Table of Contents
Preface vii
Chapter 1 Perturbed Oscillations and Waves: Introductory Examples 1
1.1 Quasi-Harmonic Oscillators 1
1.1.1 Linear oscillator with damping 1
1.1.2 Oscillator with cubic nonlinearity 4
1.1.3 An active system: Van der Pol oscillator 6
1.2 Quasi-Harmonic Waves 9
1.2.1 Linear wave equation with dissipation 10
1.2.2 Nonlinear wave 14
1.3 Concluding Remarks 16
References 16
Chapter 2 Perturbation Method for Quasi-Harmonic Waves 18
2.1 General Scheme 18
2.2 Resonant Interaction of Waves 22
2.3 Geometrical Acoustics 23
2.4 Nonlinear Electromagnetic Waves in a Dispersive Dielectric 31
2.4.1 Resonance triplet 35
2.5 Concluding Remarks 37
References 38
Chapter 3 Perturbation Method for Non-Sinusoidal Waves 39
3.1 General Scheme 39
3.2 Lagrangian Systems 46
3.3 Averaged Lagrangian and Whitham's Variational Principle 50
3.4 Linear Waves 53
3.5 Concluding Remarks 55
References 55
Chapter 4 Nonlinear Waves of Modulation 57
4.1 Simple Envelope Waves 57
4.2 Nonlinear Klein-Gordon Equation 61
4.2.1 Averaged equations 61
4.2.2 Attenuation of a periodic wave 65
4.3 Korteweg-de Vries Equation 69
4.3.1 Stationary waves 69
4.3.2 Slowly varying cnoidal waves 72
4.3.3 Conservation equations 74
4.3.4 Evolution of step function 76
4.4 Concluding Remarks 79
References 79
Chapter 5 Perturbation Methods for Solitary Waves and Fronts 80
5.1 Quasi-Stationary Theory 81
5.2 Lagrangian Description 86
5.3 Non-Stationary Perturbations 88
5.4 Inverse Scattering Perturbation Scheme for Solitons 92
5.5 "Equivalence Principle" 95
5.6 Concluding Remarks 96
References 97
Chapter 6 Perturbed Solitons 98
6.1 Perturbed Kd V Equation 99
6.1.1 Equation for solilon amplitude 99
6.1.2 KdV equation with dissipation 103
6.1.3 Radiation from the soliton 104
6.2 Nonlinear Klein-Gordon Equation 108
6.3 Nonlinear Schrödinger Equation 111
6.4 Rotational KdV Equation 115
6.4.1 Terminal damping of solitons 115
6.4.2 Radiation 118
6.5 Refraction of Solitons 123
6.5.1 Geometrical theory of solitons 123
6.5.2 Transverse stability of a soliton 125
6.5.3 Circular fronts. Nonlinear self-refraction of solitons 127
6.4 Concluding Remarks 130
References 130
Chapter 7 Interaction and Ensembles of Solitons and Kinks 132
7.1 General Scheme 133
7.2 Lagrangian Description 136
7.3 Types of Soliton Interactions: Repulsion, Attraction, and Bound States 140
7.4 A Generalized KdV Equation 143
7.5 Soliton Lattices and their Stochastization 147
7.6 Stable and Unstable Soliton Lattices 150
7.7 Interaction of Solitons in Electromagnetic Lines 153
7.8 Interaction of Flat-Top Solitons and Kinks 156
7.8.1 Compound solitons in the Gardner equation 156
7.8.2 Two-soliton interaction 163
7.8.3 Envelope solitons and kinks 166
7.8.4 Physical example. Large-amplitude internal waves in the ocean 169
7.9 Two-and Three-Dimensional Solitons 172
7.10 Concluding Remarks 176
References 176
Chapter 8 Dissipative and Active Systems. Auto waves 179
8.1 Burgers Equation and Taylor Shocks 179
8.2 Autosolitons and Explosive Instability 184
8.3 Parametric Amplification of Solitons 186
8.3.1 Solitons as accelerated particles 186
8.3.2 Parametric generation of solitons 189
8.3.3 Two-dimensional resonators 192
8.4 Autowaves in Reaction-Diffusion Systems 194
8.4.1 KPP-Fisher model 194
8.4.2 Two-component models 196
8.5 Concluding Remarks 201
References 202
Index 205
