Non-Linear Waves in Dispersive Media (eBook)

Front Cover 1
Non-Linear Waves in Dispersive Media 4
Copyright Page 5
Table of Contents 6
PREFACE 8
INTRODUCTION 10
Chapter 1. LINEAR APPROXIMATION 14
§ 2 General Solution of the Linearized Equations 14
§ 3 Linearized Korteweg–de Vries Equation 17
Chapter 2. EXAMPLES OF DISPERSIVE MEDIA 21
§ 4 Gravitational Waves on Fluid Surfaces 21
§ 5 The Boussinesq Equation 23
§ 6 Ion-sound Waves in Unmagnetized Plasma 28
§ 7 Non-linear Waves in Magnetized Plasma 31
§ 8 Non-linear Electromagnetic Waves in Isotropie Dielectrics 38
§ 9 Sound Waves with Dispersion 45
Chapter 3. NON-LINEAR STATIONARY WAVES 50
§ 10 Steady Solutions of the Boussinesq Equations 50
§ 11 Stationary Waves Propagating Transversely to the Magnetic Field in Rarefied Plasma (34–37, 3) 57
§ 12 Other Examples of Stationary Waves 60
Chapter 4. NON-LINEAR WAVES IN WEAKLY DISPERSIVE MEDIA 66
§ 13 The Burgers Equation 66
§ 14 Solution of the Burgers Equation 72
§15 The Korteweg–de Vries Equation 75
§ 16 Conservation Laws for the Korteweg-de Vries Equation 79
§ 17 General Pattern of the Evolution of Initial Perturbations in Weakly Dispersive Media 82
§ 18 Analytical Solution of the Korteweg-de Vries Equation 85
§ 19 Asymptotic Expressions for the Amplitudes of Solitons and "Tails" for Large Values of s 93
§ 20 Self-similar Solutions of the Korteweg-de Vries Equation 96
§ 21 Quasi-linear Solutions of the Korteweg-de Vries Equation 98
§ 22 Flow Around a Thin Body in a Dispersive Medium 105
§ 23 Shock Waves in Dispersive Media 114
Chapter 5. WAVES OF ENVELOPES 119
§ 24 Non-linear Geometrical Optics 119
§ 25 Instability Criteria for Stationary Waves 122
§ 26 Evolution of the Wave Envelopes in the "Hydrodynamic Approximation" 125
§ 27 Non-linear Parabolic Equation 133
§ 28 Self-modulation of Waves (Modulational Instability) 141
§ 29 Self-focusing and Self-channelling of Waves 148
§ 30 Electro-acoustic Waves in Plasma 154
APPENDIX A: NON-LINEAR WAVES WITH SLOWLY VARYING PARAMETERS (ADIABATIC APPROXIMATION OF WHITHAM) 160
A 1 Variation Principle 160
A 2 Adiabatic Invariants 165
A 3 Non-linear Geometrical Optics 169
APPENDIX B: EVOLUTION OF ELECTRO-ACOUSTIC WAVES IN PLASMA WITH NEGATIVE DIELECTRIC PERMITTIVITY 171
B 1 Boundary Conditions 171
B 2 Excitation and Evolution of Electro-acoustic Waves 174
B 3 Solution of the Boundary-value Problem 183
B 4 General Solution of the Fundamental Equations 187
REFERENCES 190
INDEX 194
OTHER TITLES IN THE SERIES IN NATURAL PHILOSOPHY 198