Localization in Periodic Potentials: From Schr?dinger Operators to the GrossPit [Paperback]

Describes modern methods in the analysis of reduced models of BoseEinstein condensation in periodic lattices.This comprehensive book describes modern methods in the analysis of reduced models of BoseEinstein condensation in periodic lattices. Aimed at researchers and graduate students working in applied mathematics and physical sciences where nonlinear waves arise, its unique focus is on localized nonlinear waves in periodic potentials and lattices.This comprehensive book describes modern methods in the analysis of reduced models of BoseEinstein condensation in periodic lattices. Aimed at researchers and graduate students working in applied mathematics and physical sciences where nonlinear waves arise, its unique focus is on localized nonlinear waves in periodic potentials and lattices.This book provides a comprehensive treatment of the GrossPitaevskii equation with a periodic potential; in particular, the localized modes supported by the periodic potential. It takes the mean-field model of the BoseEinstein 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 Schr?dinger equations, the nonlinear Dirac equations, and the discrete nonlinear Schr?dinger 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 GrossPitaevskii equation. It also serves as a reference for theoretical physicists interested in localization in periodic potentials.Preface; 1. Formalism of the nonlinear Schr?dinger equations; 2. Justification of the nonlinear Schr?dinger equations; 3. Existence of localized modes in pl³I