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Tables of Contents for Nonlinearity and Disorder
Chapter/Section Title
Page #
Page Count
Preface
xiii
Poem
xvii
Contributing Authors
xix
Part I Nonlinear Waves in Random Media
Exponential Localization Versus Soliton Propagation
3
16
Introduction
3
1
Random linear media
4
4
Solitary waves
8
1
Solitons in random media
9
2
Numerical simulations
11
1
Conclusion
12
7
13
1
14
5
Pulse Confinement in Optical Fibers with Random Dispersion
19
10
Introduction
19
1
Basic modelling
20
4
Numerical results
24
2
Conclusion
26
3
Propagation of an Optical Pulse in a Fiber with Random Dispersion - management
29
8
Introduction
29
1
The model
30
3
Numerical simulations
33
2
Conclusion
35
2
Spatial Solitons in Random Quadratic Media
37
18
Introduction
37
3
Description of the model. Equations for the mean fields
40
3
Analytical model for soliton evolution based on the MFM equations
43
2
Numerical simulations of solitons propagation in random quadratic media
45
6
Conclusions
51
4
Correlation of the Amplitude and Phase Random Modulations in Nonlinear Medium
55
6
Spatio-temporal Chaos in Time Evolution of BEC with Attractive Two-body Interactions
61
8
Dynamics of a Bose-Einstein Condensate in Optical Trap
69
12
Introduction
69
1
Basic equations and the method of analysis
70
3
Numerical modelling
73
3
Conclusion
76
5
Part II Nonlinear Waves in Discrete, Including Disordered, Systems
Aspects of Discrete Breathers and New Directions
81
18
Introduction
81
2
Existence proofs for discrete breathers
83
9
Perspectives on discrete breathers applications
92
7
Complexity at Mesoscale
99
16
Introduction
99
1
Pattern formation in a driven nonlinear lattice
100
4
Dry friction
104
3
Patterns arising from elastic compatibility
107
4
Conclusions
111
4
Resonant Mode Interactions in Nonlinear Lattices
115
16
Properties of a linear lattice
116
3
Nonlinear model
119
1
Second harmonic generation in a two-dimensional lattice
120
4
Quadratic solitons in a lattice
124
1
Dynamics of a double chains
125
4
Conclusion
129
2
Discrete Solitons in Nonlinear Waveguide Arrays
131
28
Introduction
131
3
Basic mathematical models
134
2
Discrete diffraction and modulational instability
136
5
Solitons in cubic nonlinear media
141
9
Solitons in quadratic nonlinear media
150
5
Conclusions
155
4
Quantization of Intrinsic Nonlinear Localized Modes in Solids
159
14
Introduction
159
2
Formulation of the problem in the rotating-wave approximation
161
2
Effective linear lattices and quantization
163
1
Solutions of one-ILM problem
164
3
Self-consistent theory
167
6
Experiments with Discrete Breathers in Josephson Arrays
173
16
Introduction
173
1
Josephson junction as a nonlinear oscillator
174
3
Theoretical model for Josephson ladder arrays
177
1
Experimental methods and results
177
6
Future work
183
2
Summary
185
4
Breather Compactification in Nonlinear Anharmonic Lattices with Nonlocal Dispersive Interactions
189
8
Introduction
189
2
Analytic solutions
191
1
Numerical analysis
192
2
Conclusions
194
3
Growth and Decay of Weakly Perturbed Discrete Breathers
197
8
Introduction
197
1
Model and results
198
4
Summary
202
3
Standing Waves in 1D Nonlinear Lattices
205
8
Introduction
205
1
The model
205
1
Descriptions of standing waves
206
3
Stability of standing waves
209
4
Wannier-Stark Solitons in Waveguide Arrays with Linear Potential
213
10
Linear properties
214
2
Experimental results
216
1
Wannier-Stark solitons
217
2
Conclusions
219
4
Part III Optical Solitons in Inhomogeneous Media
The Stimulated Raman Scattering System: a Numerical Study
223
16
Introduction
223
1
The model
224
2
Saturation of Raman effect with distance l
226
2
Observation of Raman solitons:
228
5
Phase effects
233
6
Chaotic Dynamics of Topological Defects in Parametrically Excited Waves
239
16
Introduction
239
3
Dynamics of topological defects in parametrically excited waves
242
9
Conclusion
251
4
Wavebreaking in Hydrodynamics of the Defocusing NLS Equation
255
8
Introduction
255
1
Dispersionless limit of the NLS equation, its periodic solutions and their modulations
256
3
Exact solution of the Whitham equations and description of the region of oscillations after wavebreaking point
259
4
Propagation and Interaction of Extremely Short Electromagnetic Pulses in Non-Linear Media
263
16
Extremely short pulse propagation in resonance medium
264
3
Extremely short pulse propagation under quasi-resonance condition
267
1
Extremely short pulse propagation in non-resonance medium
268
9
Conclusion
277
2
Nonlinear Impurity Modes in Homogeneous and Periodic Media
279
12
Introduction
279
1
General analysis
280
3
Impurity-induced localization in a homogeneous medium
283
2
Impurity-induced localization in a periodic superlattice
285
2
Concluding remarks
287
4
Effects of Linear and Nonlinear Photo-elasticity in Super-lattices Near Interband Resonances
291
6
Introduction
291
1
Calculation of the dielectric constant and the linear and nonlinear photo-elastic constants
292
2
Discussion
294
1
Conclusion
295
2
Resonant Optics of Impurities in Photonic Crystals
297
10
Introduction
297
2
Two-photon relaxation processes in the field of coherent wave
299
1
The density matrix equation for impurity atom
300
1
Trapping of resonant excitation of impurity
301
1
Deactivation of level inside the gap by non resonant coherent wave
302
2
Population inversion
304
1
Conclusion
304
3
Bragg-grating Solitons in Dual-core Semi-linear Systems
307
8
Introduction
307
2
The Bragg grating on the nonlinear core
309
2
The Bragg grating on the linear core
311
2
Conclusion
313
2
Transverse Patterns in Type-II Optical Parametric Oscillators
315
8
Introduction
315
1
Mean field equations for a type-II OPO
316
1
Walk-off effects
317
2
Polarization coupling and domain walls
319
2
Summary
321
2
Self-trapping and Localized Modes in 2-D Photonic Crystals
323
18
Introduction
323
1
Model
324
1
Effective discrete equations
325
2
A Linear defect mode
327
1
Nonlinear localized modes
327
4
Soliton Self-shaping Through Transverse Electrostrictive Modulation in a Brillouin Fiber Ring Laser
Introduction
331
2
4-wave SBS-Kerr-CBS model
333
2
Pulse's dynamics
335
1
Stability map
336
2
Stability enhancement
338
2
Conclusion
340
1
Acknowledgements
340
1
Quantum Dynamics in Kerr Nonlinear Coupler via Phase-space Representation
341
10
Introduction
341
1
Quantum dynamics of the three mode NLC
342
3
Quantum dynamics of the two mode NLC in the cavity
345
3
Acknowledgments
348
3
Exact Cluster Expansions for a One-dimensional δ-function Bose Gas
351
14
Introduction
351
2
Formulation and two-particle case
353
2
N-particle case
355
2
Thermodynamic properties
357
3
Summary and Discussions
360
5
Part IV Nonlinear Optical Waves and Nonlinear Waves in General
Nonlinear Optical Shocks and Missiles. Theory, Applications and Numerical Methods
365
22
Basic propagation equations
367
2
Formation of carrier shocks in nonlinear transparent dielectrics
369
2
Optical missiles from diffracted optical shocks
371
6
Effects of material dispersion on shock and missile formation
377
3
Further remarks: experiments and alternative applications
380
2
Numerical methods for electromagnetic pulse propagation
382
5
Nonlinear Analysis of Heart Rate Variability
387
10
Introduction
387
1
Dynamic systems and time series
388
4
Experimental
392
1
Results:
393
1
Discussion and conclusions
393
4
Phenomenon of ``Quantized'' Oscillation Excitation
397
14
Introduction
397
3
Excitation of ``quantized'' pendulum oscillations
400
2
General conditions for ``quantized'' oscillation excitation
402
4
Conclusion
406
5
The Doppler Effect and Form of the Profile of Radiation Spectral Lines
411
4
Introduction
411
1
Distribution function of fluxons
412
1
The Profile of the radiation spectral line
413
1
Short summary
414
1
Dissipative soliton dynamics in strongly driven ferromagnets
415
8
Introduction
415
1
Model
416
1
Multiple-time scale method
417
1
Bloch domain wall
418
2
Numerical results
420
1
Summary and discussion
421
2
Nonlinear Eigenmodes in Quadratic Film Waveguides
423
12
Accelerating Solitons and Photorefractivity
435
8
The accelerating similarity variable
435
3
Photorefractive accelerating solitons
438
2
Numerical procedure and results
440
3
Index
443