Long time Anderson localization for nonlinear random Schroedinger equation

TL;DR

This paper proves long-term Anderson localization for a nonlinear random Schrödinger equation by using a Birkhoff normal form transform to create an energy barrier, allowing for the treatment of rough data without moment conditions.

Contribution

It introduces a novel Birkhoff normal form approach in a small neighborhood to establish localization for nonlinear equations with rough initial data.

Findings

Long time Anderson localization is achieved for nonlinear random Schrödinger equations.

The method handles rough data without moment conditions.

The approach is inspired by the RAGE theorem.

Abstract

We prove long time Anderson localization for nonlinear random Schroedinger equation in $\ell^2$ by making a Birkoff normal form type transform to creat an energy barrier where there is essentially no mode propagation. One of the new features is that this transform is in a small neighborhood enabling us to treat "rough" data, where there are no moment conditions. The formulation of the present result is inspired by the RAGE theorem.