ISBN: 3-540-65919-6
TITLE: Waves Called Solitons
AUTHOR: Remoissenet, Michel
TOC:

1 Basic Concepts and the Discovery of Solitons 1
1.1 A look at linear and nonlinear signatures 1
1.2 Discovery of the solitary wave 3
1.3 Discovery of the soliton 7
1.4 The soliton concept in physics 11
2 Linear Waves in Electrical Transmission Lines 12
2.1 Linear nondispersive waves 12
2.2 Sinusoidal-wave characteristics 15
2.2.1 Wave energy density and power 18
2.3 The group-velocity concept 19
2.4 Linear dispersive waves 21
2.4.1 Dispersive transmission lines 21
2.4.2 Electrical network 23
2.4.3 The weakly dispersive limit 26
2.5 Evolution of a wavepacket envelope 27
2.6 Dispersion-induced wavepacket broadening 31
Appendix 2A. General solution for the envelope evolution 34
Appendix 2B. Evolution of the envelope of a Gaussian wavepacket 35
3 Solitons in Nonlinear Transmission Lines 37
3.1 Nonlinear and dispersionless transmission lines 37
3.2 Combined effects of dispersion and nonlinearity 41
3.3 Electrical solitary waves and pulse solitons 42
3.4 Laboratory experiments on pulse solitons 46
3.4.1 Experimental arrangement 46
3.4.2 Series of experiments 48
3.5 Experiments with a pocket version of the electrical network 52
3.6 Nonlinear transmission lines in the microwave range 56
Appendix 3A. Calculation of the effect of nonlinearity on wave propagation 58
Appendix 3B. Derivation of the solitary-wave solution 60
Appendix 3C. Derivation of the KdV equation and its soliton solution 62
Appendix 3D. Details of the electronics: switch driver and pulse generator 64
4 More on Transmission-Line Solitons 65
4.1 Lattice solitons in the electrical Toda network 65
4.1.1 Lattice solitons 67
4.2 Experiments on lattice solitons 68
4.2.1 Collisions of two lattice solitons moving in opposite directions 70
4.2.2 The Fermi-Pasta-Ulam recurrence phenomenon 70
4.3 Periodic wavetrains in transmission lines 71
4.3.1 The solitary wave limit and sinusoidal limit of the cnoidal wave 72
4.4 Modulated waves and the nonlinear dispersion relation 72
4.5 Envelope and hole solitons 74
4.5.1 Experiments on envelope and hole solitons 76
4.6 Modulational instability 77
4.7 Laboratory experiments on modulational instability 82
4.7.1 Model equations 82
4.7.2 Experiments 84
4.8 Modulational instability of two coupled waves 86
4.9 Microwave solitons in magnetic transmission lines 88
4.9.1 Nonlinear spin waves 88
4.9.2 NLS model equation for spin waves 88
4.9.3 Observation of magnetic envelope solitons 89
4.10 Solitons and signal processing 91
Appendix 4A. Periodic wavetrain solutions 93
Appendix 4B. The Jacobi elliptic functions 95
4B.1 Asymptotic limits 96
4B.2 Derivatives and integrals 98
Appendix 4C. Envelope and hole soliton solutions 98
5 Hydrodynamic Solitons 103
5.1 Equations for surface water waves 103
5.1.1 Reduced fluid equations 104
5.2 Small-amplitude surface gravity waves 100
5.3 Linear shallow- and deep-water waves 108
5.3.1 Shallow-water waves 108
5.3.2 Deep-water waves 109
5.4 Surface-tension effects: capillary waves 110
5.5 Solitons in shallow water 112
5.6 Experiments on solitons in shallow water 115
5.6.1 Experimental arrangement 116
5.6.2 Experiments 116
5.7 Stokes waves and soliton wavepackets in deep water 120
5.7.1 Stokes waves 120
5.7.2 Soliton wavepackets 121
5.7.3 Experiments on solitons in deep water 122
5.8 Experiments on modulational instability in deep water 123
5.9 Some applications of the KdV model 126
5.9.1 Blood pressure wave propagation 126
5.9.2 Nonlinear modes of liquid drops 127
Appendix 5A. Basic equations of fluid mechanics 127
5A.1 Conservation of mass 127
5A.2 Conservation of momentum 129
5A.3 Conservation of entropy 130
Appendix 5B. Basic definitions and approximations 130
5B.1 Streamline 130
5B.2 Irrotational and incompressible flow 131
5B.3 Two-dimensional flow: the stream function 132
5B.4 Boundary conditions 134
5B.5 Surface tension 135
Appendix 5C. Derivation of the KdV equation: the perturbative approach 136
Appendix 5D. Derivation of the nonlinear dispersion relation 139
Appendix 5E. Details of the probes and the electronics 142
6 Mechanical Solitons 143
6.1 An experimental mechanical transmisssion line 143
6.1.1 General description of the line 143
6.1.2 Construction of the line 145
6.2 Mechanical kink solitons 145
6.2.1 Linear waves in the low-amplitude limit 146
6.2.2 Large amplitude waves: kink solitons 147
6.2.3 Lorentz contraction of the kink solitons 149
6.3 Particle properties of the kink solitons 151
6.4 Kinkkink and kinkantikink collisions 152
6.5 Breather solitons 154
6.6 Experiments on kinks and breathers 156
6.7 Helical waves, or kink array 157
6.8 Dissipative effects 159
6.9 Envelope solitons 161
6.10 Lattice effects 163
6.10.1 Pocket version of the pendulum chain, lattice effects 163
6.10.2 Pendulum chain with weak coupling 164
6.11 A mechanical tranmsission line with two equilibrium states 165
6.11.1 Periodic and double-well substrate potentials 165
6.11.2 General description of the mechanical chain 166
6.11.3 Kink-soliton solutions 169
6.11.4 Compacton-like kinks or compactons 170
6.11.5 Experiments 173
6.12 Solitons, compactons and nanopterons 175
Appendix 6A. Kink-soliton and antikink-soliton solutions 178
Appendix 6B. Calculation of the energy and the mass of a kink soliton 179
Appendix 6C. Solutions for kink-kink and kink-antikink collisions, and breathers 180
6C.1 Kink solutions 182
6C.2 Kinkkink collisions 182
6C.3 Breather solitons 183
6C.4 Kinkantikink collision 184
Appendix 6D. Solutions for helical waves 185
Appendix 6E. Pendulum with torsion and gravity 187
Appendix 6F Model equation for the pendulum chain 187
7 Fluxons in Josephson Transmission Lines 189
7.1 The Josephson effect in a short junction 189
7.1.1 The small Josephson junction 190
7.2 The long Josephson junction as a transmission line 192
7.3 Dissipative effects 196
7.4 Experimental observations of fluxons 198
7.4.1 Indirect observation 198
7.4.2 Direct observation 199
7.4.3 Lattice effects 201
Appendix 7A. Josephson equations 201
8 Solitons in Optical Fibers 203
8.1 Optical-fiber characteristics 203
8.1.1 Linear dispersive effects 204
8.1.2 Nonlinear effects 206
8.1.3 Effect of losses 207
8.2 Wave-envelope propagation 208
8.3 Bright and dark solitons 210
8.3.1 Bright solitons 211
8.3.2 Dark solitons 213
8.4 Experiments on optical solitons 214
8.5 Perturbations and soliton communications 216
8.5.1 Effect of losses 216
8.5.2 Soliton communications 217
8.6 Modulational instability of coupled waves 218
8.7 A look at quantum-optical solitons 219
8.8 Some other kinds of optical solitons: spatial solitons 221
Appendix 8A. Electromagnetic equations in a nonlinear medium 222
9 The Soliton Concept in Lattice Dynamics 225
9.1 The one-dimensional lattice in the continuum approximation 225
9.2 The quasi-continuum approximation for the monatomic lattice 230
9.3 The Toda lattice 232
9.4 Envelope solitons and localized modes 233
9.5 The one-dimensional lattice with transverse nonlinear modes 235
9.6 Motion of dislocations in a one-dimensional crystal 238
9.7 The one-dimensional lattice model for structural phase transitions 239
9.7.1 The orderdisorder transition 241
9.7.2 The displacive transition 242
9.8 Kink-soliton solutions for generalized on-site potentials 244
9.9 A lattice model with an exact kink-soliton solution 247
9.10 Energy localization in nonlinear lattices. 250
9.10.1 Self-trapped states: polaron and conformon 250
9.10.2 Intrinsic localized modes or discrete breathers 251
9.11 Observation of discrete breathers 253
9.11.1 Discrete pendulum chains 253
9.11.2 Mechanical chain with torsion and gravity 254
9.11.3 A chain of magnetic pendulums 256
Appendix 9A. Solutions for transverse displacements 257
Appendix 9B. Kink-soliton or domain-wall solutions 259
Appendix 9C Construction of a double-well potential 260
10 A Look at Some Remarkable Mathematical Techniques 262
10.1 Lax equations and the inverse scattering transform method 262
10.1.1 The Fourier-transform method for linear equations 263
10.1.2 The Lax pair for nonlinear evolution equations 264
10.2 The KdV equation and the spectral problem 266
10.3 Time evolution of the scattering data 267
10.3.1 Discrete eigenvalues 267
10.3.2 Continuous spectrum 269
10.4 The inverse scattering problem 270
10.4.1 Discrete spectrum only: soliton solution 271
10.5 Response of the KdV model to an initial disturbance 273
10.5.1 The delta function potential 273
10.5.2 The rectangular potential well 274
10.5.3 The sech-squared potential well 274
10.6 The inverse scattering transform for the NLS equation 275
10.7 The Hirota method for the KdV equation 277
10.8 The Hirota method for the NLS equation 280
1 1 Diffusive solitons 284
11.1 Combined effects of dissipation and nonlinearity 285
11.1.1 A diffusive electrical transmission line 285
11.1.2 Linear diffusive waves 287
11.1.3 Kink-shaped diffusive solitons 288
11.1.4 Experiments on electrical diffusive solitons 290
11.2 Reaction diffusion processes 291
11.2.1 Reaction diffusion equations 291
11.2.2 A chemical model with reaction diffusion 293
11.2.3 An electrical lattice with reaction diffusion 296
11.2.4 Experiments with an electrical lattice 298
11.3 A mechanical analog with diffusive solitons 299
11.3.1 Chain with flexion and gravity 299
11.3.2 Experimental chain 300
11.4 Reaction diffusion processes in lattices 301
11.4.1 Propagation failure 301
11.4.2 Discrete reaction diffusion model with exact solution 302
Appendix 11A. Derivation of the Burgers equation. 303
Appendix 11B. Solution of the reaction diffusion equation 304
Appendix 11C. Equation of motion of an Euler strut 305
References 307
Subject Index 325
END
