Nonlinear Dynamics

Advanced Texts in Physics

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M. Lakshmanan S. Rajasekar

Nonlinear Dynamics Integrability, Chaos, and Patterns

With 193 Figures, 17 Tables, Numerous Examples and Problems

" Springer

Professor M. Lakshmanan Department of Physics and Center for Nonlinear Dynamics Bharathidasan University Tiruchirapalli - 620 024 India

Library of Congress Cataloging-in-Publication Data.

Lakshmanan, M. (Muthuswamy)

Dr. S. Rajasekar Department of Physics Manonmaniam Sundaranar University Tirunelveli - 627 012 India

Nonlinear dynamics : integrability, chaos, and patterns / M. Lakshmanan; S. Rajasekar. p. cm - (Advanced texts in physics, ISSN 1439-2674) Includes bibliographical references and index. ISBN 978-3-642-62872-6 ISBN 978-3-642-55688-3 (eBook) DOI 10.1007/978-3-642-55688-3

1. Dynamics. 2. Nonlinear theories. 1. Rajasekar, S. (Shanmuganathan), 1963- II. Title. III Series. QA845.L252002 531'.11-dc21 2002030441

ISSN 1439-2674

ISBN 978-3-642-62872-6

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Preface

Nonlinearity is ubiquitous and all pervading in the physical world. For a long time nonlinear systems were essentially studied under linear approxi­mations, barring a few exceptions. However, the famous Fermi-Pasta-Ulam numerical experiments of the year 1955 on energy sharing between modes in anharmonic lattices triggered the golden era of modern nonlinear dynamics. Several path-breaking discoveries followed in the subsequent decades. Two of these developments during the 1960s stand apart as having radically changed our outlook on nonlinear systems and the underlying dynamics.

In the year 1963, E.N. Lorenz numerically integrated a simplified system of the three coupled first-order nonlinear equations of the fluid convection model describing the atmospheric weather conditions. The bounded nonpe­riodic trajectories of the equations started from two nearby initial states diverged exponentially until they become completely uncorrelated resulting in unpredictability of future state in a fully deterministic dynamical system. Such a solution became known as chaotic and, with this discovery, the field of chaotic dynamics was born.

Not much later, in 1965, Zabusky and Kruskal numerically analysed the initial value problem of the Korteweg-de Vries (KdV) equation, which repre­sents a nonlinear dispersive system. They observed a phenomenon completely opposite to that of chaos. In their experiments solitary waves interacted among themselves and re-emerged unchanged in form and speed. Because of the particle-like nature of the collision of solitary waves, Zabusky and Kruskal coined the name soliton to describe such a solitary wave. Kruskal and coworkers went on to develop a completely analytic procedure called the inverse scattering transform (1ST) to solve the initial value problem of the KdV equation. This marked the advent of the modern era of integrable nonlinear systems.

Independent of these developments, various important studies on non­linear diffusive and dissipative systems and the underlying patterns were pursued during this period. Subsequently, starting in the 1970s, the study of nonlinear dynamical systems experienced an explosive growth. Analytical and numerical tools were developed and fascinating results obtained. In re­cent times increasing attention has been focussed on exploring real technolog­ical applications of nonlinear dynamics: Controlling of chaos, synchronization of chaos and secure communication, cryptography, optical-soliton-based com­munication, magnetoelectronics, spatio-temporal patterns to name but a few.

VIII Preface

Applications of nonlinear dynamics have been found throughout the realms of physics, engineering, chemistry and biology. Numerous mathematical ideas and techniques have been used to study nonlinear systems, and these, in turn, have enriched the field of mathematics itself. The field of nonlinear dynamics has hence emerged as a highly interdisciplinary endeavour.

Considering its multidisciplinary nature and important practical applica­tions, the topic of nonlinear dynamics has now been introduced as part of advanced level undergraduate, graduate and masters-level courses in many countries and the number of researchers on this field is continuing to grow. The present authors have spent a major part of their academic careers in studying and understanding the many-faceted and facinating features of non­linear dynamics. They have also been giving courses on various aspects of nonlinear dynamics to masters-level and research students in physics and mathematics for several years. The authors have now endeavoured to de­velop their teaching materials into an advanced level text book containing 16 chapters and 10 appendices, hopefully catering to the needs of advanced undergraduate, graduate and masters-level course students in physics, math­ematics and engineering, who aspire to obtain a sound basic knowledge of nonlinear dynamics. The material covered includes in a rather unified way the three major themes of nonlinear dynamics: Chaos, integrability (includ­ing solitons) and spatio-temporal patterns. Ideally, the material can be fully covered in a two semester course: One possibility is to have one semester on chaos and one semester on integrability and patterns. Another is to have a basic course on chaos and integrability (Chaps. 1-5, 7, 11-13) and an ad­vanced course on the remaining materials (covered in Chaps. 6,8-10,14-16). A single semester course can also be devised by including relevant chapters and the lecturer will be able to identify the suitable parts very easily since the text is clearly structured.

Regarding the content, the first nine chapters are concerned with bifurca­tions and chaos. Starting from the basic notions of nonlinearity and nonlinear dynamical systems we have introduced the concept of attractors, bifurcations, chaos and characterization of regular and chaotic motions. These aspects are analysed in simple dissipative and conservative systems and in electronic cir­cuits. Additional compact discussions of specialized and advanced topics on chaos are presented in Chap. 9. Chapters 10-14 deal with integrability and integrable systems. In particular, Chap. 10 is devoted to integrable finite di­mensional nonlinear systems. Starting from the notion of integrability, we describe the analytical methods used to identify integrability and employ them to study typical nonlinear systems. Chapters 11-14 are concerned with the study of solitonic systems. After introducing the notion of solitary waves and solitons, the inverse scattering transform method, Hirota's bilinearization and the Backlund transformation techniques are presented and the notion of completely integrable infinite-dimensional nonlinear dynamical systems are elucidated. These methods are applied to many nonlinear wave equations of

Preface IX

contemporary interest. Then in Chap. 15, spatio-temporal patterns in non­linear reaction-diffusion systems are presented. Some of the potential techno­logical applications of chaos and solitons are discussed in Chap. 16. The book also contains ten appendices which deal with several important concepts and methods which could not be presented in the main text without disrupting the continuity and style of presentation of the topics concerned.

The book also contains numerous Exercises and Problems which the the authors hope will enhance the understanding of the subject to the level of present-day active research. In our classification the 'Exercises' are essentially meant to augment the discussion and derivation in the text, while the 'Prob­lems' are generally of a more advanced nature. Some of the problems even relate to current research investigations and we hope that they will motivate the student to become involved in contemporary research in nonlinear dy­namics. They will also help students and teachers to identify suitable projects for further investigations. Necessary references to the literature are included chapter-by-chapter (towards the end of the book), among them publications that will be helpful for tackling the Exercises and Problems.

During the preparation of this book we have received considerable support from many colleagues, students and friends. In particular we are grateful to Prof. A. Kundu, Dr. K.M. Tamizhmani, Dr. R. Sahadevan, Dr. S.N. Pandey, Dr. S. Parthasarathy, Dr. M. Senthil Velan, Dr. K. Murali, Dr. P. Muru­ganandam, and Dr. R. Sankaranarayanan among others for their critical reading and suggestions on various parts of the book. It is a pleasure to thank Dr. A. Venkatesan, Dr. P. Philominathan, Dr. V. Chinnathambi, Mr. K. Thamilmaran, Mr. T. Kanna, Mr. P. Palaniyandi, Ms. P.S. Bindu, Mr. C. Senthil Kumar, Mr. D.V. Senthil Kumar and Mr. V.K. Chandrasekar for their assistance in the preparation of some of the figures, checking the derivations and problems, etc. However, the authors are solely responsible for any shortcomings, errors or misconceptions that remain. A few of the illustrations are reproduced from other sources and appropriate references are given at the relevant places. We sincerely thank the respective authors and publishers for granting us permission to use these figures. It is also a pleasure for us to record our thanks to the Department of Science of Tech­nology, Government of India for providing support under various research projects, which enabled us to undertake this task. We also appreciate very much the various suggestions of Springer-Verlag, especially Dr. Angela Lahee of the Physics Editorial Department, on the manuscript. Finally, we thank our family members for their unflinching support and encouragement during the course of this project, without which it would not have been possible to complete it.

Tiruchirapalli, July 2002

M. Lakshmanan S. Rajasekar

Contents

1. What is Nonlinearity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Dynamical Systems: Linear and Nonlinear Forces. . . . . . . . . . 2 1.2 Mathematical Implications of Nonlinearity. . . . . . . . . . . . . . . . 5

1.2.1 Linear and Nonlinear Systems. . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Linear Superposition Principle . . . . . . . . . . . . . . . . . . . . 7

1.3 Working Definition of Nonlinearity. . . . . . . . . . . . . . . . . . . . . .. 10 1.4 Effects of Nonlinearity ................................. 11

2. Linear and Nonlinear Oscillators. . . . . . . . . . . . . . . . . . . . . . . . .. 17 2.1 Linear Oscillators and Predictability. . . . . . . . . . . . . . . . . . . .. 17

2.1.1 Free Oscillations ................................ 18 2.1.2 Damped Oscillations ............................ 19 2.1.3 Damped and Forced Oscillations .................. 20

2.2 Damped and Driven Nonlinear Oscillators. . . . . . . . . . . . . . .. 21 2.2.1 Free Oscillations ................................ 22 2.2.2 Damped Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23 2.2.3 Forced Oscillations - Primary Resonance

and Jump Phenomenon (Hysteresis) ... . . . . . . . . . . .. 23 2.2.4 Secondary Resonances

(Subharmonic and Superharmonic) ................ 26 2.3 Nonlinear Oscillations and Bifurcations. . . . . . . . . . . . . . . . . .. 27 Problems .................................................. 29

3. Qualitative Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31 3.1 Autonomous and Nonautonomous Systems ............... 32 3.2 Dynamical Systems as Coupled First-Order

Differential Equations: Equilibrium Points. . . . . . . . . . . . . . .. 34 3.3 Phase Space/Phase Plane and Phase Trajectories:

Stability, Attractors and Repellers . . . . . . . . . . . . . . . . . . . . . .. 36 3.4 Classification of Equilibrium Points: Two-Dimensional Case. 38

3.4.1 General Criteria for Stability ..................... 38 3.4.2 Classification of Equilibrium (Singular) Points ...... 40

3.5 Limit Cycle Motion - Periodic Attractor ................. 50 3.5.1 Poincare-Bendixson Theorem. . . . . . . . . . . . . . . . . . . .. 52

XII Contents

3.6 Higher Dimensional Systems. . . . . . . . . . . . . . . . . . . . . . . . . . .. 54 3.6.1 Example: Lorenz Equations. . . . . . . . . . . . . . . . . . . . . .. 55

3.7 More Complicated Attractors . . . . . . . . . . . . . . . . . . . . . . . . . .. 58 3.7.1 Torus.......................................... 59 3.7.2 Quasiperiodic Attractor ......................... 62 3.7.3 Poincare Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63 3.7.4 Chaotic Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64

3.8 Dissipative and Conservative Systems. . . . . . . . . . . . . . . . . . .. 65 3.8.1 Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . . . . . . .. 68

3.9 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69

4. Bifurcations and Onset of Chaos in Dissipative Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75 4.1 Some Simple Bifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76

4.1.1 Saddle-Node Bifurcation ......................... 77 4.1.2 The Pitchfork Bifurcation. . . . . . . . . . . . . . . . . . . . . . .. 80 4.1.3 Transcritical Bifurcation ......................... 83 4.1.4 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85

4.2 Discrete Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89 4.2.1 The Logistic Map ............................... 90 4.2.2 Equilibrium Points and Their Stability. . . . . . . . . . . .. 91 4.2.3 Stability When the First Derivative

Equals to +1 or -1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92 4.2.4 Periodic Solutions or Cycles ...................... 94 4.2.5 Period Doubling Phenomenon. . . . . . . . . . . . . . . . . . . .. 96 4.2.6 Onset of Chaos: Sensitive Dependence

on Initial Conditions - Lyapunov Exponent. . . . . . . .. 98 4.2.7 Bifurcation Diagram ............................. 101 4.2.8 Bifurcation Structure in the Interval 3.57 ::; a ::; 4 ... 103 4.2.9 Exact Solution at a = 4 .......................... 104 4.2.10 Logistic Map: A Geometric Construction

of the Dynamics - Cobweb Diagrams .............. 105 4.3 Strange Attractor in the Henon Map ..................... 107

4.3.1 The Period Doubling Phenomenon ................. 108 4.3.2 Self-Similar Structure ............................ 110

4.4 Other Routes to Chaos ................................. 111 4.4.1 Quasiperiodic Route to Chaos ..................... 111 4.4.2 Intermittency Route to Chaos ..................... 113 4.4.3 Type-I Intermittency ............................ 114 4.4.4 Standard Bifurcations in Maps .................... 116

Problenls .................................................. 118

Contents XIII

5. Chaos in Dissipative Nonlinear Oscillators and Criteria for Chaos ................................... 123 5.1 Bifurcation Scenario in Duffing Oscillator ................. 124

5.l.1 Period Doubling Route to Chaos .................. 126 5.l.2 Intermittency Transition ......................... 130 5.l.3 Quasiperiodic Route to Chaos ..................... 132 5.l.4 Strange Nonchaotic Attractors (SNAs) ............. 133

5.2 Lorenz Equations ...................................... 135 5.2.1 Period Doubling Bifurcations and Chaos ........... 136

5.3 Some Other Ubiquitous Chaotic Oscillators ............... 142 5.3.1 Driven van der Pol Oscillator ..................... 142 5.3.2 Damped, Driven Pendulum ...................... 142 5.3.3 Morse Oscillator ................................ 145 5.3.4 Rossler Equations ............................... 146

5.4 Necessary Conditions for Occurrence of Chaos ............ 147 5.4.1 Continuous Time Dynamical Systems

(Differential Equations) .......................... 147 5.4.2 Discrete Time Systems (Maps) .................... 148

5.5 Computational Chaos, Shadowing and All That ........... 151 5.6 Conclusions ........................................... 153 Problems .................................................. 153

6. Chaos in Nonlinear Electronic Circuits ................... 159 6.1 Linear and Nonlinear Circuit Elements ................... 159 6.2 Linear Circuits: The Resonant RLC Circuit ............... 161 6.3 Nonlinear Circuits ..................................... 165

6.3.1 Chua's Diode: Autonomous Case .................. 165 6.3.2 A Simple Practical Implementation of Chua's Diode. 167 6.3.3 Bifurcations and Chaos .......................... 167

6.4 Chaotic Dynamics of the Simplest Dissipative Nonautonomous Circuit: Murali-Lakshmanan-Chua (MLC) Circuit ................ 171 6.4.1 Experimental Realization ......................... 171 6.4.2 Stability Analysis ............................... 172 6.4.3 Explicit Analytical Solutions ...................... 173 6.4.4 Experimental and Numerical Studies ............... 174

6.5 Analog Circuit Simulations ............................. 178 6.6 Some Other Useful Nonlinear Circuits .................... 181

6.6.1 RL Diode Circuit ................................ 181 6.6.2 Hunt's Nonlinear Oscillator ....................... 182 6.6.3 p-n Junction Diode Oscillator ..................... 182 6.6.4 Modified Chua Circuit ........................... 182 6.6.5 Colpitt's Oscillator .............................. 184

6.7 Nonlinear Circuits as Dynamical Systems ................. 185 Problems .................................................. 185

XIV Contents

7. Chaos in Conservative Systems ........................... 191 7.1 Poincare Cross Section or Surface of Section .............. 192 7.2 Possible Orbits in Conservative Systems .................. 196

7.2.1 Regular Trajectories ............................. 197 7.2.2 Irregular Trajectories ............................ 201 7.2.3 Canonical Perturbation Theory:

Overlapping Resonances and Chaos ................ 202 7.3 Henon-Heiles System .................................. 204

7.3.1 Equilibrium Points .............................. 206 7.3.2 Poincare Surface of Section of the System .......... 207 7.3.3 Numerical Results .............................. 208

7.4 Periodically Driven Undamped Duffing Oscillator .......... 213 7.5 The Standard Map .................................... 216

7.5.1 Linear Stability and Invariant Curves .............. 217 7.5.2 Numerical Analysis: Regular and Chaotic Motions ... 222

7.6 Kolmogorov-Arnold-Moser Theorem ..................... 226 7.7 Conclusions ........................................... 227 Problenls .................................................. 228

8. Characterization of Regular and Chaotic Motions ..................................... 235 8.1 Lyapunov Exponents ................................... 235 8.2 Numerical Computation of Lyapunov Exponents .......... 238

8.2.1 One-Dimensional Map ........................... 238 8.2.2 Computation of Lyapunov Exponents

for Continuous Time Dynamical Systems ........... 239 8.3 Power Spectrum ....................................... 245

8.3.1 The Power Spectrum and Dynamical Motion ........ 245 8.4 Autocorrelation ....................................... 250 8.5 Dimension ............................................ 253 8.6 Criteria for Chaotic Motion ............................. 255 Problenls .................................................. 258

9. Further Developments in Chaotic Dynamics .............. 259 9.1 Time Series Analysis ................................... 260

9.1.1 Estimation of Time-Delay and Embedding Dimension . . . . . . . . . . . . . . . . . . . . . . . 260

9.1.2 Largest Lyapunov Exponent ...................... 261 Problems .................................................. 261 9.2 Stochastic Resonance .................................. 262 Problenls .................................................. 264 9.3 Chaotic Scattering .................................... 266 Problems .................................................. 268 9.4 Controlling of Chaos ................................... 269

9.4.1 Controlling and Controlling Algorithms ............ 270

Contents XV

9.4.2 Stabilization of UPO ............................. 271 Problenls .................................................. 274 9.5 Synchronization of Chaos ............................... 277

9.5.1 Chaos in the DVP Oscillator ..................... 277 9.5.2 Synchronization of Chaos in the DVP Oscillator .... 278 9.5.3 Chaotic Signal Masking and Transmission

of Analog Signals ............................... 280 9.5.4 Chaotic Digital Signal Transmission ............... 282

Problems .................................................. 284 9.6 Quantum Chaos ....................................... 284

9.6.1 Quantum Signatures of Chaos .................... 284 9.6.2 Rydberg Atoms and Quantum Chaos .............. 287 9.6.3 Hydrogen Atom in a

Generalized van der Waals Interaction ............. 289 9.6.4 Outlook ....................................... 291

Problems .................................................. 293 9.7 Conclusions ........................................... 293

10. Finite Dimensional Integrable Nonlinear Dynamical Systems ............................ 295 10.1 What is Integrability? .................................. 296 10.2 The Notion of Integrability ............................. 297 10.3 Complete Integrability -

Complex Analytic Integrability .......................... 300 10.3.1 Real Time and Complex Time Behaviours .......... 301 10.3.2 Partial Integrability and Constrained Integrability ... 302 10.3.3 Integrability and Separability ..................... 302

10.4 How to Detect Integrability: Painleve Analysis ............ 305 10.4.1 Classification of Singular Points ................... 306 10.4.2 Historical Development of the Painleve Approach

and Integrability of Ordinary Differential Equations . 307 10.4.3 Painleve Method of Singular Point Analysis

for Ordinary Differential Equations ................ 311 10.5 Painleve Analysis and Integrability

of Two-Coupled Nonlinear Oscillators .................... 317 10.5.1 Quartic Anharmonic Oscillators .................. 317

10.6 Symmetries and Integrability ........................... 321 10.6.1 Invariance Conditions,

Determination of Infinitesimals and First Integrals of Motion ..................... 323

10.6.2 Application - The Henon-Heiles System ............ 326 10.7 A Direct Method of Finding Integrals of Motion ........... 330 10.8 Integrable Systems with Degrees

of Freedom Greater Than Two .......................... 331 10.9 Integrable Discrete Systems ............................. 333

XVI Contents

10.10 Integrable Dynamical Systems on Discrete Lattices .................................... 335

10.11 Conclusion ........................................... 336 Problems .................................................. 337

11. Linear and Nonlinear Dispersive Waves .................. 341 11.1 Linear Waves ......................................... 341 11.2 Linear Nondispersive Wave Propagation .................. 342 11.3 Linear Dispersive Wave Propagation ..................... 343 11.4 Fourier Transform and Solution

of Initial Value Problem ............................... 345 11.5 Wave Packet and Dispersion ............................ 348 11.6 Nonlinear Dispersive Systems ........................... 350

11.6.1 An Illustration of the Wave of Permanence ......... 350 11.6.2 John Scott Russel's Great Wave of Translation ...... 350

11. 7 Cnoidal and Solitary Waves ............................. 352 11.7.1 Korteweg-de Vries Equation and the Solitary Waves

and Cnoidal Waves .............................. 352 11.8 Conclusions ........................................... 355 Problems .................................................. 355

12. Korteweg-de Vries Equation and Solitons ................. 359 12.1 The Scott Russel Phenomenon and KdV Equation ......... 359 12.2 The Fermi-Pasta-Ulam Numerical Experiments

on Anharmonic Lattices ............................... 366 12.2.1 The FPU Lattice ............................... 366 12.2.2 FPU Recurrence Phenomenon .................... 368

12.3 The KdV Equation Again! .............................. 369 12.3.1 Asymptotic Analysis and the KdV Equation ........ 369

12.4 Numerical Experiments of Zabusky and Kruskal: The Birth of Solitons .................................. 372

12.5 Hirota's Direct or Bilinearization Method for Soliton Solutions of KdV Equation ................... 375

12.6 Conclusions ........................................... 380

13. Basic Soliton Theory of KdV Equation ................... 381 13.1 The Miura Transformation and Linearization

of KdV: The Lax Pair .................................. 382 13.1.1 The Miura Transformation ....................... 382 13.1.2 Galilean Invariance

and Schrodinger Eigenvalue Problem .............. 383 13.l.3 Linearization of the KdV Equation ................ 384 13.l.4 Lax Pair ....................................... 385

13.2 Lax Pair and the Method of Inverse Scattering: A New Method to Solve the Initial Value Problem ......... 386

Contents XVII

13.2.1 The Inverse Scattering Transform (1ST) Method for KdV Equation ............................... 386

13.3 Explicit Soliton Solutions ............................... 390 13.3.1 One-Soliton Solution (N = 1) ..................... 390 13.3.2 Two-Soliton Solution ............................ 392 13.3.3 N-Soliton Solution .............................. 393 13.3.4 Soliton Interaction ............................... 394 13.3.5 Nonreflectionless Potentials ...................... 395

13.4 Hamiltonian Structure of KdV Equation .................. 395 13.4.1 Dynamics of Continuous Systems .................. 396 13.4.2 KdV as a Hamiltonian Dynamical System .......... 398 13.4.3 Complete Integrability of the KdV Equation ........ 399

13.5 Infinite Number of Conserved Densities .................. 402 13.6 Backlund Transformations .............................. 403 13.7 Conclusions ........................................... 405

14. Other Ubiquitous Soliton Equations ...................... 407 14.1 Identification of Some Ubiquitous

Nonlinear Evolution Equations from Physical Problems .... 408 14.1.1 The Nonlinear Schrodinger Equation

in Optical Fibers ................................ 409 14.1.2 The Sine-Gordon Equation

in Long Josephson Junctions ...................... 410 14.1.3 Dynamics of Ferromagnets:

Heisenberg Spin Equations ....................... 412 14.1.4 The Lattice with Exponential Interaction:

The Toda Equation .............................. 414 14.2 The Zakharov-Shabat (ZS)j

Ablowitz-Kaup-Newell-Segur (AKNS) Linear Eigenvalue Problem and NLEES .................. 414 14.2.1 The AKNS Linear Eigenvalue Problem

and AKNS Equations ............................ 415 14.2.2 The Standard Soliton Equations .................. 416

14.3 Solitary Wave Solutions and Basic Solitons ............... 418 14.3.1 The MKdV Equation: Pulse Soliton ............... 418 14.3.2 The sine-Gordon Equation:

Kink, Antikink and Breathers ..................... 419 14.3.3 The Nonlinear Schrodinger Equation:

Envelope Soliton ................................ 424 14.3.4 The Heisenberg Spin Equation: The Spin Soliton .... 425 14.3.5 The Toda Lattice: Discrete Soliton ................ 426

14.4 Hirota's Method and Soliton Nature of Solitary Waves ...................................... 427 14.4.1 The Modified KdV Equation ..................... 427 14.4.2 The NLS Equation .............................. 429

XVIII Contents

14.4.3 The sine-Gordon Equation ........................ 431 14.4.4 The Heisenberg Spin System ...................... 432

14.5 Solutions via 1ST Method .............................. 434 14.5.1 Direct and Inverse Scattering ..................... 434 14.5.2 Time Evolution of the Scattering Data ............. 435 14.5.3 Soliton Solutions ................................ 436

14.6 Backlund Transformations ............................. 438 14.7 Conservation Laws and Constants of Motion .............. 440 14.8 Hamiltonian Structure and Integrability .................. 444

14.8.1 Hamiltonian Structure ........................... 444 14.8.2 Complete Integrability of the NLS Equation ........ 445

14.9 Conclusions ........................................... 448 Problenls .................................................. 451

15. Spatio-Temporal Patterns ................................ 455 15.1 Linear Diffusion Equation .............................. 456 15.2 Nonlinear Diffusion and Reaction-Diffusion Equations ...... 458

15.2.1 Nonlinear Reaction-Diffusion Equations ............ 459 15.2.2 Dissipative Systems .............................. 461

15.3 Spatio-Temporal Patterns in Reaction-Diffusion Systems ... 462 15.3.1 Homogeneous Patterns ........................... 463 15.3.2 Autowaves: Travelling Wave Fronts, Pulses, etc ...... 463 15.3.3 Ring Waves, Spiral Waves and Scroll Waves ........ 468 15.3.4 Turing Instability and Turing Patterns ............. 471 15.3.5 Localized Structures ............................. 477 15.3.6 Spatio-Temporal Chaos .......................... 478

15.4 Cellular Neural/Nonlinear Networks (CNNs) .............. 482 15.4.1 Cellular Nonlinear Networks (CNNs) ............... 482 15.4.2 Arrays of MLC Circuits: Simple Examples of CNN .. 484 15.4.3 Active Wave Propagation and its Failure

in One-Dimensional CNN s . . . . . . . . . . . . . . . . . . . . . . . . 485 15.4.4 Turing Patterns ................................. 487 15.4.5 Spatio-Temporal Chaos .......................... 488

15.5 Some Exactly Solvable Nonlinear Diffusion Equations ...... 492 15.5.1 The Burgers Equation ........................... 492 15.5.2 The Fokas-Yortsos-Rosen Equation ................ 492 15.5.3 Generalized Fisher's Equation ..................... 493

15.6 Conclusion ........................................... 494 Problenls .................................................. 494

16. Nonlinear Dynamics: From Theory to Technology .............................. 497 16.1 Chaotic Cryptography ................................. 498

16.1.1 Basic Idea of Cryptography ....................... 498 16.1.2 An Elementary Chaotic Cryptographic System ...... 498

Contents XIX

16.2 Using Chaos (Controlling) to Calm the Web .............. 500 16.3 Some Other Possibilities of Using Chaos .................. 504

16.3.1 Communicating by Chaos ........................ 504 16.3.2 Chaos and Financial Markets ..................... 505

16.4 Optical Soliton Based Communications ................... 506 16.5 Soliton Based Optical Computing ....................... 508

16.5.1 Photo-Refractive Materials and the Manakov Equation ....................... 508

16.5.2 Soliton Solutions and Shape Changing Collisions .... 509 16.5.3 Optical Soliton Based Computation ................ 513

16.6 Micromagnetics and Magnetoelectronics .................. 519 16.7 Conclusions ........................................... 521

A. Elliptic Functions and Solutions of Certain Nonlinear Equations ........................... 523 Problenls .................................................. 530

B. Perturbation and Related Approximation Methods .................................. 532 B.1 Approximation Methods

for Nonlinear Differential Equations ...................... 532 B.2 Canonical Perturbation Theory

for Conservative Systems ............................... 536 B.2.1 One Degree of Freedom Hamiltonian Systems ....... 536 B.2.2 Two Degrees of Freedom Systems ................. 538

Problenls .................................................. 540

C. A Fourth-Order Runge-Kutta Integration Method ....................................... 542 Problenls .................................................. 544

D. Nature of Phase Space Trajectories for AI, A2 < 0 and Al < 0 < A2 (Sect. 3.4.2) .............. 545 Problenls .................................................. 546

E. Fractals and Multifractals ................................ 547 Problenls .................................................. 551

F. Spectrum of the sech2 ax Potential ....................... 553 Problenls .................................................. 555

G. Inverse Scattering Transform for the Schrodinger Spectral Problem .................... 556 G.1 The Linear Eigenvalue Problem ......................... 556 G.2 The Direct Scattering Problem .......................... 557 G.3 The Inverse Scattering Problem ......................... 559

XX Contents

G.4 Reconstruction of the Potential. ......................... 561 Problenls .................................................. 561

H. Inverse Scattering Transform for the Zakharov-Shabat Eigenvalue Problem ............ 562 H.l The Linear Eigenvalue Problem ......................... 562 H.2 The Direct Scattering Problem .......................... 563 H.3 Inverse Scattering Problem ............................. 565 H.4 Reconstruction of the Potentials ......................... 566 ProblenlS .................................................. 567

I. Integrable Discrete Soliton Systems ...................... 568 1.1 Integrable Finite Dimensional N-Particles System

on a Line: Calogero-Moser System ....................... 568 1.2 The Toda Lattice ...................................... 570 1.3 Other Discrete Lattice Systems .......................... 572 1.4 Solitary Wave (Soliton) Solution of the Toda Lattice ....... 573 Problems .................................................. 575

J. Painleve Analysis for Partial Differential Equations ........................ 576 J.l The Painleve Property for PDEs ........................ 576

J .1.1 Painleve Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 J.2 Exanlples ............................................. 578

J.2.1 KdV Equation .................................. 578 J.2.2 The Nonlinear Schrodinger Equation ............... 581

Problems .................................................. 584

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

References .................................................... 597

Index ......................................................... 611