Localization in Periodic Potentials

Localization in Periodic Potentials : From Schrodinger Operators to the Gross-Pitaevskii Equation

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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 Schrodinger equations, the nonlinear Dirac equations, and the discrete nonlinear Schrodinger 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.show more

Product details

  • Electronic book text | 416 pages
  • Cambridge University Press (Virtual Publishing)
  • Cambridge, United Kingdom
  • English
  • 35 b/w illus. 165 exercises
  • 1139154028
  • 9781139154024

Table of contents

Preface; 1. Formalism of the nonlinear Schrodinger equations; 2. Justification of the nonlinear Schrodinger 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.show more

Review quote

"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 Reviewsshow more

About Dmitry E. Pelinovsky

Dmitry E. Pelinovsky is a Professor in the Department of Mathematics at McMaster University, Canada.show more