Nonlinear waves, solitons, and chaos

Title......Page 1
Contents ......Page 3
Foreword ......Page 8
1.1.1 Nonlinear phenomena in our everyday experience ......Page 10
1.1.2 Nonlinear phenomena in the laboratory ......Page 11
1.2.1 The Korteweg-de Vries and Kadomtsev-Petviashvili equations and a first look at solitons ......Page 15
1.2.2 The nonlinear Schrodinger equation ......Page 18
1.3 What is a plasma? ......Page 20
1.4 Wave modes on a water surface ......Page 23
1.4.1 Mathematical iheory ......Page 24
1.4.2 Comments ......Page 26
1.5 Linear stability analysis and its limitations ......Page 28
1.6 Chaos, turbulence and strange attractors ......Page 32
1.7 Contents of Chapters 2-10 ......Page 35
2.2 Plasma waves ......Page 38
2.3 CMA diagrams ......Page 44
2.4 Instabilities......Page 48
2.5 The Vlasov equation ......Page 53
2.6 Weak instabilities ......Page 59
Exercises ......Page 64
3.1 Introduction ......Page 65
3.2 Kinematics of unstable wave packets ......Page 67
3.3 Moving coordinate syslems ......Page 72
3.4 Higher dimensional systems ......Page 75
Exercise ......Page 76
4.1 Introduction ......Page 77
4.2 Simple surface waves ......Page 80
4.3 The Rayieigh -Taylor instability ......Page 86
4.4 The Keivin Helmholtz instability ......Page 88
4.5 Solid-liquid interface instabilities ......Page 92
4.6.1 The small amplitude onset of wave instability ......Page 94
4.6.2 Further numerical results ......Page 102
Exercises ......Page 104
5.1.1 Some physical equations ask for surgery ......Page 105
5.1.2 Examples ......Page 106
5.2 A few model equations as derived by introducing a small parameter ......Page 108
5.2.1 Shallow water, weak amplitude gravity waves ......Page 109
5.2.2 Weak amplitude ion acoustic waves in an unmagnetized plasma ......Page 113
5.2.3 Weak amplitude ion acoustic waves in a magnetized plasma ......Page 115
5.3.1 Spreading, splitting and instabilities ......Page 117
5.3.2 The story of deep water waves ......Page 124
5.3.3 Mystery of the missing term ......Page 126
5.3.4 Dynamics of a wavepacket ......Page 129
5.3.5 Some generalizations ......Page 131
5.4 A general look at two families of model equations ......Page 136
5.5 A natural extension to finite amplitude waves due to Hayes ......Page 139
5.6 Temporal development of instabilities and wave-wave coupling ......Page 143
5.7 Concluding remarks ......Page 149
Exercises ......Page 150
6.1 Introduction ......Page 153
6.2.1 One stationary wave in a dispersionless medium ......Page 154
6.2.2 A two-ftuid layer soliton pair ......Page 161
6.3.1 Statistical description of a plasma and BGK waves ......Page 167
6.3.2 No trapped particles ......Page 168
6.3.3 Various limits ......Page 169
6.3.4 Trapped particle equilibria ......Page 171
6.3.5 Stability: subsequent developments ......Page 174
6.4 Lagrangian methods ......Page 177
6.5 Lagrangian interpolation ......Page 185
Exercises ......Page 191
7.1 Introduction ......Page 193
7.2 The direct method ......Page 196
7.3 Constants of motion ......Page 200
7.4 Inverse scattering method ......Page 201
7.5 Baeklund transformations ......Page 205
7.6 Entr'acte ......Page 207
7.7 Breathers and boundary effects ......Page 208
7.8 Experimental evidence ......Page 211
7.9.1 Introducing the trace method ......Page 212
7.9.2 One and two soliton solutions ......Page 214
7.9.3 Some other developments and summary ......Page 222
7.10 Integrable equations in two space dimensions as treated by the Zakharov-Shabat method ......Page 225
7.10.1 Lax pairs and the partial differential equations they represent ......Page 226
7.10.2 Extension to x,y,t ......Page 227
7.10.3 How to proceed from the Lax pair to the general solution ......Page 228
7.10.4 An example: the Kadomtsev-Petviashvili equation ......Page 229
7.11 Summary ......Page 232
Exercises ......Page 233
8.1 A brief historical survey of large amplitude nonlinear wave studies ......Page 235
8.1.1 Solitons ......Page 237
8.1.2 Water waves are unstable ......Page 240
8.1.3 The geometrical optics limit ......Page 241
8.1.4 Recent results ......Page 246
8.1.5 What the remainder of Chapter 8 is about ......Page 248
8.2 Four methods as illustrated by the nonlinear Klein-Gordon equation ......Page 249
8.2.1 Whitham I ......Page 250
8.2.2 Whitham II ......Page 256
8.2.3 K expansion ......Page 257
8.3 Higher dimemional dynamics ......Page 261
8.3.1 Kadomlsev-Petviashvili as analysed by Whitham II ......Page 262
8.3.3 Common features of the weak amplitude and soliton limits for \psi=0. ......Page 271
8.3.4 Group velocity ......Page 272
8.3.5 Zakharov-Kuznetsov as analysed by K expansion ......Page 275
.3.6 The variational method ......Page 284
8.4 A more physical approach leading to an assessment of models ......Page 285
8.4.1 Form of the waves considered ......Page 286
.4.2 Unmagnetized plasmas, \Omega_c=0 ......Page 287
8.4.3 Magnetized plasmas, \Omega_c>0 ......Page 291
8.5 Dynamics of nonlinear wave, shock and solilon solutions to the cubic nonlinear Schrodinger equation ......Page 296
8.5.1 Results of a general stability calculation ......Page 297
8.5.2 One-dimensional dynamics: \psi=0 ......Page 300
8.5.3 Oblique and perpendicular propagation of perturbations ......Page 301
8.6 Some general conclusions and possible future lines of investigation ......Page 302
Exercises ......Page 305
9.1 Interest in higher dimensional plasma solitons ......Page 307
9.2.1 Model equations in non-Cartesian geometry ......Page 308
9.2.2 Cylindrical soliton equations CI and CII......Page 309
9.2.4 Summary ......Page 311
9.3.1 Integrability by inverse scattering ......Page 312
9.3.2 Conservation laws ......Page 313
9.4.1 Exact solutions to CI ......Page 317
9.4.2 Initial value problem and experiments ......Page 321
9.4.3 Reflection from the axis (centre) ......Page 324
9.4.4 Models ......Page 326
9.5 Langmuir solitons ......Page 328
9.5.2 Stability of Langmuir solitons ......Page 330
9.6 Interacting solitons and some conclusions ......Page 333
9.7 Epilogue: some other examples of spherical and cylindrical solitons ......Page 335
Exercises ......Page 337
10.1 Introduction ......Page 338
10.2 Bifurcation sequences and chaos ......Page 347
10.3 Flows and maps ......Page 365
10.4 Strange attractors ......Page 370
10.5 Effect of external noise ......Page 383
10.6 Experimental evidence for strange attractors ......Page 384
10.8 Conclusions ......Page 387
Exercises ......Page 388
1.1 Ion acoustic waves in an unmagnetized plasma, \Omega_c=0 ......Page 390
1.2 Magnetized plasmas, \Omega_c>0 ......Page 391
2 Relation between the trace method and the inverse scattering method ......Page 393
3.1 No magnetic field ......Page 394
3.2 \Omega_c>0 ......Page 395
4 Colliding soliton theory ......Page 397
5 A model equation for spherical solitons ......Page 399
References ......Page 401
Author index ......Page 422
Subject index ......Page 428