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Tables of Contents for Nonlinear Science
Chapter/Section Title
Page #
Page Count
List of Figures
xxi
1 THE BIRTH OF A PARADIGM
1
27
1.1 From the Great Wave to the Great War
1
7
1.1.1 Hydrodynamics
1
2
1.1.2 Nonlinear diffusion
3
3
1.1.3 Bäcklund transformation theory
6
1
1.1.4 A theory of matter
7
1
1.2 Between the wars
8
3
1.3 Nonlinear research from 1945 to 1985
11
10
1.3.1 Nerve studies
11
1
1.3.2 Autocatalytic chemical reactions
12
2
1.3.3 Solitons
14
5
1.3.4 Local modes in molecules and molecular crystals
19
1
1.3.5 Elementary particle research
20
1
1.4 Recent developments
21
2
References
23
5
2 LINEAR WAVE THEORY
28
27
2.1 Dispersionless linear equations
28
2
2.2 Dispersive linear equations
30
1
2.3 The linear diffusion equation
31
2
2.4 Driven systems
33
4
2.4.1 Green's method
33
2
2.4.2 Fredholm's theorem
35
2
2.5 Stability
37
3
2.5.1 General definitions
37
1
2.5.2 Linear stability
38
1
2.5.3 Signaling problems
39
1
2.6 Scattering theory
40
8
2.6.1 Solutions of Schrödinger's equation
40
3
2.6.2 Gel'fand-Levitan theory
43
5
2.6.3 A reflectionless potential
48
1
2.7 Problems
48
5
References
53
2
3 THE CLASSICAL SOLITON EQUATIONS
55
55
3.1 The Korteweg-de Vries (KdV) equation
57
14
3.1.1 Long water waves
57
1
3.1.2 Solitary wave solutions
58
1
3.1.3 Periodic solutions
59
2
3.1.4 A Bäcklund transformation for KdV
61
6
3.1.5 N-soliton formulas
67
4
3.2 The sine-Gordon (SG) equation
71
17
3.2.1 Long Josephson junctions
71
1
3.2.2 Solitary waves
72
2
3.2.3 Periodic waves
74
3
3.2.4 Nonlinear standing waves
77
4
3.2.5 Two-soliton solutions
81
4
3.2.6 More spatial dimensions
85
3
3.3 The nonlinear Schrödinger (NLS) equation
88
10
3.3.1 Nonlinear wave packets
88
2
3.3.2 Modulated traveling-wave solutions of NLS(+)
90
2
3.3.3 Dark soliton solutions of NLS(-)
92
1
3.3.4 A BT for NLS(+)
93
2
3.3.5 Transverse phenomena
95
3
3.4 Summary
98
1
3.5 Problems
98
8
References
106
4
4 REACTION-DIFFUSION SYSTEMS
110
66
4.1 Simple reaction-diffusion equations
111
6
4.1.1 The Zeldovich-Frank-Kamenetsky (Z-F) equation
111
5
4.1.2 The Burgers equation
116
1
4.2 The Hodgkin-Huxley (H-H) system
117
10
4.2.1 Space-clamped squid membrane dynamics
118
6
4.2.2 The H-H impulse
124
3
4.3 Simplified nerve models
127
11
4.3.1 The Markin-Chizmadzhev (M-C) model
127
2
4.3.2 The FitzHugh-Nagumo (F-N) model
129
5
4.3.3 Morris-Lecar (M-L) models
134
4
4.4 Stability analyses
138
5
4.4.1 The Z-F equation
138
1
4.4.2 The M-C model
139
1
4.4.3 The F-N model
140
3
4.4.4 The H-H and M-L systems
143
1
4.5 Decremental conduction
143
4
4.6 Nonuniform fibers
147
7
4.6.1 Tapered fibers
147
2
4.6.2 Leading-edge charge and impulse ignition
149
1
4.6.3 Dendritic logic
150
4
4.7 More space dimensions
154
7
4.7.1 Two-dimensional nonlinear diffusion
154
2
4.7.2 Nonlinear diffusion in three dimensions
156
3
4.7.3 Turing patterns
159
1
4.7.4 Hypercycles
160
1
4.8 Summary
161
1
4.9 Problems
162
9
References
171
5
5 NONLINEAR LATTICES
176
62
5.1 Spring-mass lattices
177
8
5.1.1 The Toda-lattice soliton
178
1
5.1.2 Lattice solitary waves
179
1
5.1.3 Existence of lattice solitary waves
180
2
5.1.4 Intrinsic localized modes and intrinsic gap modes
182
3
5.2 Lattices with nonlinear on-site potentials
185
17
5.2.1 The discrete sine-Gordon equation
187
3
5.2.2 Nonlinear Schrödinger lattices
190
7
5.2.3 The discrete self trapping equation
197
5
5.3 Biological solitons
202
8
5.3.1 Alpha-helix solitons in protein
202
3
5.3.2 Self-trapping in globular proteins
205
2
5.3.3 Solitons in DNA
207
3
5.4 Nonconservative lattices
210
11
5.4.1 Quasiharmonic lattices
210
5
5.4.2 Myelinated nerves
215
4
5.4.3 Emergence of form by replication
219
2
5.5 Assemblies of neurons
221
2
5.6 Summary
223
1
5.7 Problems
224
6
References
230
8
6 INVERSE SCATTERING METHODS
238
49
6.1 Linear scattering revisited
240
10
6.1.1 Scattering solutions, bound states, and upper half plane poles
240
2
6.1.2 Why the upper half plane poles must be simple
242
3
6.1.3 The Gel'fand-Levitan equation again
245
4
6.1.4 Any questions?
249
1
6.2 Inverse scattering method for KdV
250
8
6.2.1 General description
250
2
6.2.2 Some examples
252
5
6.2.3 Reduction to Fourier analysis in the small amplitude limit
257
1
6.3 Two-component scattering theory
258
8
6.3.1 Linear theory
258
6
6.3.2 ISMs for two-component scattering
264
2
6.4 The sine-Gordon equation
266
5
6.5 The nonlinear Schrödinger equation
271
2
6.6 Conservation laws
273
4
6.6.1 Conservation laws for the KdV equation
274
2
6.6.2 Conserved densities for matrix scattering
276
1
6.7 Summary
277
1
6.8 Problems
278
7
References
285
2
7 PERTURBATION THEORY
287
50
7.1 Perturbed matrices
288
2
7.2 A damped harmonic oscillator
290
3
7.2.1 Energy analysis
290
1
7.2.2 Multiple time scales
291
2
7.3 Energy analysis of soliton dynamics
293
8
7.3.1 Korteweg-de Vries solbons
294
2
7.3.2 Sine-Gordon solitons
296
3
7.3.3 Nonlinear Schrödinger solitons
299
2
7.4 More general soliton analyses
301
8
7.4.1 Multiple scale analysis of an SG kink
301
5
7.4.2 Variational analysis of an NLS soliton
306
3
7.5 Multisoliton perturbation theory
309
10
7.5.1 General theory
310
4
7.5.2 Kink-antikink collisions
314
3
7.5.3 Radiation from a fluxon
317
2
7.6 Neural perturbations
319
7
7.6.1 The FitzHugh-Nagumo system
320
2
7.6.2 Electrodynamic (ephaptic) coupling of nerves
322
4
7.7 Summary
326
1
7.8 Problems
327
8
References
335
2
8 QUANTUM LATTICE SOLITONS
337
87
8.1 Quantum oscillators
337
16
8.1.1 A classical nonlinear oscillator
337
2
8.1.2 The birth of quantum theory
339
3
8.1.3 A quantum linear oscillator
342
3
8.1.4 The rotating wave approximation
345
2
8.1.5 The Born-Oppenheimer approximation
347
3
8.1.6 Dirac's notation
350
1
8.1.7 Pump-probe measurements
351
2
8.2 Self trapping in the dihalomethanes
353
8
8.2.1 Classical analysis
354
2
8.2.2 Quantum analysis
356
4
8.2.3 Comparison with experiments
360
1
8.3 Boson lattices
361
16
8.3.1 The discrete self trapping equation
361
4
8.3.2 A lattice nonlinear Schrödinger equation
365
5
8.3.3 Soliton wave packets
370
2
8.3.4 The Hartree approximation
372
5
8.4 More general quanta
377
9
8.4.1 The Ablowitz-Ladik equation
377
3
8.4.2 Salerno's equation
380
1
8.4.3 A fermionic polaron model
381
3
8.4.4 The Hubbard model
384
2
8.5 Energy transport in protein
386
15
8.5.1 Dynamic equations
386
4
8.5.2 Experimental observations
390
8
8.5.3 Recent comments
398
3
8.6 A quantum lattice sine-Gordon equation
401
2
8.7 Theoretical perspectives
403
6
8.7.1 Number state method
403
1
8.7.2 Quantum inverse scattering method
404
2
8.7.3 QISM analysis of the DST dimer
406
1
8.7.4 Comparison of the NSM and the QISM
407
2
8.8 Summary
409
1
8.9 Problems
409
11
References
420
4
9 LOOKING AHEAD
424
9
References
431
2
APPENDIX A CONSERVATION LAWS AND CONSERVATIVE SYSTEMS
433
5
References
437
1
APPENDIX B MULTISOLITON FORMULAS
438
5
B.1 The KdV equation
438
1
B.2 The SG equation
438
2
B.3 The NLS equation
440
1
B.4 The Toda lattice 440 References
441
2
APPENDIX C ELLIPTIC FUNCTIONS
443
5
References
447
1
APPENDIX D STABILITY OF NERVE IMPULSES
448
8
References
454
2
APPENDIX E PERIODIC TODA-LATTICE SOLITONS
456
2
References
457
1
APPENDIX F ANALYTIC APPROXIMATIONS FOR LONG LATTICE SOLITARY WAVES
458
2
Reference
459
1
APPENDIX G MULTIPLE-SCALE ANALYSIS OF A DAMPED-HARMONIC OSCILLATOR
460
3
References
462
1
APPENDIX H GREEN FUNCTIONS FOR SOLITON RADIATION
463
6
References
467
2
INDEX
469