Nonlinear Science Emergence and Dynamics of Coherent Structures

Scott,Alwyn. Nonlinear Science Emergence and Dynamics of Coherent Structures (Oxford Texts in AEM 8)(2 ed,OUP,2003) ......Page 4
Copyright ......Page 5
Preface to the second edition ......Page 7
Preface to the first edition ......Page 9
Acknowledgements in the first edition ......Page 13
Contents ......Page 14
List of Figures xxi ......Page 20
1.1 A hydrodynamic soliton created in a wave tank by John Scott Russell in the 1830s 2 ......Page 24
1.2 Measurement of the change in membrane conductance (band) and membrane voltage (line) with time during the passage of a nerve impulse on a squid axon 9......Page 31
1.3.1 Nerve studies 11 ......Page 33
1.4 Self-oscillatory ring waves in a two-dimensional chemical reaction diffusion system 13......Page 35
2.1 Sketch of the boundary conditions on K(x,?) in the (x,?)-plane 45......Page 67
2.2 Contours in the complex A;-plane for computing B(x — r) from the inverse Fourier transform of Equation (2.38) 46......Page 68
3.1.3 Periodic solutions 59 ......Page 81
3.2 A periodic solution of Equation (3.20) 60 ......Page 82
3.3 A plot of a 2-soliton collision: —uz{x,t) from Equation (3.36) 64......Page 86
3.2.1 Long Josephson junctions 71 ......Page 93
3.2.3 Periodic waves 74 ......Page 96
3.7 Surface plot of a kink-antikink solution plotted from Equation (3.83) with ^ = 1/2 82......Page 104
3.2.6 More spatial dimensions 85 ......Page 107
3.9 A plot of \un\S2((x,t)\ fr°m Equation (3.113) 94......Page 116
4.1.1 The Zeldovich-Frank-Kamenetsky (Z-F) equation 111 ......Page 133
4.2 Finding a traveling-wave solution of Equation (4.2) 113......Page 135
4.1.2 The Burgers equation 116 ......Page 138
4.2.1 Space-clamped squid membrane dynamics 118 ......Page 140
4.5 From Equations (4.19), the stationary values of the switching variables: no(F), mo(F), and ho{V) (upper panel), and the corresponding switching times: rn, rm, and (lower panel) 122......Page 144
4.6 Schematic representation of a homoclinic trajectory in the (V, W, ra, ft, n) traveling-wave phase space defined in Equations (4.21) 125 ......Page 147
4.7 A full-sized action potential (atv = 18.8m/s) and an unstable threshold impulse (at 5.66 m/s) for the H-H axon at 18.5°C 126 ......Page 148
4.8 (a) Ionic currents in the M-C model as a function of the traveling-wave variable (?). (b) Structure of the associated nerve impulse 128 ......Page 150
4.3.2 The FitzHugh-Nagumo (F-N) model 129 ......Page 151
4.10 In the F-N system with 0 < e 1, V and R are shown as functions of the traveling-wave variable ?. 131 ......Page 153
4.11 The homoclinic solution trajectory of Equations (4.27) corresponding to the traveling-wave impulse in Figure 4.10. 132 ......Page 154
4.12 Propagation speeds for impulse solutions of a FitzHugh-Nagumo (F-N) system plotted against the temperature parameter e 133 ......Page 155
4.3.3 Morris-Lecar (M-L) models 134 ......Page 156
4.14 Plots of the initial ionic current (Ji) and the steady state current (Jss) under the M-L model 136 ......Page 158
4.15 Amplitude of a traveling-wave impulse on an H-H axon plotted against a narcotization factor—77—reducing the maximum sodium and potassium conductances 144 ......Page 166
4.16 Propagation of a decremental impulse on an H-H axon narcotized by the factor 77 = 0.25 145 ......Page 167
4.18 (a) Abrupt widening of a nerve fiber, (b) Branching region 151 ......Page 173
4.7.2 Nonlinear diffusion in three dimensions 156 ......Page 178
4.20 A computer generated plot of a scroll ring 158......Page 180
4.21 Lichens growing on a wall in Scotland may be a biological example of spiral waves 170 ......Page 192
5.1 Spring-mass lattices 177 ......Page 199
5.2 (a) A plot of the acoustic mode dispersion relation for a uniform lattice given in Equation (5.14) with 10 masses and periodic boundary conditions, (b) A corresponding plot for two atoms in each of 10 unit cells, leading to a band of optical modes 183 ......Page 205
5.3 The relations between 7 and uj for stationary solutions of DST with three lattice sites, M as in Equation (5.51), and e — 1. Branch designations are explained in the text 200 ......Page 222
5.4 A short section of alpha helix. The dashed lines indicate relatively weak hydrogen bonds 203 ......Page 225
5.5 A cartoon of a short section of DNA double helix 208 ......Page 230
5.6 A unit cell of the two-dimensional tunnel diode array 211 ......Page 233
5.7 A 4 x 4 array (N2 = 16) of the unit cells in Figure 5.6 212 ......Page 234
5.8 A Necker cube 214 ......Page 236
5.9 Equivalent circuit for a myelinated axon 216 ......Page 238
5.4.3 Emergence of form by replication 219 ......Page 241
6.1 Amplitudes of scattering solutions 241 ......Page 263
6.2.3 Reduction to Fourier analysis in the small amplitude limit 257 ......Page 279
7.1 A (superconducting) Josephson junction of length I with a fluxon (kink) traveling in the x-direction at velocity v 298 ......Page 320
7.5.3 Radiation from a fluxon 317 ......Page 339
7.3 First-order corrections to impulse speeds under the influence of mutual coupling. (Data courtesy of Steve Luzader [17].) 325 ......Page 347
8.1.1 A classical nonlinear oscillator 337 ......Page 359
8.1.4 The rotating wave approximation 345 ......Page 367
8.1.7 Pump-probe measurements 351 ......Page 373
8.4 Energy eigenvalues of the QDNLS equation in the / —> oo limit, with e = 1 and several values of 7 368 ......Page 390
8.5 Two quantum energy eigenvalues against propagation number for (a) the quantum Ablowitz-Ladik equation and (b) the fermionic polaron model with 7 < 2 379 8.6 Infrared absorption spectra of crystalline ACN in the Amide-I (CO stretching) region at three different temperatures 391 ......Page 413
8.7 Infrared absorption spectrum of crystalline ACN in the NH stretching region at two different temperatures 400 ......Page 422
8.6 A quantum lattice sine-Gordon equation 401 ......Page 423
C.2 Plots of sn(C, fc) and cn(?, fc) as functions of Ct/K{k) 445 ......Page 467
C.3 Plots of dn(?, fc) as functions of ?/K(k). (Note that dn(?,0) = 1.) 446 ......Page 468
1.1.1 Hydrodynamics 1 ......Page 23
1.1.2 Nonlinear diffusion 3 ......Page 25
1.1.3 Backlund transformation theory 6 ......Page 28
1.1.4 A theory of matter 7 ......Page 29
1.2 Between the wars 8 ......Page 30
1.3.2 Autocatalytic chemical reactions 12 ......Page 34
1.3.3 Solitons 14 ......Page 36
1.3.4 Local modes in molecules and molecular crystals 19 ......Page 41
1.3.5 Elementary particle research 20 ......Page 42
1.4 Recent developments 21 ......Page 43
References 23 ......Page 45
2.1 Dispersionless linear equations 28 ......Page 50
2.2 Dispersive linear equations 30 ......Page 52
2.3 The linear diffusion equation 31 ......Page 53
2.4.1 Green’s method 33 ......Page 55
2.4.2 Fredholm’s theorem 35 ......Page 57
2.5.1 General definitions 37 ......Page 59
2.5.2 Linear stability 38 ......Page 60
2.5.3 Signaling problems 39 ......Page 61
2.6.1 Solutions of Schrodinger’s equation 40 ......Page 62
2.6.2 Gel’fand-Levitan theory 43 ......Page 65
2.7 Problems 48 ......Page 70
References 53 ......Page 75
3 THE CLASSICAL SOLITON EQUATIONS 55 ......Page 77
3.1.1 Long water waves 57 ......Page 79
3.1.2 Solitary wave solutions 58 ......Page 80
3.1.4 A Backlund transformation for KdV 61 ......Page 83
3.1.5 AT-soliton formulas 67 ......Page 89
3.2.2 Solitary waves 72 ......Page 94
3.2.4 Nonlinear standing waves 77 ......Page 99
3.2.5 Two-soliton solutions 81 ......Page 103
3.3.1 Nonlinear wave packets 88 ......Page 110
3.3.2 Modulated traveling-wave solutions of NLS(+) 90 ......Page 112
3.3.3 Dark soliton solutions of NLS(—) 92 ......Page 114
3.3.4 A BT for NLS(+) 93 ......Page 115
3.3.5 Transverse phenomena 95 ......Page 117
3.5 Problems 98 ......Page 120
References 106 ......Page 128
4 REACTION-DIFFUSION SYSTEMS 110 ......Page 132
4.2 The Hodgkin-Huxley (H-H) system 117 ......Page 139
4.2.2 The H-H impulse 124 ......Page 146
4.3.1 The Markin-Chizmadzhev (M-C) model 127 ......Page 149
4.4.1 The Z-F equation 138 ......Page 160
4.4.2 The M-C model 139 ......Page 161
4.4.3 The F-N model 140 ......Page 162
4.5 Decrementai conduction 143 ......Page 165
4.6.1 Tapered fibers 147 ......Page 169
4.6.2 Leading-edge charge and impulse ignition 149 ......Page 171
4.6.3 Dendritic logic 150 ......Page 172
4.7.1 Two-dimensional nonlinear diffusion 154 ......Page 176
4.7.3 Turing patterns 159 ......Page 181
4.7.4 Hypercycles 160 ......Page 182
4.8 Summary 161 ......Page 183
4.9 Problems 162 ......Page 184
References 171 ......Page 193
5 NONLINEAR LATTICES 176 ......Page 198
5.1.1 The Toda-lattice soliton 178 ......Page 200
5.1.2 Lattice solitary waves 179 ......Page 201
5.1.3 Existence of lattice solitary waves 180 ......Page 202
5.1.4 Intrinsic localized modes and intrinsic gap modes 182 ......Page 204
5.2 Lattices with nonlinear on-site potentials 185 ......Page 207
5.2.1 The discrete sine-Gordon equation 187 ......Page 209
5.2.2 Nonlinear Schrodinger lattices 190 ......Page 212
5.2.3 The discrete self-trapping equation 197 ......Page 219
5.3.1 Alpha-helix solitons in protein 202 ......Page 224
5.3.2 Self-trapping in globular proteins 205 ......Page 227
5.3.3 Solitons in DNA 207 ......Page 229
5.4.1 Quasiharmonic lattices 210 ......Page 232
5.4.2 Myelinated nerves 215 ......Page 237
5.5 Assemblies of neurons 221 ......Page 243
5.6 Summary 223 ......Page 245
5.7 Problems 224 ......Page 246
References 230 ......Page 252
6 INVERSE SCATTERING METHODS 238 ......Page 260
6.1.1 Scattering solutions, bound states, and upper half plane poles 240 ......Page 262
6.1.2 Why the upper half plane poles must be simple 242 ......Page 264
6.1.3 The Gel’fand-Levitan equation again 245 ......Page 267
6.1.4 Any questions? 249 ......Page 271
6.2.1 General description 250 ......Page 272
6.2.2 Some examples 252 ......Page 274
6.3.1 Linear theory 258 ......Page 280
6.3.2 ISMs for two-component scattering 264 ......Page 286
6.4 The sine-Gordon equation 266 ......Page 288
6.5 The nonlinear Schrodinger equation 271 ......Page 293
6.6 Conservation laws 273 ......Page 295
6.6.1 Conservation laws for the KdV equation 274 ......Page 296
6.6.2 Conserved densities for matrix scattering 276 ......Page 298
6.8 Problems 277 ......Page 299
References 285 ......Page 307
7 PERTURBATION THEORY 287 ......Page 309
7.1 Perturbed matrices 288 ......Page 310
7.2.1 Energy analysis 290 ......Page 312
7.2.2 Multiple time scales 291 ......Page 313
7.3 Energy analysis of soliton dynamics 293 ......Page 315
7.3.1 Korteweg-de Vries solitons 294 ......Page 316
7.3.2 Sine-Gordon solitons 296 ......Page 318
7.3.3 Nonlinear Schrodinger solitons 299 ......Page 321
7.4.1 Multiple scale analysis of an SG kink 301 ......Page 323
7.4.2 Variational analysis of an NLS soliton 306 ......Page 328
7.5 Multisoliton perturbation theory 309 ......Page 331
7.5.1 General theory 310 ......Page 332
7.5.2 Kink-antikink collisions 314 ......Page 336
7.6 Neural perturbations 319 ......Page 341
7.6.1 The FitzHugh-Nagumo system 320 ......Page 342
7.6.2 Electrodynamic (ephaptic) coupling of nerves 322 ......Page 344
7.7 Summary 326 ......Page 348
7.8 Problems 327 ......Page 349
References 335 ......Page 357
8.1.2 The birth of quantum theory 339 ......Page 361
8.1.3 A quantum linear oscillator 342 ......Page 364
8.1.5 The Born-Oppenheimer approximation 347 ......Page 369
8.1.6 Dirac’s notation 350 ......Page 372
8.2 Self-trapping in the dihalomethanes 353 ......Page 375
8.2.1 Classical analysis 354 ......Page 376
8.2.2 Quantum analysis 356 ......Page 378
8.2.3 Comparison with experiments 360 ......Page 382
8.3.1 The discrete self-trapping equation 361 ......Page 383
8.3.2 A lattice nonlinear Schrodinger equation 365 ......Page 387
8.3.3 Soliton wave packets 370 ......Page 392
8.3.4 The Hartree approximation 372 ......Page 394
8.4.1 The Ablowitz-Ladik equation 377 ......Page 399
8.4.2 Salerno’s equation 380 ......Page 402
8.4.3 A fermionic polaron model 381 ......Page 403
8.4.4 The Hubbard model 384 ......Page 406
8.5.1 Dynamic equations 386 ......Page 408
8.5.2 Experimental observations 390 ......Page 412
8.5.3 Recent comments 398 ......Page 420
8.7.1 Number state method 403 ......Page 425
8.7.2 Quantum inverse scattering method 404 ......Page 426
8.7.3 QISM analysis of the DST dimer 406 ......Page 428
8.7.4 Comparison of the NSM and the QISM 407 ......Page 429
8.9 Problems 409 ......Page 431
References 420 ......Page 442
9 LOOKING AHEAD 424 ......Page 446
References 431 ......Page 453
APPENDIX A CONSERVATION LAWS AND CONSERVATIVE SYSTEMS 433 ......Page 455
References 437 ......Page 459
B.2 The SG equation 438 ......Page 460
B.4 The Toda lattice 440 ......Page 462
References 441 ......Page 463
APPENDIX C ELLIPTIC FUNCTIONS 443 ......Page 465
References 447 ......Page 469
APPENDIX D STABILITY OF NERVE IMPULSES 448 ......Page 470
References 454 ......Page 476
APPENDIX E PERIODIC TODA-LATTICE SOLITONS 456 ......Page 478
References 457 ......Page 479
APPENDIX F ANALYTIC APPROXIMATIONS FOR LONG LATTICE SOLITARY WAVES 458 ......Page 480
Reference 459 ......Page 481
APPENDIX G MULTIPLE-SCALE ANALYSIS OF A DAMPED-HARMONIC OSCILLATOR 460 ......Page 482
References 462 ......Page 484
APPENDIX H GREEN FUNCTIONS FOR SOLITON RADIATION 463 ......Page 485
References 467 ......Page 489
INDEX 469 ......Page 491
cover ......Page 1