Standing waves of the Anderson-Gross-Pitaevskii equation

TL;DR

This paper investigates the existence, regularity, localization, and stability of standing wave solutions for the one- and two-dimensional Anderson-Gross-Pitaevskii equation, a nonlinear Schrödinger equation with random potential.

Contribution

It introduces a variational approach to construct standing wave solutions for the stochastic nonlinear Schrödinger equation with white noise potential in low dimensions.

Findings

Existence of standing wave solutions in 1D and 2D.

Results on regularity and localization of solutions.

Analysis of stability properties.

Abstract

In this paper, we study standing waves for the Anderson-Gross-Pitaevskii equation in dimension 1 and 2. The Anderson-Gross-Pitaevskii equation is a nonlinear Schr\"odinger equation with a confining potential and a multiplicative spatial white noise. Standing waves are characterized by a profile which is invariant by the dynamic and solves a nonlinear elliptic equation with spatial white noise potential. We construct such solutions via variational methods and obtain some results on their regularity, localization and stability.