Nonlinear Waves, Solitons and Chaos

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ISBN: 9780521635578 | 0521635578
Cover: Paperback
Copyright: 7/31/2000

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This revised and updated second edition of a highly successful book is the only text at this level to embrace a universal approach to three major developments in classical physics; namely nonlinear waves, solitons and chaos. The authors now include new material on biology and laser theory, and go on to discuss important recent developments such as soliton metamorphosis. A comprehensive treatment of basic plasma and fluid configurations and instabilities is followed by a study of the relevant nonlinear structures. Each chapter concludes with a set of problems. This text will be particularly valuable for students taking courses in nonlinear aspects of physics. In general, it will be of value to final year undergraduates and beginning graduate students studying fluid dynamics, plasma physics and applied mathematics.

Foreword to the first edition

Foreword to the second edition

xiii

Introduction

(24)

Occurrence of nonlinear waves and instabilities in Nature

(4)

Nonlinear phenomena in our everyday experience

(2)

Nonlinear phenomena in the laboratory

(2)

Universal wave equations

(4)

The Korteweg--de Vries and Kadomtsev--Petviashvili equations and a first look at solitons

(3)

The nonlinear Schrodinger equation

(1)

Nonlinear optics

(1)

What is a plasma?

(3)

Wave modes on a water surface

(6)

Mathematical theory

(2)

Comments

(4)

Linear stability analysis and its limitations

(1)

Nonlinear structures

(5)

Coherent structures and pattern formation

(1)

Contents of Chapters 2--11

(2)

Linear waves and instabilities in infinite media

(23)

Introduction

(1)

Plasma waves

(5)

CMA diagrams

(3)

Instabilities

(4)

The Vlasov equation

(6)

Weak instabilities

(5)

Exercises

(1)

Convective and non-convective instabilities; group velocity in unstable media

(11)

Introduction

(2)

Kinematics of unstable wavepackets

(4)

Moving coordinate systems

(2)

Higher dimensional systems

(1)

Summary

(2)

Exercise

(2)

A first look at surface waves and instabilities

(23)

Introduction

(2)

Simple surface waves

(5)

The Rayleigh--Taylor instability

(2)

The Kelvin--Helmholtz instability

(3)

Solid--liquid interface instabilities

(1)

A first look at gravity wave instabilities

(9)

The small amplitude onset of wave instability

(2)

Further numerical results

(6)

Summary

(1)

Exercises

(1)

Model equations for small amplitude waves and solitons; weakly nonlinear theory

(41)

Introduction

(3)

Some physical equations ask for surgery

(1)

Examples

(2)

A few model equations as derived by introducing a small parameter

(7)

Shallow water, weak amplitude gravity waves

(4)

Weak amplitude ion acoustic waves in an unmagnetized plasma

(2)

Weak amplitude ion acoustic waves in a magnetized plasma

(1)

Weakly nonlinear waves

(15)

Spreading, splitting and instabilities

(6)

The story of deep water waves

(2)

Mystery of the missing term

100

(2)

Dynamics of a wavepacket

102

(2)

Some generalizations

104

(3)

A general look at two families of model equations

107

(4)

A natural extension to finite amplitude waves due to Hayes

111

(3)

Temporal development of instabilities and wave-wave coupling

114

(5)

Concluding remarks

119

(4)

Exercises

119

(4)

Exact methods for fully nonlinear waves and solitons

123

(43)

Introduction

123

(1)

Phase plane analysis and other methods

124

(19)

One stationary wave in a dissipationless medium

124

(5)

A two-fluid layer soliton pair

129

(3)

Weak ion acoustic shock waves in a collisional plasma

132

(2)

Solitons generated by laser fields

134

(2)

Solitons and domains in dipole chains

136

(3)

Discrete equations

139

(4)

Bernstein--Greene--Kruskal waves

143

(9)

Statistical description of a plasma and BGK waves

144

(1)

No trapped particles

145

(1)

Various limits

146

(1)

Trapped particle equilibria

147

(3)

Stability; subsequent developments

150

(2)

Lagrangian methods

152

(7)

Lagrangian interpolation

159

(7)

Exercises

164

(2)

Cartesian solitons in one and two space dimensions

166

(36)

Introduction

166

(2)

The direct method

168

(3)

Constants of motion

171

(2)

Inverse scattering method

173

(2)

Backlund transformations

175

(2)

Entr'acte

177

(1)

Breathers and boundary effects

178

(2)

Experimental evidence

180

(1)

Plane soliton interaction in two space dimensions

181

(11)

Introducing the trace method

181

(2)

One and two soliton solutions

183

(5)

Some other developments and summary

188

(4)

Integrable equations in two space dimensions as treated by the Zakharov--Shabat method

192

(7)

Lax pairs and the PDEs they represent

193

(1)

Extension to x,y,t

194

(1)

How to proceed from the Lax pair to the general solution

195

(1)

An example: the Kadomtsev-Petviashvili equation

196

(3)

Summary

199

(3)

Exercises

200

(2)

Evolution and stability of initially one-dimensional waves and solitons

202

(66)

A brief historical survey of large amplitude nonlinear wave studies

202

(11)

Solitons

204

(2)

Water waves are unstable

206

(1)

The geometrical optics limit

207

(4)

More recent results

211

(1)

What the remainder of Chapter 8 is about

212

(1)

Four methods as illustrated by the nonlinear Klein--Gordon equation

213

(11)

Whitham I

214

(5)

Whitham II

219

(1)

K expansion

220

(3)

Hayes

223

(1)

Higher dimensional dynamics

224

(21)

Kadomtsev--Petviashvili as analysed by Whitham II

224

(6)

Various limits

230

(1)

Common features of the weak amplitude and soliton limits for ψ = 0

231

(2)

Group velocity

233

(3)

Zakharov-Kuznetsov as analysed by K expansion

236

(8)

The variational method

244

(1)

A more physical approach leading to an assessment of models

245

(7)

Form of the waves considered

245

(1)

Unmagnetized plasmas, Ωc = 0

246

(3)

Magnetized plasmas, Ωc > 0

249

(3)

Dynamics of nonlinear wave, shock and soliton solutions to the cubic nonlinear Schrodinger equation

252

(6)

Results of a general stability calculation

253

(1)

One-dimensional dynamics: ψ = 0

254

(3)

Oblique and perpendicular propagation of perturbations

257

(1)

The direct K method

258

(6)

Transverse instability of Zakharov--Kuznetsov solitons

259

(4)

The Cahn--Hilliard equation

263

(1)

Some general conclusions and possible future lines of investigation

264

(4)

Exercises

265

(3)

Cylindrical and spherical solitons in plasmas and other media

268

(28)

Interest in higher dimensional plasma solitons

268

(1)

Unidirectional cylindrical and spherical ion acoustic solitons

269

(3)

Model equations in non-Cartesian geometry

269

(1)

Cylindrical soliton equations CI and CII

269

(2)

Spherical solitons

271

(1)

Summary

272

(1)

Properties of unidirectional soliton equations

272

(3)

Integrability by inverse scattering

272

(1)

Conservation laws

273

(2)

Soliton solutions as compared with numerics and experiments

275

(12)

Exact solutions to CI

277

(1)

Initial value problem and experiments

277

(3)

Reflection from the axis (centre)

280

(4)

Models

284

(3)

Stability of cylindrical solitons

287

(1)

Langmuir solitons

287

(4)

Integrability

288

(1)

Stability of Langmuir solitons

288

(3)

Interacting solitons and some conclusions

291

(1)

Epilogue. Some other examples of spherical and cylindrical solitons

292

(4)

Exercises

294

(2)

Soliton metamorphosis

296

(8)

The next step in investigating soliton behaviour

296

(1)

Decay of line KPI solitons in two dimensions

297

(4)

Decay of 2D solitons in three dimensions

301

(2)

2D solitons perturbed perpendicular to the motion

301

(1)

2D solitons perturbed parallel to the velocity

302

(1)

Conclusions

303

(1)

Exercises

303

(1)

Non-coherent phenomena

304

(42)

Introduction

304

(7)

Bifurcation sequences and chaos

311

(15)

Flows and maps

326

(3)

Strange attractors

329

(11)

Effect of external noise

340

(1)

Experimental evidence for strange attractors

341

(2)

Other theories of turbulence

343

(1)

Conclusions

344

(2)

Exercises

345

(1)

Appendices

346

(14)

A1 Parameter stretching as suggested by the linear dispersion relations

346

(3)

A1.1 Ion acoustic waves in an unmagnetized plasma, Ωc = 0

346

(1)

A1.2 Magnetized plasmas, Ωc > 0

347

(2)

A2 Relation between the trace method and the inverse scattering method

349

(2)

A3 Some formulae for perturbed nonlinear ion acoustic waves and solitons

351

(3)

A3.1 No magnetic field

351

(1)

A3.2 Ωc > 0

352

(2)

A4 Colliding soliton theory

354

(2)

A5 A model equation for spherical solitons

356

(2)

A6 Stability calculation for 2D KPI soliton in 3D

358

(2)

References

360

(19)

Author index

379

(8)

Subject index

387

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Nonlinear Waves, Solitons and Chaos

Summary

Table of Contents

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