Table of Contents
Preface vii
List of Figures xv
Chapter 1 Wave Phenomena in Linear Systems 1
1.1 Wave propagation in uniform medium 2
1.2 Dispersive dielectrics 5
1.3 Modes in linear systems (superposition applicable) 9
1.3.1 Analytical approaches 12
1.3.2 Numerical display 15
1.4 Transfer function and impulse response function of the system 18
1.4.1 Impulse response for time harmonic pulses 18
1.4.2 Impulse response for ultra-short pulses 19
Problems 20
Chapter 2 Wave Propagation in Linear Inhomogeneous Media 23
2.1 WKB solution 23
2.2 Solution of the wave equation near a turning point 25
2.3 Ray tracing in inhomogeneous media 27
2.4 General formulation of ray trajectory equations 28
2.5 Method of characteristics 34
2.6 Mode method for time harmonic systems 37
Problems 41
Chapter 3 Waves Traversing a Temporal Discontinuity Interface Between Two Media 43
3.1 Space-time duality of wave phenomena at a discontinuity interface between media 43
3.2 Wave propagation in suddenly created unmagnetized plasma 44
3.3 Wave propagation in suddenly created magneto plasma 48
3.3.1 Branches of modes 48
3.3.2 Continuity conditions at temporal discontinuity interface 50
3.3.3 Momentum and energy conservation 55
Problems 59
Chapter 4 Slow Varying Systems (One Dimensional Lumped Systems) 62
4.1 Introduction 62
4.2 Initial value problem for a one-dimensional lumped system-Duffing equation 64
4.3 Source excited oscillatory problem (forced Duffing oscillator) 67
4.4 Oscillatory problem with friction (one dimensional lumped systems with damping) 72
4.5 Forced bistate oscillator with friction (one dimensional nonlinear systems with three equilibria - from deterministic to chaotic) 80
4.6 The Van der Pol equation 85
Problems 87
Chapter 5 Lagrangian and Hamiltonian Method in One Dimension 89
5.1 Equations of motion 89
5.2 Average Lagrangian and Hamiltonian method for approximate response 91
5.2.1 Examples 92
5.3 Averaging for strongly nonlinear variable parameter systems 94
5.4 Analytical approach for strongly nonlinear variable parameter lumped systems 97
Problems 104
Chapter 6 Nonlinear Waves 106
6.1 Introduction 106
6.2 "Mode" types in nonlinear systems (Riemann invariants) 108
6.3 Equations for self-consistent description of nonlinear waves in plasma 112
6.4 Formulation of nonlinear wave equations 113
6.4.1 Nonlinear Schrödinger equation for electromagnetic wave 113
6.4.2 Nonlinear Schrödinger equation for electron plasma (Langmuir) wave 115
6.4.3 Korteweg-de Vries (KdV) equation for ion acoustic wave 117
6.4.4 Burgers equation for dissipated ion acoustic wave 119
Problems 121
Chapter 7 Analytical Solutions of Nonlinear Wave Equations 122
7.1 Nonlinear Schrodinger equation (NLSE) 122
7.1.1 Characteristic features of solutions 122
A Conservation laws 123
B Scaling symmetry 123
C Galilean invariance 124
D Virial theorem (Variance identity) 124
7.1.2 Analyses 124
A Periodic solutions 125
B Solitary solution 127
7.2 Korteweg-de Vries (K-dV) equation 129
7.2.1 Conservation laws 129
7.2.2 Potential and modified Korteweg-de Vries (p & mK-dV) equations 130
7.2.3 Propagating modes 130
A Periodic Solution 131
B Soliton trapped in self-induced potential well 133
7.2.4 Soliton solution with Bäcklund transform 134
7.2.5 Transition from nonstationary to stationary 135
A Inverse scattering transform (IST) 135
B Example, a two-soliton solution 138
C Asymptotic form of the two-soliton solution 140
D Pulse behavior in the transition region 142
7.3 Burgers equation 143
7.3.1 Analytical solution via the Cole-Hopf transformation 143
7.3.2 Propagating modes 145
Problems 146
Chapter 8 Wave-wave and Wave-particle Interactions 147
8.1 Vlasov-Poisson system 148
8.2 Velocity diffusion 149
8.3 Mode coupling 149
8.4 Quasi-linear diffusion and equivalent temperature 154
8.5 Renormalization of quasilinear diffusion equation-resonance broadening 157
8.6 Collapse of nonlinear waves 161
Problems 163
Answers to Problems 164
Bibliography 181
Index 185
