Nonlinear science: emergence and dynamics of coherent structures
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford <<[u.a.]>>
Oxford Univ. Press
2003
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Oxford texts in applied and engineering mathematics
8 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXIII, 480 S. |
| ISBN: | 0198528523 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804140508840198144 |
|---|---|
| adam_text | Titel: Nonlinear science
Autor: Scott, Alwyn
Jahr: 2003
CONTENTS
List of Figures xxi
1 THE BIRTH OF A PARADIGM 1
1.1 From the Great Wave to the Great War 1
1.1.1 Hydrodynamics 1
1.1.2 Nonlinear diffusion 3
1.1.3 Backlund transformation theory 6
1.1.4 A theory of matter 7
1.2 Between the wars 8
1.3 Nonlinear research from 1945 to 1985 11
1.3.1 Nerve studies 11
1.3.2 Autocatalytic chemical reactions 12
1.3.3 Solitons 14
1.3.4 Local modes in molecules and molecular crystals 19
1.3.5 Elementary particle research 20
1.4 Recent developments 21
References 23
2 LINEAR WAVE THEORY 28
2.1 Dispersionless linear equations 28
2.2 Dispersive linear equations 30
2.3 The linear diffusion equation 31
2.4 Driven systems 33
2.4.1 Green s method 33
2.4.2 Fredholm s theorem 35
2.5 Stability 37
2.5.1 General definitions 37
2.5.2 Linear stability 38
2.5.3 Signaling problems 39
2.6 Scattering theory 40
2.6.1 Solutions of Schrodinger s equation 40
2.6.2 Gel fand-Levitan theory 43
2.6.3 A reflectionless potential 48
2.7 Problems 48
References 53
xvi
CONTENTS
71
71
72
74
THE CLASSICAL SOLITON EQUATIONS 55
3.1 The Korteweg-de Vries (KdV) equation
3.1.1 Long water waves
3.1.2 Solitary wave solutions ^8
3.1.3 Periodic solutions
3.1.4 A Backlund transformation for KdV 61
3.1.5 N-soliton formulas ^7
3.2 The sine-Gordon (SG) equation
3.2.1 Long Josephson junctions
3.2.2 Solitary waves
3.2.3 Periodic waves
3.2.4 Nonlinear standing waves 77
3.2.5 Two-soliton solutions 81
3.2.6 More spatial dimensions 85
3.3 The nonlinear Schrodinger (NLS) equation 88
3.3.1 Nonlinear wave packets 88
3.3.2 Modulated traveling-wave solutions of NLS(+) 90
3.3.3 Dark soliton solutions of NLS(—) 92
3.3.4 A BT for NLS(+) 93
3.3.5 Transverse phenomena 95
3.4 Summary 98
3.5 Problems 98
References 106
REACTION-DIFFUSION SYSTEMS 110
4.1 Simple reaction-diffusion equations 111
4.1.1 The Zeldovich-Frank-Kamenetsky (Z-F) equation 111
4.1.2 The Burgers equation 116
4.2 The Hodgkin-Huxley (H-H) system 117
4.2.1 Space-clamped squid membrane dynamics 118
4.2.2 The H-H impulse 124
4.3 Simplified nerve models 127
4.3.1 The Markin-Chizmadzhev (M-C) model 127
4.3.2 The FitzHugh-Nagumo (F-N) model 129
4.3.3 Morris-Lecar (M-L) models 134
4.4 Stability analyses
4.4.1 The Z-F equation
4.4.2 The M-C model
4.4.3 The F-N model
4.4.4 The H-H and M-L systems
4.5 Decremental conduction
4.6 Nonuniform fibers
4.6.1 Tapered fibers
4.6.2 Leading-edge charge and impulse ignition
138
138
139
140
143
143
147
147
149
CONTENTS
xvii
4.6.3 Dendritic logic 150
4.7 More space dimensions 154
4.7.1 Two-dimensional nonlinear diffusion 154
4.7.2 Nonlinear diffusion in three dimensions 156
4.7.3 Turing patterns 159
4.7.4 Hypercycles 160
4.8 Summary 161
4.9 Problems 162
References 171
5 NONLINEAR LATTICES 176
5.1 Spring-mass lattices 177
5.1.1 The Toda-Iattice soliton 178
5.1.2 Lattice solitary waves 179
5.1.3 Existence of lattice solitary waves 180
5.1.4 Intrinsic localized modes and intrinsic gap modes 182
5.2 Lattices with nonlinear on-site potentials 185
5.2.1 The discrete sine-Gordon equation 187
5.2.2 Nonlinear Schrodinger lattices 190
5.2.3 The discrete self-trapping equation 197
5.3 Biological solitons 202
5.3.1 Alpha-helix solitons in protein 202
5.3.2 Self-trapping in globular proteins 205
5.3.3 Solitons in DNA 207
5.4 Nonconservative lattices 210
5.4.1 Quasiharmonic lattices 210
5.4.2 Myelinated nerves 215
5.4.3 Emergence of form by replication 219
5.5 Assemblies of neurons 221
5.6 Summary 223
5.7 Problems 224
References 230
6 INVERSE SCATTERING METHODS 238
6.1 Linear scattering revisited 240
6.1.1 Scattering solutions, bound states, and upper half
plane poles 240
6.1.2 Why the upper half plane poles must be simple 242
6.1.3 The Gel fand-Levitan equation again 245
6.1.4 Any questions? 249
6.2 Inverse scattering method for KdV 250
6.2.1 General description 250
6.2.2 Some examples 252
xviii
CONTENTS
257
258
258
266
271
273
6.2.3 Reduction to Fourier analysis in the small amplitude
limit
G.3 Two-component scattering theory
6.3.1 Linear theory
6.3.2 ISMs for two-component scattering 264
6.4 The sine-Gordon equation
6.5 The nonlinear Schrodinger equation
6.6 Conservation laws
6.6.1 Conservation laws for the KdV equation 274
6 6 2 Conserved densities for matrix scattering 276
977
6.7 Summary
6.8 Problems ^78
References
7 PERTURBATION THEORY 287
7.1 Perturbed matrices 288
7.2 A damped harmonic oscillator 290
7.2.1 Energy analysis 290
7.2.2 Multiple time scales 291
7.3 Energy analysis of soliton dynamics 293
7.3.1 Korteweg-de Vries solitons 294
7.3.2 Sine-Gordon solitons 296
7.3.3 Nonlinear Schrodinger solitons 299
7.4 More general soliton analyses 301
7.4.1 Multiple scale analysis of an SG kink 301
7.4.2 Variational analysis of an NLS soliton 306
7.5 Multisoliton perturbation theory 309
7.5.1 General theory 310
7.5.2 Kink-antikink collisions 314
7.5.3 Radiation from a fluxon 317
7.6 Neural perturbations 319
7.6.1 The FitzIIugh-Nagumo system 320
7.6.2 Electrodynamic (ephaptic) coupling of nerves 322
7.7 Summary 326
7.8 Problems 327
References 335
8 QUANTUM LATTICE SOLITONS 337
8.1 Quantum oscillators 337
8.1.1 A classical nonlinear oscillator 337
8.1.2 The birth of quantum theory
8.1.3 A quantum linear oscillator
339
342
8.1.4 The rotating wave approximation 345
8.1.5 The Born-Oppenheimer approximation 347
CONTENTS
xix
8.1.6 Dirac s notation 350
8.1.7 Pump-probe measurements 351
8.2 Self-trapping in the dihalomethanes 353
8.2.1 Classical analysis 354
8.2.2 Quantum analysis 356
8.2.3 Comparison with experiments 360
8.3 Boson lattices 361
8.3.1 The discrete self-trapping equation 361
8.3.2 A lattice nonlinear Schrodinger equation 365
8.3.3 Soliton wave packets 370
8.3.4 The Hartree approximation 372
8.4 More general quanta 377
8.4.1 The Ablowitz-Ladik equation 377
8.4.2 Salerno s equation 380
8.4.3 A fermionic polaron model 381
8.4.4 The Hubbard model 384
8.5 Energy transport in protein 386
8.5.1 Dynamic equations 386
8.5.2 Experimental observations 390
8.5.3 Recent comments 398
8.6 A quantum lattice sine-Gordon equation 401
8.7 Theoretical perspectives 403
8.7.1 Number state method 403
8.7.2 Quantum inverse scattering method 404
8.7.3 QISM analysis of the DST dimer 406
8.7.4 Comparison of the NSM and the QISM 407
8.8 Summary 409
8.9 Problems 409
References 420
9 LOOKING AHEAD 424
References 431
APPENDIX A CONSERVATION LAWS AND
CONSERVATIVE SYSTEMS 433
References 437
APPENDIX B MULTISOLITON FORMULAS 438
B.l The KdV equation 438
B.2 The SG equation 438
B.3 The NLS equation 440
B.4 The Toda lattice 440
References 441
xx CONTENTS
APPENDIX C ELLIPTIC FUNCTIONS 443
References ^47
APPENDIX D STABILITY OF NERVE IMPULSES 448
References 454
APPENDIX E PERIODIC TODA-LATTICE SOLITONS 456
References 457
APPENDIX F ANALYTIC APPROXIMATIONS FOR
LONG LATTICE SOLITARY WAVES 458
Reference 459
APPENDIX G MULTIPLE-SCALE ANALYSIS OF A
DAMPED-HARMONIC OSCILLATOR 460
References 462
APPENDIX H GREEN FUNCTIONS FOR SOLITON
RADIATION 463
References 4G7
INDEX
469
|
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| discipline | Physik Informatik |
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| language | English |
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| physical | XXIII, 480 S. |
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| publisher | Oxford Univ. Press |
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| spelling | Scott, Alwyn 1931-2007 Verfasser (DE-588)124032400 aut Nonlinear science emergence and dynamics of coherent structures Alwyn Scott 2. ed. Oxford <<[u.a.]>> Oxford Univ. Press 2003 XXIII, 480 S. txt rdacontent n rdamedia nc rdacarrier Oxford texts in applied and engineering mathematics 8 Nichtlineare Dynamik (DE-588)4126141-0 gnd rswk-swf Soliton (DE-588)4135213-0 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 gnd rswk-swf Nichtlineare Dynamik (DE-588)4126141-0 s DE-604 Nichtlineares dynamisches System (DE-588)4126142-2 s Soliton (DE-588)4135213-0 s Oxford texts in applied and engineering mathematics 8 (DE-604)BV017577755 8 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018484203&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Scott, Alwyn 1931-2007 Nonlinear science emergence and dynamics of coherent structures Oxford texts in applied and engineering mathematics Nichtlineare Dynamik (DE-588)4126141-0 gnd Soliton (DE-588)4135213-0 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd |
| subject_GND | (DE-588)4126141-0 (DE-588)4135213-0 (DE-588)4126142-2 |
| title | Nonlinear science emergence and dynamics of coherent structures |
| title_auth | Nonlinear science emergence and dynamics of coherent structures |
| title_exact_search | Nonlinear science emergence and dynamics of coherent structures |
| title_full | Nonlinear science emergence and dynamics of coherent structures Alwyn Scott |
| title_fullStr | Nonlinear science emergence and dynamics of coherent structures Alwyn Scott |
| title_full_unstemmed | Nonlinear science emergence and dynamics of coherent structures Alwyn Scott |
| title_short | Nonlinear science |
| title_sort | nonlinear science emergence and dynamics of coherent structures |
| title_sub | emergence and dynamics of coherent structures |
| topic | Nichtlineare Dynamik (DE-588)4126141-0 gnd Soliton (DE-588)4135213-0 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd |
| topic_facet | Nichtlineare Dynamik Soliton Nichtlineares dynamisches System |
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