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Nonlinearity and Disorder: Theory and Applications

Author(s): NATO Advanced Research Workshop on Nonlinearity and Disorder: Theory and Applications (2001 : Tashkent, Uzbekistan); Bang, Ole; Sorensen, Mads Peter

ISBN10: 1402001916

ISBN13: 9781402001918

Cover: Hardcover

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An up to date exposition of nonlinear phenomena in random and inhomogeneous media which provides recent results on the combined effects of nonlinearity and inhomogeneity, including random inhomogeneity. Topics covered include recent developments within such popular areas as nonlinear photonic crystals, inhomogeneous optical fibres (dispersion management), discrete nonlinear lattices, discrete breathers, Bose-Einstein condensates, ultra-short optical pulse, Josephson lattices, various types of inhomogeneous waveguides and nonlinear quantization.

Preface

xiii

Poem

xvii

M.R. Djumaev

Contributing Authors

xix

Part I Nonlinear Waves in Random Media

Exponential Localization Versus Soliton Propagation

(16)

J. Garnier

Introduction

(1)

Random linear media

(4)

Solitary waves

(1)

Solitons in random media

(2)

Numerical simulations

(1)

Conclusion

(7)

Appendix: An introduction to the inverse scattering transform

(1)

Appendix: Proof of Proposition 5

(5)

Pulse Confinement in Optical Fibers with Random Dispersion

(10)

M. Chertkov

I. Gabitov

J. Moeser

Introduction

(1)

Basic modelling

(4)

Numerical results

(2)

Conclusion

(3)

Propagation of an Optical Pulse in a Fiber with Random Dispersion - management

(8)

B.A. Malomed

A. Berntston

Introduction

(1)

The model

(3)

Numerical simulations

(2)

Conclusion

(2)

Spatial Solitons in Random Quadratic Media

(18)

A. Kobyakov

F. Kh. Abdullaev

S.A. Darmanyan

E. Schmidt

F. Lederer

Introduction

(3)

Description of the model. Equations for the mean fields

(3)

Analytical model for soliton evolution based on the MFM equations

(2)

Numerical simulations of solitons propagation in random quadratic media

(6)

Conclusions

(4)

Correlation of the Amplitude and Phase Random Modulations in Nonlinear Medium

(6)

M.R. Djumaev

N.K. Nosirova

A.A. Abdumalikov

Spatio-temporal Chaos in Time Evolution of BEC with Attractive Two-body Interactions

(8)

A. Gammal

V.S. Filho

L. Tomio

T. Frederico

Dynamics of a Bose-Einstein Condensate in Optical Trap

(12)

F.Kh. Abdullaev

B.B. Baizakov

V.V. Konotop

Introduction

(1)

Basic equations and the method of analysis

(3)

Numerical modelling

(3)

Conclusion

(5)

Part II Nonlinear Waves in Discrete, Including Disordered, Systems

Aspects of Discrete Breathers and New Directions

(18)

S. Aubry

G. Kopidakis

Introduction

(2)

Existence proofs for discrete breathers

(9)

Perspectives on discrete breathers applications

(7)

Complexity at Mesoscale

(16)

A.R. Bishop

K.O. Rasmussen

J. Roder

T. Lookman

A. Saxena

A. Vanossi

P. Kevrekidis

Introduction

(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)

V.V. Konotop

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)

F. Lederer

S. Darmanyan

A. Kobyakov

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)

S. Takeno

V.V. Konotop

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)

A.V. Ustinov

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)

B. Dey

M. Eleftheriou

G.P. Tsironis

Introduction

189

(2)

Analytic solutions

191

(1)

Numerical analysis

192

(2)

Conclusions

194

(3)

Growth and Decay of Weakly Perturbed Discrete Breathers

197

(8)

M. Johansson

Introduction

197

(1)

Model and results

198

(4)

Summary

202

(3)

Standing Waves in 1D Nonlinear Lattices

205

(8)

A.M. Morgante

M. Johansson

G. Kopidakis

S. Aubry

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)

T. Pertsch

A. Brauer

U. Peschel

F. Lederer

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)

J.G. Caputo

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)

A.B. Ezersky

S.V. Kiyashko

A. V. Nazarovsky

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)

A.M. Kamchatnov

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)

A.I. Maimistov

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)

A.A. Sukhorukov

Yu.S. Kivshar

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)

R.A. Ayukhanov

G. N. Shkerdin

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)

A.M. Basharov

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)

J. Atai

B.A. Malomed

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)

G. Izus

M. Santagiustina

M. San Miguel

P. Colet

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)

S.F. Mingaleev

Yu.S. Kivshar

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

C. Montes

I. Bongrand

E. Picholle

J. Botineau

A. Picozzi

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)

B.A. Umarov

Abdel-Baset M.A. Ibrahim

M.R.B. Wahiddin

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)

M. Wadati

G. Kato

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)

L. Vazquez

M.A. Porras

F. Salazar-Bloise

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)

G. Celebi

I.S. Uzun

M. Pehlivan

M.H. Asyali

A. Turkoglu

I. Soydan

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)

V. Damgov

P. Trenchev

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)

M.R. Djumaev

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)

A.S. Kirakosyan

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)

M. Neviere

S. A. Darmanyan

Accelerating Solitons and Photorefractivity

435

(8)

D.F. Parker

M. Facao

Ch. Radha

The accelerating similarity variable

435

(3)

Photorefractive accelerating solitons

438

(2)

Numerical procedure and results

440

(3)

Index

443

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