ISBN: 3-540-66918-3
TITLE: Nonlinear Science at the Dawn of the 21st Century
AUTHOR: Christiansen, P.L.; Sorensen, M.P.; Scott, A.C. (Eds.)
TOC:

I Nonlinear Science 1
1 Nonlinear Coherent Phenomena in Continuous Media 3
E.A. Kuznetsov and V.E. Zakharov
1 Introduction 3
2 Phase randomization in nonlinear media 4
3 Nonlinear Schrdinger equation 9
4 Solitons in the focusing NSLE 11
5 Collapses in the NLSE 15
6 Weak, strong and superstrong collapses 18
7 Anisotropic black holes 21
8 Structure in media with weak dispersion 26
9 Singularities on a fluid surface 32
10 Solitons and collapses in the generalized KP equation 34
11 Self-focusing in the boundary layer 38
12 References 42
2 Perturbation Theories for Nonlinear Waves 47
L. Ostrovsky and K. Gorshkov
1 Introduction 47
2 Modulated waves 49
3 Direct perturbation method 51
4 Perturbation theories for solitary waves 52
4.1 Direct perturbation method for solitons: quasistationary approach 52
4.2 Nonstationary approach 54
4.3 Inverse-scattering perturbation scheme 55
4.4 "Equivalence principle" 57
4.5 Example: soliton interaction in Lagrangian systems 58
4.6 Radiation from solitons 59
5 Asymptotic reduction of nonlinear wave equations 60
6 Conclusions 61
7 References 62
II Superconductivity and Magnetism 67
3 Josephson Devices 69
A. Barone and S. Pagano
1 Introduction 69
2 Elements of the Josephson effect 70
3 SQUIDs 72
4 Digital devices 75
5 Detectors 77
6 Voltage standard 78
7 Microwave oscillators 80
8 Conclusions 83
9 References 83
4 Josephson Flux-Flow Oscillators in Microwave Fields 87
M. Salerno and M. Samuelsen
1 Introduction 87
2 Flux-flow oscillators in uniform microwave fields 88
3 Flux-flow oscillators in non-uniform microwave fields 91
4 Numerical experiment 94
5 Conclusions 98
6 Appendix 98
7 References 100
5 Coupled Structures of Long Josephson Junctions 103
G. Carapella and G. Costabile
1 Stacks of two long Josephson junctions 103
1.1 The physical system and its model 103
1.2 Experiments on stacks of two long Josephson junctions 106
2 Parallel arrays of Josephson junctions 109
2.1 The physical system and its model 109
2.2 Numerical and experimental results on five-junctions parallel arrays 112
3 Triangular cells of long Josephson junctions 113
3.1 The model 114
3.2 Numerical and experimental results 115
4 References 118
6 Stacked Josephson Junctions 121
N.F. Pedersen
1 Introduction 121
2 Short summary of fluxon properties 121
3 Stacked junctions 123
4 Fluxon solutions, selected examples 125
4.1 The coherent 2-fluxon mode 125
4.2 The two modes of the two fluxon case 127
5 Stacked junction plasma oscillation solutions 128
6 Conclusion 135
7 References 135
7 Dynamics of Vortices in Two-Dimensional Magnets 137
F.G. Mertens and A.R. Bishop
1 Introduction 137
2 Collective variable theories at zero temperature 141
2.1 Thiele equation 141
2.2 Vortex mass 143
2.3 Hierarchy of equations of motion 146
2.4 An alternative approach: coupling to magnons 152
3 Effects of thermal noise on vortex dynamics 155
3.1 Equilibrium and non-equilibrium situations 155
3.2 Collective variable theory and Langevin dynamics simulations 155
3.3 Noise-induced transitions between opposite polarizations 161
4 Dynamics above the Kosterlitz-Thouless transition 163
4.1 Vortex-gas approach 163
4.2 Comparison with simulations and experiments 164
4.3 Vortex motion in Monte Carlo simulations 165
5 Conclusion 166
6 References 167
III Nonlinear Optics 171
8 Spatial Optical Solitons 173
Yu.S. Kivshar
1 Introduction 173
2 Spatial vs. temporal solitons 175
3 Basic equations 176
4 Stability of solitary waves 178
4.1 One-parameter solitary waves 179
4.2 Two-parameter solitary waves 180
5 Experiments on self-focusing 182
6 Soliton spiralling 184
7 Multi-hump solitons and solitonic gluons 186
8 Discrete spatial optical solitons 189
9 References 191
9 Nonlinear Fiber Optics 195
G.P. Agrawal
1 Introduction 195
2 Fiber characteristics 196
2.1 Single-mode fibers 196
2.2 Fiber nonlinearities 196
2.3 Group-velocity dispersion 197
3 Pulse propagation in fibers 198
3.1 Nonlinear Schrodinger equation 198
3.2 Modulation instability 198
4 Optical solitons 199
4.1 Bright solitons 200
4.2 Dark solitons 201
4.3 Loss-managed solitons 202
4.4 Dispersion-managed solitons 203
5 Nonlinear optical switching 205
5.1 SPM-based optical switching 205
5.2 XPM-based optical switching 207
6 Concluding remarks 209
7 References 209
10 Self-Focusing and Collapse of Light Beams in Nonlinear Dispersive Media 213
L. Berg and J. Juul Rasmussen
1 Introduction 213
2 General properties of self-focusing with anomalous group velocity dispersion 214
2.1 Basic properties 214
2.2 Self-similar wave collapses 216
3 Self-focusing with normal group velocity dispersion 220
4 Discussion of the general properties, outlook 224
5 References 226
11 Coherent Structures in Dissipative Nonlinear Optical Systems 229
J.V. Moloney
1 Introduction 229
2 Nonlinear waveguide channeling in air 234
2.1 Dynamic spatial replenishment of femtosecond pulses propagating in air 236
3 Control of optical turbulence in semiconductor lasers 239
3.1 The control 240
4 Summary and conclusions 243
5 References 245
12 Solitons in Optical Media with Quadratic Nonlinearity 247
B.A. Malomed
1 Introduction 247
2 The basic theoretical models 249
3 The solitons 25 3
4 Conclusion 260
5 References 261
IV Lattice Dynamics and Applications 263
13 Nonlinear Models for the Dynamics of Topological Defects in Solids 265
Yu.S. Kivshar, H. Benner and O.M. Braun
1 Introduction 265
2 The FK model and the SG equation 266
3 Physical models 269
3.1 Dislocations in solids 269
3.2 Magnetic chains 270
3.3 Josephson junctions 271
3.4 Hydrogen-bonded chains 273
3.5 Surface physics and adsorbed atomic layers 275
3.6 Models of the DNA dynamics 276
4 Properties of kinks 277
4.1 On-site potential of general form 277
4.2 Discreteness effects 279
4.3 Kinks in external fields 280
4.4 Compacton kinks 281
5 Experimental verifications 281
5.1 Josephson junctions 281
5.2 Magnetic systems 283
6 Concluding remarks 285
7 References 286
14 2-D Breathers and Applications 293
J.L. Marn, J.C. Eilbeck and F.M. Russell
1 Introduction 293
2 Deciphering the lines in mica 294
3 Numerical and analogue studies 296
4 Longitudinal moving breathers in 2D lattices 299
5 Breather collisions 302
6 Conclusions and further applications 303
6.1 Application to sputtering 303
6.2 Application to layered HTSC materials 303
7 References 304
15 Scale Competition in Nonlinear Schrdinger Models 307
Yu. B. Gaididei, P.L. Christiansen and S.F. Mingaleev
1 Introduction 307
2 Excitations in nonlinear Kronig-Penney models 308
3 Discrete NLS models with long-range dispersive interactions 311
4 Stabilization of nonlinear excitations by disorder 316
5 Summary 319
6 References 320
16 Demonstration Systems for Kink-Solitons 323
M. Remoissenet
1 Introduction 323
2 Mechanical chains with double-well potential 325
2.1 Chain with torsion and gravity 325
2.2 Chain with flexion and gravity 329
2.3 Numerical simulations 330
3 Experiments 331
3.1 Chain with torsion and gravity 331
3.2 Chain with flexion and gravity 332
4 Lattice effects 333
5 Conclusion 334
6 Appendix 335
7 References 336
17 Quantum Lattice Solitons 339
A. C. Scott
1 Introduction 339
2 Local modes in the dihalomethanes 339
2.1 Classical analysis 340
2.2 Quantum analysis 341
2.3 Comparison with experiments 344
3 A lattice nonlinear Schrdinger equation 344
4 Local modes in crystalline acetanilide 348
5 Conclusions 354
6 References 355
18 Noise in Molecular Systems 357
G.P. Tsironis
1 Introduction 35 7
2 Additive correlated ratchets 35 8
3 Current reversal 363
4 Synthetic motor protein motion 364
5 Targeted energy transfer and nonequilibrium fluctuations in bioenergetics 368
6 References 369
V Biomolecular Dynamics and Biology 371
19 Nonlinear Dynamics of DNA 373
L.V. Yakushevich
1 Introduction 373
2 General description of DNA dynamics.
Classification of the internal motions 375
3 Mathematical modeling of DNA dynamics. Hierarchy of the models 376
3.1 Principles of modeling 376
3.2 Structural hierarchy 376
3.3 Dynamical hierarchy 377
4 Nonlinear mathematical models. Solved and unsolved problems 379
4.1 Ideal models 379
4.2 Nonideal models 380
4.3 Statistics of solitons in DNA 380
5 Nonlinear DNA models and experiment 381
5.1 Hydrogen-tritium (or hydrogen-deuterium) exchange 381
5.2 Resonant microwave absorption 382
5.3 Scattering of neutrons and light 383
5.4 Fluorescence depolarization 384 
6 Nonlinear conception and mechanisms of DNA functioning 385
6.1 Nonlinear mechanism of conformational transitions 385
6.2 Nonlinear conformational waves and long-range effects 385
6.3 Direction of transcription process 386
7 References 387
20 From the FPU Chain to Biomolecular Dynamics 393
A.V. Zolotaryuk, A.V. Savin and P.L. Christiansen
1 Introduction 393
2 Helices in two and three dimensions 394
3 Equations of motion for a helix backbone 397
4 Small-amplitude limit 398
5 Three-component soliton solutions 400
5.1 3D case: solitons of longitudinal compression 402
5.2 2D case: other types of solutions 404
6 Conclusions 405
7 References 407
21 Mutual Dynamics of Swimming Microorganisms and Their Fluid Habitat 409
J.O. Kessler, G.D. Burnett and K.E. Remick
1 Introduction 409
2 Bioconvection (I) 411
2.1 Observations 411
2.2 Continuum theory 412
3 Bacteria in constraining environments (II) 415
4 Possibilities for computer simulation 417
5 Conclusion 421
6 Appendix I: Statistical methods 423
7 Appendix II 424
8 References 425
22 Nonlinearities in Biology: The Brain as an Example 427
H. Haken
1 Introduction 427
2 Some salient features of neurons 427
3 The noisy lighthouse model of a neural network 429
4 The special case of two neurons 430
5 Time-averages 433
6 The averaged neural equations 434
7 How to make contact with experimental data?
Synergetics as a guide 440
8 Concluding remarks and outlook 443
9 References 444
Index 447
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