Localization in Periodic Potentials

by Pelinovsky, Dmitry E.

• ISBN13: 9781107621541
• ISBN10: 1107621542
• Format: Paperback
• Copyright: 2011-11-28
• Publisher: Cambridge Univ Pr

Summary

This book provides a comprehensive treatment of the Gross-Pitaevskii equation with a periodic potential; in particular, the localized modes supported by the periodic potential. It takes the mean-field model of the Bose-Einstein condensation as the starting point of analysis and addresses the existence and stability of localized modes. The mean-field model is simplified further to the coupled nonlinear Schrödinger equations, the nonlinear Dirac equations, and the discrete nonlinear Schrödinger equations. One of the important features of such systems is the existence of band gaps in the wave transmission spectra, which support stationary localized modes known as the gap solitons. These localized modes realise a balance between periodicity, dispersion and nonlinearity of the physical system. Written for researchers in applied mathematics, this book mainly focuses on the mathematical properties of the Gross-Pitaevskii equation. It also serves as a reference for theoretical physicists interested in localization in periodic potentials.

Table of Contents

Preface	p. ix
Formalism of the nonlinear Schrodinger equations	p. 1
Asymptotic multi-scale expansion methods	p. 3
Hamiltonian structure and conserved quantities	p. 15
Well-posedness and blow-up	p. 26
Integrable equations and solitons	p. 37
Justification of the nonlinear Schrodinger equations	p. 51
Schrodinger operators with periodic potentials	p. 52
Justification of the nonlinear Dirac equations	p. 73
Justification of the nonlinear Schrodinger equation	p. 88
Justification of the DNLS equation	p. 105
Existence of localized modes in periodic potentials	p. 118
Variational methods	p. 119
Lyapunov-Schmidt reductions	p. 131
Other analytical methods	p. 159
Numerical methods	p. 173
Stability of localized modes	p. 193
Schrodinger operators with decaying potentials	p. 198
Unstable and stable eigenvalues in the spectral stability problem	p. 215
Spectral stability of localized modes	p. 229
Other methods in stability analysis	p. 266
Traveling localized modes in lattices	p. 291
Differential advance-delay operators	p. 294
Integrable discretizations for stationary localized modes	p. 301
Melnikov integral for traveling localized modes	p. 311
Normal forms for traveling localized modes	p. 326
Stokes constants for traveling localized modes	p. 335
Traveling localized modes in periodic potentials	p. 346
Mathematical notation	p. 361
Selected topics of applied analysis	p. 365
Banach algebra	p. 365
Banach Fixed-Point Theorem	p. 366
Floquet Theorem	p. 367
Fredholm Alternative Theorem	p. 369
Gagliardo-Nirenberg inequality	p. 371
Gronwall inequality	p. 371
Implicit Function Theorem	p. 372
Mountain Pass Theorem	p. 374
Noether Theorem	p. 375
Sobolev Embedding Theorem	p. 376
Spectral Theorem	p. 377
Sturm Comparison Theorem	p. 378
Unstable Manifold Theorem	p. 379
Weak Convergence Theorem	p. 380
Weyl Theorem	p. 381
Wiener algebra	p. 384
References	p. 385
Index	p. 395
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