Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons

Cover......Page 1
Nonlinear Dispersive Waves......Page 5
ISBN 9781107012547  ISBN 9781107664104......Page 6
Contents......Page 7
Preface......Page 11
Acknowledgements......Page 16
PART I FUNDAMENTALS AND BASIC APPLICATIONS......Page 17
1 Introduction......Page 19
1.1 Solitons: Historical remarks......Page 27
Exercises......Page 30
2.1 Fourier transform method......Page 33
2.2 Terminology: Dispersive and non-dispersive equations......Page 35
2.4 Conservation laws......Page 38
2.5 Multidimensional dispersive equations......Page 39
2.6 Characteristics for first-order equations......Page 40
2.7 Shock waves and the Rankine–Hugoniot conditions......Page 43
2.8 Second-order equations: Vibrating string equation......Page 49
2.9 Linear wave equation......Page 51
2.10 Characteristics of second-order equations......Page 53
2.11 Classification and well-posedness of PDEs......Page 54
Exercises......Page 58
3 Asymptotic analysis of wave equations: Properties and analysis of Fourier-type integrals......Page 61
3.1 Method of stationary phase......Page 62
3.2 Linear free Schrodinger¨ equation......Page 64
3.3 Group velocity......Page 67
3.4 Linear KdV equation......Page 70
3.5 Discrete equations......Page 77
3.6 Burgers’ equation and its solution: Cole–Hopf transformation......Page 84
3.7 Burgers’ equation on the semi-infinite interval......Page 87
Exercises......Page 89
4 Perturbation analysis......Page 91
4.1 Failure of regular perturbation analysis......Page 92
4.2 Stokes–Poincaré frequency-shift method......Page 94
4.3 Method of multiple scales: Linear example......Page 97
4.4 Method of multiple scales: Nonlinear example......Page 100
4.5 Method of multiple scales: Linear and nonlinear pendulum......Page 102
Exercises......Page 112
5 Water waves and KdV-type equations......Page 114
5.1 Euler and water wave equations......Page 115
5.2 Linear waves......Page 119
5.3 Non-dimensionalization......Page 121
5.4 Shallow-water theory......Page 122
5.5 Solitary wave solutions......Page 134
Exercises......Page 144
6.1 NLS from Klein–Gordon......Page 146
6.2 NLS from KdV......Page 149
6.3 Simplified model for the linear problem and \"universality\"......Page 154
6.4 NLS from deep-water waves......Page 157
6.5 Deep-water theory: NLS equation......Page 164
6.6 Some properties of the NLS equation......Page 168
6.7 Higher-order corrections to the NLS equation......Page 172
6.8 Multidimensional water waves......Page 174
Exercises......Page 183
7.1 Maxwell equations......Page 185
7.2 Polarization......Page 187
7.3 Derivation of the NLS equation......Page 190
7.4 Magnetic spin waves......Page 198
Exercises......Page 201
PART II INTEGRABILITY AND SOLITONS......Page 203
8.1 Traveling wave solutions of the KdV equation......Page 205
8.2 Solitons and the KdV equation......Page 208
8.3 The Miura transformation and conservation laws for the KdV equation......Page 209
8.4 Time-independent Schrödinger equation and a compatible linear system......Page 213
8.5 Lax pairs......Page 214
8.6 Linear scattering problems and associated nonlinear evolution equations......Page 215
8.7 More general classes of nonlinear evolution equations......Page 221
Exercises......Page 226
9 The inverse scattering transform for the Korteweg–de Vries (KdV) equation......Page 230
9.1 Direct scattering problem for the time-independent Schrödinger equation......Page 231
9.2 Scattering data......Page 235
9.3 The inverse problem......Page 238
9.4 The time dependence of the scattering data......Page 240
9.5 The Gel’fand–Levitan–Marchenko integral equation......Page 241
9.6 Outline of the inverse scattering transform for the KdV equation......Page 243
9.7 Soliton solutions of the KdV equation......Page 244
9.8 Special initial potentials......Page 248
9.9 Conserved quantities and conservation laws......Page 251
9.10 Outline of the IST for a general evolution system – including the nonlinear Schrödinger equation with vanishing boundary conditions......Page 255
Exercises......Page 272
PART III APPLICATIONS OF NONLINEAR WAVES IN OPTICS......Page 275
10.1 Communications......Page 277
10.2 Multiple-scale analysis of the NLS equation......Page 285
10.3 Dispersion-management......Page 290
10.4 Multiple-scale analysis of DM......Page 293
10.5 Quasilinear transmission......Page 314
10.6 WDM and soliton collisions......Page 319
10.7 Classical soliton frequency and timing shifts......Page 322
10.8 Characteristics of DM soliton collisions......Page 325
10.9 DM soliton frequency and timing shifts......Page 326
11 Mode-locked lasers......Page 329
11.1 Mode-locked lasers......Page 330
11.2 Power-energy-saturation equation......Page 333
References......Page 350
Index......Page 361