Semiclassical asymptotics for weakly nonlinear Bloch waves

TL;DR

This paper develops a rigorous two-scale WKB analysis to understand semiclassical and adiabatic asymptotics in weakly nonlinear Schrödinger equations with periodic potentials, revealing a modification of the Berry phase.

Contribution

It introduces a novel two-scale WKB method for nonlinear Schrödinger equations with periodic potentials, highlighting the nonlinear modification of the Berry phase.

Findings

Rigorous derivation of semiclassical asymptotics

Identification of nonlinear Berry phase modification

Application to weakly nonlinear periodic Schrödinger equations

Abstract

We study the simultaneous semi-classical and adiabatic asymptotics for a class of (weakly) nonlinear Schroedinger equations with a fast periodic potential and a slowly varying confinement potential. A rigorous two-scale WKB-analysis, locally in time, is performed. The main nonlinear phenomenon is a modification of the Berry phase.