Nonlinear dynamics of semiclassical coherent states in periodic potentials

TL;DR

This paper studies the behavior of semiclassical coherent states in nonlinear Schrödinger equations with periodic potentials, deriving effective models and novel averaging results for nonlocal nonlinearities.

Contribution

It constructs asymptotic solutions for nonlinear Schrödinger equations in periodic potentials and establishes a new averaging result for nonlocal nonlinearities in the critical case.

Findings

Asymptotic solutions are concentrated in phase space around the semiclassical flow.

Effective mass models govern the dynamics of generalized coherent states.

A novel averaging result is established for nonlocal nonlinearities in the critical regime.

Abstract

We consider nonlinear Schrodinger equations with either local or nonlocal nonlinearities. In addition, we include periodic potentials as used, for example, in matter wave experiments in optical lattices. By considering the corresponding semiclassical scaling regime, we construct asymptotic solutions, which are concentrated both in space and in frequency around the effective semiclassical phase-space flow induced by Bloch's spectral problem. The dynamics of these generalized coherent states is governed by a nonlinear Schrodinger model with effective mass. In the case of nonlocal nonlinearities we establish a novel averaging type result in the critical case.