The Nonlinear Schroedinger Equation with a random potential: Results and Puzzles

TL;DR

This paper reviews recent results on the nonlinear Schrödinger equation with random potentials, highlighting conflicting findings and proposing future research directions to clarify the interplay between nonlinearity and randomness.

Contribution

It summarizes recent advances and unresolved issues in the study of the NLSE with random potentials, emphasizing the need for targeted research to resolve contradictions.

Findings

Contradictory results in the behavior of NLSE with random potential

Identification of key unresolved problems in the field

Proposal of specific research questions to address outstanding issues

Abstract

The Nonlinear Schroedinger Equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a random (Anderson like) potential has been done at various levels of control: numerical, analytical and rigorous. Yet, this model equation presents us with a highly inconclusive and often contradictory picture. We will describe the main recent results obtained in this field and propose a list of specific problems to focus on, that we hope will enable to resolve these outstanding questions.