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1 Basic Concepts and the Discovery of Solitons1

1.1 A look at linear and nonlinear signatures1

1.2 Discovery of the solitary wave3

1.3 Discovery of the soliton6

1.4 The soliton concept in physics10

2 Linear Waves in Electrical Transmission Lines12

2.1 Linear nondispersive waves12

2.2 Sinusoidal-wave characteristics15

2.2.1 Wave energy density and power18

2.3 The group-velocity concept19

2.4 Linear dispersive waves21

2.4.1 Dispersive transmission lines21

2.4.2 Electrical network23

2.4.3 The weakly dispersive limit26

2.5 Evolution of a wavepacket envelope27

2.6 Dispersion-induced wavepacket broadening31

Appendix 2A.General solution for the envelope evolution34

Appendix 2B. Evolution of the envelope of a Gaussian wavepacket35

3 Solitons in Nonlinear Transmission Lines37

3.1 Nonlinear and dispersionless transmission lines37

3.2 Combined effects of dispersion and nonlinearity41

3.3 Electrical solitary waves and pulse solitons42

3.4 Laboratory experiments on pulse solitons46

3.4.1 Experimental arrangement46

3.4.2 Series of experiments48

3.5 Experiments with a pocket version of the electrical network52

3.6 Nonlinear transmission lines in the microwave range56

Appendix 3A.Calculation of the effect of nonlinearity on wave propagation58

Appendix 3B.Derivation of the solitary-wave solution60

Appendix 3C.Derivation of the KdV equation and its soliton solution62

Appendix 3D.Details of the electronics:switch driver and pulse generator64

4 More on Transmission-Line Solitons65

4.1 Lattice solitons in the electrical Toda network65

4.1.1 Lattice solitons67

4.2 Experiments on lattice solitons68

4.2.1 Collisions of two lattice solitons moving in opposite directions70

4.2.2 The Fermi-Pasta-Ulam recurTence phenomenon70

4.3 Periodic wavetrains in transmission lines71

4.3.1 The solitary wave limit and sinusoidal limit of the cnoidal wave72

4.4 Modulated waves and the nonlinear dispersion relation72

4.5 Envelope and hole solitons74

4.5.1 Experiments on envelope and hole solitons76

4.6 Modulational instability77

4.7 Laboratory experiments on modulational instability82

4.7.1 Model equations82

4.7.2 Experiments84

4.8 Modulational instability of two coupled waves86

Appendix 4A.Periodic wavetrain solutions88

Appendix 4B.The Jacobi elliptic functions90

4B.1 Asymptotic limits91

4B.2 Derivatives and integrals93

Appendix 4C.Envelope and hole soliton solutions93

5 Hydrodynamic Solitons98

5.1 Equations for surface water waves98

5.1.1 Reduced fluid equations99

5.2 Small-amplitude surface gravity waves100

5.3 Linear shallow-and deep-water waves103

5.3.1 Shallow-waterwaves103

5.3.2 Deep-water waves104

5.4 Surface-tension effects:capillary waves105

5.5 Solitons in shallow water107

5.6 Experiments on solitons in shallow water110

5.6.1 Experimental arrangement111

5.6.2 Experiments111

5.7 Stokes waves and soliton wavepackets in deep water115

5.7.1 Stokes waves115

5.7.2 Soliton wavepackets116

5.7.3 Experiments on solitons in deep water117

5.8 Experiments on modulational instability in deep water118

Appendix 5A.Basic equations of fluid mechanics121

5A.1 Conservation of mass121

5A.2 Conservation of momentum123

5A.3 Conservation of entropy124

Appendix 5B.Basic definitions and approximations124

5B.1 Streamline124

5B.2 Irrotational and incompressible flow125

5B.3 Two-dimensional flow:the stream function126

5B.4 Boundary conditions128

5B.5 Surface tension129

Appendix 5C.Derivation of the KdV equation:the perturbative approach130

Appendix 5D.Derivation of the nonlinear dispersion relation133

Appendix 5E.Details of the probes and the electronics136

6 Mechanical Solitons137

6.1 An experimental mechanical transmisssion line137

6.1.1 General description of the line137

6.1.2 Construction of the line139

6.2 Mechanical kink solitons139

6.2.1 Linear waves in the low-amplitude limit140

6.2.2 Large amplitude waves:kink solitons141

6.2.3 Lorentz contraction of the kink solitons143

6.3 Particle properties of the kink solitons145

6.4 Kink-kink and kink-antikink collisions146

6.5 Breather solitons148

6.6 Experiments on kinks and breathers150

6.7 Helical waves,or kink array151

6.8 Dissipative effects153

6.9 Envelope solitons155

6.10 Pocket version of the pendulum chain,lattice effects157

Appendix 6A.Kink soliton and antikink soliton solutions159

Appendix 6B.Calculation of the energy and the mass of a kink soliton160

Appendix 6C.Solutions for kink-kink and kink-antikink collisions,and breathers161

6C.1 Kink solutions163

6C.2 Kink-kink collisions163

6C.3 Breather solitons164

6C.4 Kink-antikink collision165

Appendix 6D.Solutions for helical waves166

7 Fluxons in Josephson Transmission Lines168

7.1 The Josephson effect in a short junction168

7.1.1 The small Josephson junction169

7.2 The long Josephson junction as a transmission line171

7.3 Dissipative effects175

7.4 Experimental observations of fluxons177

7.4.1 Indirect observation177

7.4.2 Direct observation178

7.4.3 Latrice effects180

Appendix 7A．Josephson equations180

8 Solitons in Optical Fibers182

8.1 Optical-fiber characteristics182

8.1.1 Linear dispersive effects183

8.1.2 Nonlinear effects185

8.1.3 Effect of losses186

8.2 Wave-envelope propagation187

8.3 Bright and dark solitons189

8.3.1 Bright solitons190

8.3.2 Dark solitons192

8.4 Experiments on optical solitons193

8.5 Perturbations and soliton communications195

8.5.1 Effect of losses195

8.5.2 Soliton communications196

8.6 Modulational instability of coupled waves197

8.7 A look at quantum optical solitons198

Appendix 8A.Electromagnetic equations in a nonlinear medium199

9 The Soliton Concept in Lattice Dynamics202

9.1 The one-dimensional lattice in the continuum approximation202

9.2 The quasi-continuum approximation for the monatomic lattice207

9.3 The Toda lattice209

9.4 Envelope solitons and localized modes210

9.5 The one-dimensional lattice with transverse nonlinear modes212

9.6 Motion of dislocations in a one-dimensional crystal215

9.7 The one-dimensional lattice model for structural phase transitions216

9.7.1 The order-disorder transition218

9.7.2 The displacive transition219

Appendix 9A.Solutions for transverse displacements221

Appendix 9B.Kink soliton or domain-wall solutions223

10 A Look at Some Remarkable Mathematical Techniques225

10.1 Lax equations and the inverse scattering transform method225

10.1.1 The Fourier-transform method for linear equations226

10.1.2 The Lax pair for nonlinear evolution equations227

10.2 The KdV equation and the spectral problem229

10.3 Time evolution of the scattering data230

10.3.1 Discrete eigenvalues230

10.3.2 Continuous spectrum232

10.4 The inverse scattering problem233

10.4.1 Discrete spectrum only:soliton solution234

10.5 Response of the KdV model to an initial disturbance236

10.5.1 The delta function potential236

10.5.2 The rectangular potential well237

10.5.3 The sech-squared potential well237

10.6 The inverse scattering transform for the NLS equation238

10.7 The Hirota method for the KdV equation239

10.8 The Hirota method for the NLS equation243

References247

Subject Index259