Basic Concepts and the Discovery of Solitons p. 1 A look at linear and nonlinear signatures p. 1 Discovery of the solitary wave p. 3 Discovery of the

Transcription

1 Basic Concepts and the Discovery of Solitons p. 1 A look at linear and nonlinear signatures p. 1 Discovery of the solitary wave p. 3 Discovery of the soliton p. 7 The soliton concept in physics p. 11 Linear Waves in Electrical Transmission Lines p. 12 Linear nondispersive waves p. 12 Sinusoidal-wave characteristics p. 15 Wave energy density and power p. 18 The group-velocity concept p. 19 Linear dispersive waves p. 21 Dispersive transmission lines p. 21 Electrical network p. 23 The weakly dispersive limit p. 26 Evolution of a wavepacket envelope p. 27 Dispersion-induced wavepacket broadening p. 31 General solution for the envelope evolution p. 34 Evolution of the envelope of a Gaussian wavepacket p. 35 Solitons in Nonlinear Transmission Lines p. 37 Nonlinear and dispersionless transmission lines p. 37 Combined effects of dispersion and nonlinearity p. 41 Electrical solitary waves and pulse solitons p. 42 Laboratory experiments on pulse solitons p. 46 Experimental arrangement p. 46 Series of experiments p. 48 Experiments with a pocket version of the electrical network p. 52 Nonlinear transmission lines in the microwave range p. 56 Calculation of the effect of nonlinearity on wave propagation p. 58 Derivation of the solitary-wave solution p. 60 Derivation of the KdV equation and its soliton solution p. 62 Details of the electronics: switch driver and pulse generator p. 64 More on Transmission-Line Solitons p. 65 Lattice solitons in the electrical Toda network p. 65 Lattice solitons p. 67 Experiments on lattice solitons p. 68 Collisions of two lattice solitons moving in opposite directions p. 70 The Fermi-Pasta-Ulam recurrence phenomenon p. 70 Periodic wavetrains in transmission lines p. 71 The solitary wave limit and sinusoidal limit of the cnoidal wave p. 72 Modulated waves and the nonlinear dispersion relation p. 72 Envelope and hole solitons p. 74

2 Experiments on envelope and hole solitons p. 76 Modulational instability p. 77 Laboratory experiments on modulational instability p. 82 Model equations p. 82 Experiments p. 84 Modulational instability of two coupled waves p. 86 Microwave solitons in magnetic transmission lines p. 88 Nonlinear spin waves p. 88 NLS model equation for spin waves p. 88 Observation of magnetic envelope solitons p. 89 Solitons and signal processing p. 91 Periodic wavetrain solutions p. 93 The Jacobi elliptic functions p. 95 Asymptotic limits p. 96 Derivatives and integrals p. 98 Envelope and hole soliton solutions p. 98 Hydrodynamic Solitons p. 103 Equations for surface water waves p. 103 Reduced fluid equations p. 104 Small-amplitude surface gravity waves p. 100 Linear shallow- and deep-water waves p. 108 Shallow-water waves p. 108 Deep-water waves p. 109 Surface-tension effects: capillary waves p. 110 Solitons in shallow water p. 112 Experiments on solitons in shallow water p. 115 Experimental arrangement p. 116 Experiments p. 116 Stokes waves and soliton wavepackets in deep water p. 120 Stokes waves p. 120 Soliton wavepackets p. 121 Experiments on solitons in deep water p. 122 Experiments on modulational instability in deep water p. 123 Some applications of the KdV model p. 126 Blood pressure wave propagation p. 126 Nonlinear modes of liquid drops p. 127 Basic equations of fluid mechanics p. 127 Conservation of mass p. 127 Conservation of momentum p. 129 Conservation of entropy p. 130 Basic definitions and approximations p. 130

3 Streamline p. 130 Irrotational and incompressible flow p. 131 Two-dimensional flow: the stream function p. 132 Boundary conditions p. 134 Surface tension p. 135 Derivation of the KdV equation: the perturbative approach p. 136 Derivation of the nonlinear dispersion relation p. 139 Details of the probes and the electronics p. 142 Mechanical Solitons p. 143 An experimental mechanical transmisssion line p. 143 General description of the line p. 143 Construction of the line p. 145 Mechanical kink solitons p. 145 Linear waves in the low-amplitude limit p. 146 Large amplitude waves: kink solitons p. 147 Lorentz contraction of the kink solitons p. 149 Particle properties of the kink solitons p. 151 Kink-kink and kink-antikink collisions p. 152 Breather solitons p. 154 Experiments on kinks and breathers p. 156 Helical waves, or kink array p. 157 Dissipative effects p. 159 Envelope solitons p. 161 Lattice effects p. 163 Pocket version of the pendulum chain, lattice effects p. 163 Pendulum chain with weak coupling p. 164 A mechanical tranmsission line with two equilibrium states p. 165 Periodic and double-well substrate potentials p. 165 General description of the mechanical chain p. 166 Kink-soliton solutions p. 169 Compacton-like kinks or compactons p. 170 Experiments p. 173 Solitons, compactons and nanopterons p. 175 Kink-soliton and antikink-soliton solutions p. 178 Calculation of the energy and the mass of a kink soliton p. 179 Solutions for kink-kink and kink-antikink collisions, and breathers p. 180 Kink solutions p. 182 Kink-kink collisions p. 182 Breather solitons p. 183 Kink-antikink collision p. 184 Solutions for helical waves p. 185

4 Pendulum with torsion and gravity p. 187 Model equation for the pendulum chain p. 187 Fluxons in Josephson Transmission Lines p. 189 The Josephson effect in a short junction p. 189 The small Josephson junction p. 190 The long Josephson junction as a transmission line p. 192 Dissipative effects p. 196 Experimental observations of fluxons p. 198 Indirect observation p. 198 Direct observation p. 199 Lattice effects p. 201 Josephson equations p. 201 Solitons in Optical Fibers p. 203 Optical-fiber characteristics p. 203 Linear dispersive effects p. 204 Nonlinear effects p. 206 Effect of losses p. 207 Wave-envelope propagation p. 208 Bright and dark solitons p. 210 Bright solitons p. 211 Dark solitons p. 213 Experiments on optical solitons p. 214 Perturbations and soliton communications p. 216 Effect of losses p. 216 Soliton communications p. 217 Modulational instability of coupled waves p. 218 A look at quantum-optical solitons p. 219 Some other kinds of optical solitons: spatial solitons p. 221 Electromagnetic equations in a nonlinear medium p. 222 The Soliton Concept in Lattice Dynamics p. 225 The one-dimensional lattice in the continuum approximation p. 225 The quasi-continuum approximation for the monatomic lattice p. 230 The Toda lattice p. 232 Envelope solitons and localized modes p. 233 The one-dimensional lattice with transverse nonlinear modes p. 235 Motion of dislocations in a one-dimensional crystal p. 238 The one-dimensional lattice model for structural phase transitions p. 239 The order-disorder transition p. 241 The displacive transition p. 242 Kink-soliton solutions for generalized on-site potentials p. 244 A lattice model with an exact kink-soliton solution p. 247

5 Energy localization in nonlinear lattices p. 250 Self-trapped states: polaron and conformon p. 250 Intrinsic localized modes or discrete breathers p. 251 Observation of discrete breathers p. 253 Discrete pendulum chains p. 253 Mechanical chain with torsion and gravity p. 254 A chain of magnetic pendulums p. 256 Solutions for transverse displacements p. 257 Kink-soliton or domain-wall solutions p. 259 Construction of a double-well potential p. 260 A Look at Some Remarkable Mathematical Techniques p. 262 Lax equations and the inverse scattering transform method p. 262 The Fourier-transform method for linear equations p. 263 The Lax pair for nonlinear evolution equations p. 264 The KdV equation and the spectral problem p. 266 Time evolution of the scattering data p. 267 Discrete eigenvalues p. 267 Continuous spectrum p. 269 The inverse scattering problem p. 270 Discrete spectrum only: soliton solution p. 271 Response of the KdV model to an initial disturbance p. 273 The delta function potential p. 273 The rectangular potential well p. 274 The sech-squared potential well p. 274 The inverse scattering transform for the NLS equation p. 275 The Hirota method for the KdV equation p. 277 The Hirota method for the NLS equation p. 280 Diffusive solitons p. 284 Combined effects of dissipation and nonlinearity p. 285 A diffusive electrical transmission line p. 285 Linear diffusive waves p. 287 Kink-shaped diffusive solitons p. 288 Experiments on electrical diffusive solitons p. 290 Reaction diffusion processes p. 291 Reaction diffusion equations p. 291 A chemical model with reaction diffusion p. 293 An electrical lattice with reaction diffusion p. 296 Experiments with an electrical lattice p. 298 A mechanical analog with diffusive solitons p. 299 Chain with flexion and gravity p. 299 Experimental chain p. 300

6 Reaction diffusion processes in lattices p. 301 Propagation failure p. 301 Discrete reaction diffusion model with exact solution p. 302 Derivation of the Burgers equation p. 303 Solution of the reaction diffusion equation p. 304 Equation of motion of an Euler strut p. 305 References p. 307 Subject Index p. 325 Table of Contents provided by Blackwell's Book Services and R.R. Bowker. Used with permission.