Preface ..................................................... ix
Acknowledgements ........................................... xiv
PART I FUNDAMENTALS AND BASIC APPLICATIONS ..................... 1
1 Introduction ................................................. 3
1.1 Solitons: Historical remarks ........................... 11
Exercises .............................................. 14
2 Linear and nonlinear wave equations ......................... 17
2.1 Fourier transform method ............................... 17
2.2 Terminology: Dispersive and non-dispersive equations ... 19
2.3 Parseval's theorem ..................................... 22
2.4 Conservation laws ...................................... 22
2.5 Multidimensional dispersive equations .................. 23
2.6 Characteristics for first-order equations .............. 24
2.7 Shock waves and the Rankine-Hugoniot conditions ........ 27
2.8 Second-order equations: Vibrating string equation ...... 33
2.9 Linear wave equation ................................... 35
2.10 Characteristics of second-order equations .............. 37
2.11 Classification and well-posedness of PDEs .............. 38
Exercises ................................................... 42
3 Asymptotic analysis of wave equations: Properties and
analysis of Fourier-type integrals .......................... 45
3.1 Method of stationary phase ............................. 46
3.2 Linear free Schrodinger equation ....................... 48
3.3 Group velocity ......................................... 51
3.4 Linear KdV equation .................................... 54
3.5 Discrete equations ..................................... 61
3.6 Burgers' equation and its solution: Cole-Hopf
transformation ......................................... 68
3.7 Burgers' equation on the semi-infinite interval ........ 71
Exercises ................................................... 73
4 Perturbation analysis ....................................... 75
4.1 Failure of regular perturbation analysis ............... 76
4.2 Stokes-Poincare frequency-shift method ................. 78
4.3 Method of multiple scales: Linear example .............. 81
4.4 Method of multiple scales: Nonlinear example ........... 84
4.5 Method of multiple scales: Linear and nonlinear
pendulum ............................................... 86
Exercises ................................................... 96
5 Water waves and KdV-type equations .......................... 98
5.1 Euler and water wave equations ......................... 99
5.2 Linear waves .......................................... 103
5.3 Non-dimensionalization ................................ 105
5.4 Shallow-water theory .................................. 106
5.5 Solitary wave solutions ............................... 118
Exercises .................................................. 128
6 Nonlinear Schrodinger models and water waves ............... 130
6.1 NLS from Klein-Gordon ................................. 130
6.2 NLS from KdV .......................................... 133
6.3 Simplified model for the linear problem and
"universality" ........................................ 138
6.4 NLS from deep-water waves ............................. 141
6.5 Deep-water theory: NLS equation ....................... 148
6.6 Some properties of the NLS equation ................... 152
6.7 Higher-order corrections to the NLS equation .......... 156
6.8 Multidimensional water waves .......................... 158
Exercises .................................................. 167
7 Nonlinear Schrodinger models in nonlinear optics ........... 169
7.1 Maxwell equations ..................................... 169
7.2 Polarization .......................................... 171
7.3 Derivation of the NLS equation ........................ 174
7.4 Magnetic spin waves ................................... 182
Exercises .................................................. 185
PART II INTEGRABILITY AND SOLITONS ........................... 187
8 Solitons and integrable equations .......................... 189
8.1 Traveling wave solutions of the KdV equation .......... 189
8.2 Solitons and the KdV equation ......................... 192
8.3 The Miura transformation and conservation laws for
the KdV equation ...................................... 193
8.4 Time-independent Schrodinger equation and a
compatible linear system .............................. 197
8.5 Lax pairs ............................................. 198
8.6 Linear scattering problems and associated nonlinear
evolution equations ................................... 199
8.7 More general classes of nonlinear evolution
equations ............................................. 205
Exercises .................................................. 210
9 The inverse scattering transform for the Korteweg-de
Vries (KdV) equation ....................................... 214
9.1 Direct scattering problem for the time-independent
Schrodinger equation .................................. 215
9.2 Scattering data ....................................... 219
9.3 The inverse problem ................................... 222
9.4 The time dependence of the scattering data ............ 224
9.5 The Gel'fand-Levitan-Marchenko integral equation ...... 225
9.6 Outline of the inverse scattering transform for the
KdV equation .......................................... 227
9.7 Soliton solutions of the KdV equation ................. 228
9.8 Special initial potentials ............................ 232
9.9 Conserved quantities and conservation laws ............ 235
9.10 Outline of the IST for a general evolution system -
including the nonlinear Schrodinger equation with
vanishing boundary conditions ......................... 239
Exercises .................................................. 256
PART III APPLICATIONS OF NONLINEAR WAVES IN OPTICS ........... 259
10 Communications ............................................. 261
10.1 Communications ........................................ 261
10.2 Multiple-scale analysis of the NLS equation ........... 269
10.3 Dispersion-management ................................. 274
10.4 Multiple-scale analysis of DM ......................... 277
10.5 Quasilinear transmission .............................. 298
10.6 WDM and soliton collisions ............................ 303
10.7 Classical soliton frequency and timing shifts ......... 306
10.8 Characteristics of DM soliton collisions .............. 309
10.9 DM soliton frequency and timing shifts ................ 310
11 Mode-locked lasers ......................................... 313
11.1 Mode-locked lasers .................................... 314
11.2 Power-energy-saturation equation ...................... 317
References ................................................. 334
Index ...................................................... 345
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