock. 324 Blariur problem. 85.328 Bleakmy, 231 Boh-Sommerfeld mle, 385 bound state, 364.366.483 boundary-value problem, 5.59 Boltlmann consmt. 344 Boltzmann distribution, 345,518 Boltrmannnonlinear diffusion problem. 312,326 Boltzmann uanrfomation. 314 bore. 238 Bm-lnfeld equation. 107 Bowsinem. 332 Boursinesq equation. 9,95.108,

265.333, 339,341,357,533 Brauna, 324 breakers, 203 b n a t h a solution. 481,482

Index

582

hnght soliton, 414 Brillouin. 441 Brilton, 325 Bnint-VXisiiIl frequency. 266 Burgers, 95,284.296 Burgers equation. 95. 2M, 256, 284,285,289,518

BurgersHuxley equation,98 Bunon, 231 C

emnonieal form, 10 cannonid transformation. 148 Canosa. 325 Cue. 324

&why condition,5 Cawhydata. 120, 124-125. 131, 242

Chi", 324 chmatographic models, 2124208 Chu, 96. 282.421 Clorautequetion, 140, 141, 144 clasrleal SOIULIO". 3 cnoidal wave, 350-351 Cole. 289, 301 Cole-Hopf transformation,289,361 column isotherm, 205 commutator, 392 complete elliptic integral. 350 complete integral. 113 complete solution. 113 compressionwave, 164,208 conjugatevariables, 148 consewationequations. 172-173, 319,463

369 consewation laws. 159, 160-161,

COnStWStiOn Of CWrgY,

Csuchy-Poissonproblem. 56.64, 81, 89

186.359.419.517

Cauchv Drincioal value.. 97 Cauehy pmhlem, 5 , 16-17, 34, 39. 47, 50, 80. 88, 119-120, 130. ~

I

.

.

131.. 141.. 157.. 158. 178

Cauchy-Riemann equations,387 caustic, 441 Cavali-Sfowa, 95, 3uZ characteristics, 11, 117.238 charaeteristicbase curve, 117 charactcristiccuwes, 11,117, 118, 224-225.340 sharacteristledirectiion.116, 136 characteristicequation, 11. 117, 137 characteristics h n k 253 eharacterirticslrip, 139 eharaeteristiestrip equauons, 139 characteristicvelocity, 279,281, 467 Charpit equations, 137, 139, 142 Chen. 96,99 Ch&y formula, 202

conserved densities. 360 continuity equation, 186, 209,214, 219.344

conunuousspccuum,368 convolution, 33 convolutiontheorem. 33-34 Cornish, 216 Courant. 247 Cramer. 517 Crandell, 383,403 Crank, 316.323 Cnghton, 324 cylindneal KdV equation,402

D d'Alemben solution, 17, 35.76, 244, 339 D m y law. 254 dark soliton, 414,448 Davcy, 96,422

Davis, 97. 491 Debnath, 52,56-57,59.64-66.81-

82,86. 89,96, 100, 233,278, 280, 337,350, 354, 409,41 I, 413.454.474 de Broglie wave, 266 De Leonarda,482 dc Vrie~,96,333 Debye. 333 Debye length, 345 density, 186.359 denvatwe expansion method. 531 developable surface, 145 dielefmc constants, 439 diffusion coefficient. 25.212. 310 diffusion equation. 7,25,27,39, 46,51-54,61.68-69,84-85.90, 284,289-290.310,312,316, 319-320 diffusion length. 256 diffusive wave, 285,288 diffusion-reaction equation. 3132, 319-320 dipole solution, 298 D i m delta function, 32-33,37,40. 62 Dirichlct condition. 5 Dirichlct problem. 5, 29,36,45. 102 discontinuous solution. 165,172,

178-181 dispersion relation, 264, 278,407,

441,443.444.446.497.516 for Alfven gravity wave, 266 for dc Broglie wave. 43,266 for enoidal wave. 35 I for electromagnetic wave, 266 for ion-acoustic wave, 346 for nonlinear wave. 278 far solimy wave, 410 lor suing wave, 516

for water waves, 65,265,410 dispersive wave, 264 Dcdd. 396 Dradn. 379-380.384.478

hessler. 216-217.218 Dunbar, 325 Dutta. 350. 354. 409,41 I, 474

E effective mode, 418 eigenvalues. 22,26.28 eigenfunstions. 22,26,28 eikonal equation. 151, 155,441 Ekman layer, 85 elastic panicles, 482 elastic w w e equation, I10 elecuical susceptibility, 439 clectmmagnetic wave equation. 9 electronplasma frequency, 345 elliptic equation, 12. 13 ellipuc integral. 350,409 energy density, 272,275 energy flux. 272.275 Engelbrecht, 515 entropy criterion, 180 envelope. 129 envelope soliton. 336,413-414,

526 Enz. 95

equation associated Legendre, 376 aiisymmeuic wave. 60 Benjamin-Bana-Mahony (BBM).

97,265.401 Benjamin-Om (BO), 97 Bemoulli-Euler, 90 biharmonic, 9. 64,86 Bom-Infeld, 107 Boussinesq, 9,95. 108.265. 333. 339,341,357,533

584

equation Burgers, 95.200.256.284.285,

289,502.518 BurgersHurley, 98 Cauchy-Riemann. 387 charactcristie. 11, 117. 137 Charpit. 137, 139. 142 Clairaut, 140, 141, 144 conservation, 172,173,319. 463 continuity, 186,214. 219, 344 cylindrical KdV. 402 diffusion, 7.25, 27. 39.46.5154,61,68-69,84-85.90. 284, 289, 290. 310, 312. 316.319-320 diffuaon-reaction, 302, 319-320 dynamieal, 462 eikoonal. 151, 155, 441 elastic wave, 110 electnomagneticwave, 9 elliptic, 12, 13 Euler, 120,257,277.282 Euler-Darbour. 19 Eulcr-Lagrulge.IW-102, 104, 109.216 Euler-Poison-Darboux. 234, 260 evolution. 392,395.505. 508 Fisher. 95, 301 Gxdner, 362 Gdfarnd-Levitm-Mmhmko (GLM), 373,374,375,377 Ginzhurg-Landau (GL). 98 glacier flow, 211 Hamilton, 103. 146 Hamilton-lacobi, 145-147.274 heat, 1.51 Helmholtz. 8,63,71. 73. 80 hyperbolic, 11-13.231,284 hypergeomeuic, 232

quation internal wave, 265 internal-inertial wave, 266 Kadomtscv-Petvisshvilli (KP),

99,402-403 kinematic wavq 94. 161, 186,

202.465 Klein-Gordon (KG), 8,80.91.

94. IM,265.426,453.456. 458.460.496497 Kortcwcg-deVnes (Kdy),9,

35. 333-334,343,347. 356, 358. 360. 363.398, 403,449 517,519,521,539 KdV-Burgers. 502 Lagrange. 102-103 Laplace 8, 36, 38,45,56. 59. 61,65. 79.90, 101, 387 Lax, 392-393.400,403 linear p m a l differential, 2-3. Ill linear rcattning. 487 linear wave equation, 7,2l,34,

242 Maxwell. 344.438439 minimal surface. 102 modified KdV (mKdV). 96,354-

355, 361,399,403,515,517 Navier-Stokes. 214 nonlmeardiffusion. 284,312,

314. 316 nonlinear diffusion-reacuon. 319 nonlinear panial differential. 3.

93,111, 112.270 nonlinear SchrCdmgerW S ) ,

96,282,396,398,402,408, 410,412.415.422.424, 431,436,438.447.448, 451,482,525,526,533 nonlinear wave equation, 94,238 Painlevb, 357,491 parabolic, 12,284

Nonlinear Parrid Di@renrioI4~hnionr equation

paraxial, 444 Peregrine, 520 Poisson. 8. 70, 109 hdtl-Blasius. 328 quasi-limarpanial, 2, 111,282 Riccati. 364 mll wave, 216 Rossby wave. 83.266 Schriidinger. 9.42~44,109, 266,364. 371,396,401,407 shallow water, 236,257,339, 340 sine-&rdon (SG),94,386,399, 470-471,482,484,495,499 smh-Gordon, 399.498 telegraph. 8,80,83-84,232, 258 uaffic flow, 255 Tncomi, 13,260 water wave, I 0 7 wave, 3. 7, 16. 21,34.47, 55. 64, 62. 73-77.79,94, 104. 110. 238.242.439.493 Whitham, 271-273,276,278, 282.404.421 Yang-Mills. 99 Etabrook, 387. 389,396 Euler equation. 120.257.277,282 Euler-Darbourequation,19 Enler-Lagrangeequation. 100-102, 104,109,276 Eulcr-Poisson-Darbour equation. 234.264

F fan,

194,207

585

fancentend,194-195 Feir. 280,418 Fcrguson, 418 Fermi. 333,418 Fermi-Pasta-Ulam(FPU) ncumnce.334,418 Fife, 95. 323,325 Fisher, 95,301 Fisher equauon. 95,301 fission, 381 FitzHugh. 98 flood W W ~ ,ZM, 203 flux, 159,186,359 flux uansport, 319 foeusing mstability. 448 forcing term, 264 Fordy. 396 Fourier method, 5.21 Fourier solution. 47 Founer mnrform.32.44.268 fractionalderivative. 52-53 Freeman, 99.402 Fnnkcl, 95 Fnedncho,247 Fmude number. 216,218 functional,99-101 Fulton, 97 fundamentalharmonic. 23 f""danw"Ial parameters,337,344 fundmental period. 23 fundamental solution, 39-40.292, 314 G Galvin, 335 Gardner. 335,361,363,506,514 Gardnerequation,362 Gardner msformauon,361 Gardner-Morikawam f m a l i o n , 506.511,513

Index

586

Gazdag. 325 Oelfand-Levitan-Merchenko (GUI) equation, 373,374,375,377 general integral. I13 general solution, 3. 113 generalized NLS equation, 438,526 generalized simple wave, 250252 generalized solution, 3.159, 174 generalizedRankine-Hugoniot shock condition, 253 generalized Riemann invariant%252 geomeuical optics approximation, 460 Ghcz, 323 Gibbon. 95,396 Gibbons, 415.436 Ginzburg-Landau(GL) quation. 98 glacierflow, 210211 Goldstem, I50 Gordon. 8 Goner-Mellink law, 318 Gomhkov. 531 gravity wave, 65,265,410 Grwn'r function, 394.67-78.91. 453-459 p e n signal problem, 196 Griftilh. 231 Grindmd, 325 p u p velocity, 97,265,273.280 grow& Rate, 301 growth term, 161 Gurney, 323 Gurti", 323

H Haberman, 96 Hadamad example,6 Hagsuom, 325 Hahn, 492-494 Haight. 189

Hamiltmian, 102, 145. 150 Hamilton denrily, 482-483 Hamilton equations. 103. 146 Hamilton function. 102 Hamilton-lacobi equauon, 145-147, 150, 151,274 Hamilton principal function, 145 Hamilton variational principle, 102, 104.276.531 Hammack, 335.413 Hankel transform, 58-59 Hasegawa, 97,443 Hasirnoto, 96.422, 428, 431. 531 Harsan. 531 heat equation. 7.51 Helmholtz equauon, 8,63,71.73 Hilberl space, 366, 392-394 Hilben uansform, 97 Hirota, 335 Hcdgkm and Huxley equation,98 Hcdograph transformation, 247 Hopf, 289 Hoppensuadt. 307 Howanl. 320 hvdraulic iumo. 203.238 , . hyperbolicequation, 11-13.231233.284 hywrgeomeme equation. 232 hypcrgeomeuic function. 232 ~

..

I lchikawa, 97 illpored problem, 6,280 index of refracuon. 151 Infeld. 96,344,436 initial-value problem, 5-6, 17.81, 82, 119, 161, 163-174, 178, 221,267. 285. 289. 363, 392 Inouc, 537 instantons, 99 integrability condition. 137

Nonlinear Pornhi Dzferenriai Eyuotiow

integral surface. 4. 116 rnteraction suength, 418 internal wave equation. 265 internal-inotial wave equstio~,266 inverse Founer tansform, 32.44 inverse Hankel vansform, 59 inverse Laplace flansfm, 49 inverse Scattering w n s f n m , 363374,487

ion-acousuc wave, 344, 346,518 ion-plasma frequency, 345 isenmpic flow, 222.257.260 isothermal cuwes, 37

587

Keller, 325 Kelley, 96 Ken effect. 442 kine& w a v e e q ~ W h ,94,161, 186,202,465

kinetic energy, 25,43, 110 kinematic wave. 185-187 kink, 336,473, 476, 485 Kirchhoff vansformation, 316 Klcin. 8 Klein-Cordon (KG) equation,8, 80, 91. 94, 104, 105, 265. 426, 453,456.458.460. 496-497

Klettz-Seddon law, 204 Kolmogomv, 95,302

J lacobi cn-funelion, 350.351 lacobi dn-function, 41 1 lacobi sn-function, 350, 410, 474 Jafobi elliptic function, 350, 410, 474

Jacobi beta funcfim 3DD,-I54 Jeffrey, 252. 259.441.515, 533 Johnson, 96,99,325. 379-380 Jones, 268, 325 Josephson. 95 jump condition. 173. 174 jump discontinuity, 177, 314 lost SOhlionS, 371

K Kadomlsev, 99 Kadamwv-Petviashvili (KP) equation. 99,402-403

Kalmppan, 321 Kako, 96 Kakutani, 96, 502, 515 Kamelaka, 325 Kaplllm,97,351, 385,436 Kaup, 396,400,482 Kawahara, 515,531,533

Konno. 450 Kontorova. 95 Koppll, 320 Koneweg. 96,333 KorIeweg.de Vies (KdV) quation, 9, 35,265, 333-334, 343.347, 356, 358, 360.363.398, 403, 449,517,519,521,539 KdV-Burgersequation. 502 h e r s . 441 Krien. 324 Kmneker delta, 23 Krurhkal. 97 Krurkal, 334.335.361.418 Kulikovskii, 516 Kynch. 94,208,256

L Lagrangebracket, 149-150 Lagrange equation, 102-103 Lagrangian stability, 417 Lagrangian. 102-105, 107 Lagrangian dennty, 104,467,482 Lake, 96,418 Lamb. 95,482,487.489.491. 498 Lame constants, 80

Index

588

Landau, 383 ~ a n p m u isoliton, r 433 ~angmu~r isotherm model, 206 Laplace egustion, 8.36.38.45.56, 59, 61. 65, 79. 90. 101. 387

La&ian, 42,338 Lardner, 324 Larson, 307.324 L * ~ I, S I , Z ~ O - Z ~ ~ 335,391 , Lax entropy criterion, IBO.253 ax equation. 392-393.4W. 403 Lax pair. 392.403 Legendre polynomial. 376 Libouich, 96.538 I.e vequc, 254 Lie mu^, 310, 356-357

linear p d a l differentialequstion.2, 3, 111 linear scattering equation,487 linear wave equation, 7.21.34.242 Liu, 96.99 local mucn~. 269.n0, n 3 local Mach number. 246 local wavenumber. 269-270.273 Logan, 325 Lorenz, 324 Luke, 105, 107

M

magnetic wave equation. 439 Makhankov. 336 Manning fomula, 203 Manoranjm. 325 matrix operators. 395 Matsumoto, 537 Maxwell equations, 344,438-439 Marworihy. 335 McC111,492-494

McKem, 325 mean value property. 31 Men. 96,282.421 Memnne law, 23 method of characteristics.118, 135 method of multiple sale. 531-540 minimal surface equation, 107 Miura. 361,364,401 Miura uansformation,361,364 Mitchell, 325 modifiedKdV (mKdV) equation.96, 354-355, 361,399,403,515. 517

modulationalinstability. 424,426 Mollenausr, 443 moment of instability, 342 Monge axls, 116 Mongc cone, 117. 136-137 monoclinalflood wave. 203 monochromatic wave. 440 Montmll. 320 Morikawa, 506 multiple Fourier fransform. 44 Munier, 316 Murray, 314.322-325

Ma. 450

MacCamy, 323 Mach line, 246 Mach numkr, 156,246

N N-soliton solution. 376 N-wave solution. 296,297

Madsen, 96

Naguma, 98 Navicr-Stokesequations. 214

magnetic permeability,439

Nonlmror Pam01 Df@remtkzl Equtionr

Nayfeh, 97,531 Ncumann condition. 6 Neumann boundary-value problem, 6,38, 88 Newell. 400,482 Newman. 320-324 Nirbet, 323 NishBara, 516 nodes,24 nondispemivs wave. 265 nonlinear diffusion equation, 284. 312, 314,316 nonlincardiffusion-muon equauon, 319 nonlinear Dirichlet problem. 102 nonlinear evolution equauan. 392, 395 nonlinear group velocity. 280 nonlinear K I e i n - W o n (KO) equation, 94. 104,426,496 nonlinear p d a l diffennual equation, 3,93,111, 112,270 nonlinear SehrBdinger(NLS) equalion, 96. 282,396,398. 402,408-410.412.415, 422, 424,431,436,438, 447,448, 451,482,525-526, 536 nonlinear superposition principle, 389-390.486 nonlinear wave eqmuon. 94,238 normal dispersion. 448 normal modes,23 Nozaki, 516 Nye, 94.210 Nwogo. 96

0 Oikawa. 415,516 Okubo. 314 Om. 97.422, 502, 511,515.531 order. 2

589 OsWvsky, 502,527.530 Outi. 413 ovenones, 23

P Pack, 231 Painlev6 equation, 357.491 parabolic equation. 12,284 pararial equation, 444 Parker. 324 partial differenual equation, 2 par!kularsoluuon, 113,I14 Pasta. 333,418 Patlle. 320 Pawlik, 97 Pslinovsky, 502, 527. 530 Penel, 324 Peregrine. 95-96.449.450.520 Peregrine equation. 520 Perring, 477,487 Petviashvili, 99 phase velocity, 97,264.404 Phillips, 451 pislon problem, 227,234 Planck constant, 8.42.266, 386 Plateau pmhlem. 101 Platzman, 324 Poinc& thcorem. 148 point of mflerion. 321 Poisson bracket, 147 Poisson equation. 8. 70, 109 Poisson integral formula. 31,36, 40 Poisson kernel, 32 palanzation. 439 p l y m p i c gas, 228 potential, 365,400.422, 487 potential energy, 25.43, 110 Prandtl-Blasius problem, 327,328 h a d , 9697.516.538

590

prolongation smcture method, 396

Index

ruledsuface, 145 Russell. 331

Q quasi-linearpanial differential equation, 2, 108, 111.282

R Raizr. 324 Rankine-Hugannotcondmon. 174. 253 Ravindran. 96-97,538 Rayleigh, 332 Rayleigh problem. 85 Rayleigh quot!ent, 416 reaction term, 161.302 Redekopp,96 nductivc pemrbation method, 502 reflectioncnfficient, 368 refractive wave, 163,179 refracuve index. 440 Reynolds number, 291,297,299 Riabouchinsky. 340 Rieeau equation, 364 Richards. 94, 189 Riemann, 219, 228, 287 Riemann function, 75.232 &emam initial-value problem, 221 Riemann invariants. 219,226,228, 239-243.248.252, 341, 466 Robin boundmyvalue problem. 6 Robin condition, 6 Rodin, 324 Raoaler, 358 roll wave, 213,216-217 roll wave equation, 216 Rosenau, 325 Roskes, 96, 422 Rosrby wave equation, 83,266 Rowland, 436 Rowlands, 97,426

S saddle pom 304-305 Sagdeev, 347 Sand". 531 Sanuki, 438 saric, 97 Satsuma. 98-99 scanning data. 363,370 raltcring equation. 487 scattering state. 364, 368 Schiff, 94 Schimizu, 97 Schmidt. 436 SeMidinger quaban. 9 , 4 2 4 109.266. 364. 371, 396, 401, 407 Schwarr inequality. 416-417 Scott, 94-98. 324, 472 scculm (unbounded) tam, 534 Scddon. 203 Segur, 96.335.378,400.413 self-focusing instability, 442 self-inducedtransparency (SIT),492 Sen, 90. 517 sepmwion of variables, 20-32 separauix, 305 Serret-Frenet formulas, 428 Shabat, 357, 395, 405. 413, 419, 422,450 shallow water equations, 236.257, 339, 340 shallow water speed, 35,333,339 shallow water waves, 236,257,520 Shigesada. 323 shock wave, 114, 166, 174, 192195,230 similarity solution, 310. 311.491 similarity uanrformauon. 311,491

591

Nonlinear Portid Dtflerentid EqmMnr

similarity variable. 292, 311 simple wave. 179,219,222.227, 229,242 Sine-Gordon (SG) equation. 94. 386, 399,470-471. 475.482, 484,495-499 Sinh-Gordon equation. 399.498 single-soliton solution. 375. 483 singular solution, 114. 153 Skyme. 95,477.487 Slceman. 98, 325 Slobmikina,516 Smoller, 254, 325 s m o o ~function. 114 sn-function. 350,410,474 solilary wave. 335, 349,410, 421 soliton, 335, 349. 389, 390, 473. 476.485 solution complete, 113 discontinuous. 163. 178-181 general. 113 generalized,3.159, 174.177 particular. 113, 114 singular. 114, 153 weak, 3, 114, 159, 174177.254 Soremen,96 source solution, 292 source term, 302 Sparrow. 328 s p e d of shallow water waves, 333, 337 slable node, 304 standing wave, 24 stationary phase method, 58.66, 268,295 stationary point, 58,66,268. 295 Stewanson, 96, 97, 98. 422 Stokes. 2 s . 332 Stokes problem, 84 Stokes-Ekmanproblem, 85

Stokes expansion, 279, 421 m c h i n g Wansfonnation,310 suetching variable, 511 Stuart. 97, 98, 531 Storm-Liouvilleproblem, 393 Sturrock, 531 Su. 514 superpositionprinciple linear, 4-5. 22, 40, 71 nonlinear. 389. 390. 486 supersonic Row,245-247.253 surface wave problem. 87.89

T Talanov. 96 Taniuti, 96. 97. 259. 502, 511. 515.516 Taooen.. 97.443 . telegraph equation, 8. 80.83-84, 232,258 h a l wndoctivity, 52 theta function, 3W, 354 three-soliton Solution, 380 T%yagamjq415 T&, 95 traffic flow, 189-202 traffic flow equation. 255 traffic signal problem. 198 hiangular wave solution. 294 wnsform Fourier, 32.44.268 Hankel, 58-59 Hilben. 97 inverse scattering.363-374 Laplace. 49-50 Trieomi equation, 13.264 transition layer, 288,293 Vansition region, 174 transmissioncoefficient,368 uansversevibration of ruing,21

..

Inda

592

uaveling wave solution,286,303,

306 Tmllinger, 482 Turin, 319 two-soliton solution,379. 390

u Ulam, 333,418 undularbore. 350

WBVe

generalized simple. 250-252 gravity. 65.265.410 incident 368 inertial, 266 internal. 265 internal-inertial. 266 ion-acoustic. 344,346,518 kinematic, 185-187

N. 296,297 V van SaarlOrn, 98 Van Wijngarden, 96

nondirpersivc, 265 reflected, 368 refractive. 163. 179

vmauonal principle. 99, 104-110 Varma. 97 veloeity potential, 56,65, 81 velocity of sound, 220 vortex Blamcnt, 428.432

roll, 213,216217 Rossby, 83,266 shock, 114,166, 174, 192.230 shallow water, 236,520 simple. 179, 219,222, 227,229 s o l i l q , 335,349,410.427 standing, 24 Stokes. 263,280 transmitted. 368 water. 265.280

W Wadati, 95.97.450 Wahlquist, 387,389.396 Walsh. 301.325 Wan& 98 ward. 99 Washimi, 96,97, 502 water wave uluationr. 107

Watson, 531 weak solution,3. 114. 159,172,

174-177.254 WBVe

Alfvkn gravity. 266 compression. 164,208 cnoidal. 350-351 de Broglic, 266 diffusive. 285,288 dispersive, 264

eleemmgnetie. 9 expansive. 163. 170. 179,207 flood. 202,203

wave diffraction, 444

wsveenergy, 418 waveequation, 3,7. 16.21. 34.47. 55,60,62, 73-77,79,94, 104, 110, 238,242,394,439,493 wavefront. 153 wave propagation veloeity, 187 Wei, 502, 5ll.513-514 Weidman. 96.335 Welland, 97 well posed problem. 6, I7 wen12c1.441 Werl, 320 Whitham. 94. 105,107-108,185189,267, 275,281, 293.296,

353,355,409.418.459.468 Whitham averaged Lagmngian, 277.

468