Localization in Periodic Potentials
From Schrödinger Operators to the Gross–Pitaevskii Equation

Describes modern methods in the analysis of reduced models of Bose–Einstein condensation in periodic lattices.

Dmitry E. Pelinovsky (Author)

9781107621541, Cambridge University Press

Paperback / softback, published 6 October 2011

407 pages, 35 b/w illus. 165 exercises
22.8 x 15.3 x 2 cm, 0.58 kg

"The book brilliantly harnesses powerful techniques, teaches them "on-the-job" and illustrates them with a profound and beautiful analysis of these equations, unreally real as suggested by one slogan of Chapter 2, a quote by Einstein, :As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do no refer to reality."
Emma Previato, Mathematics Reviews

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.

Preface
1. Formalism of the nonlinear Schrödinger equations
2. Justification of the nonlinear Schrödinger equations
3. Existence of localized modes in periodic potentials
4. Stability of localized modes
5. Traveling localized modes in lattices
Appendix A. Mathematical notations
Appendix B. Selected topics of applied analysis
References
Index.

Subject Areas: Applied mathematics [PBW], Differential calculus & equations [PBKJ]