Long-time Anderson Localization for the Nonlinear Schrodinger Equation Revisited

TL;DR

This paper confirms that in a 1D nonlinear random Schrödinger equation, the wavefront displacement grows logarithmically over long time scales, supporting long-time Anderson localization.

Contribution

It revisits and confirms the conjecture that wavefront displacement remains logarithmic in time for the nonlinear Schrödinger equation, extending previous results to long time scales.

Findings

Wavefront displacement is logarithmic in time for the 1D nonlinear random Schrödinger equation.

Supports the long-time Anderson localization conjecture.

Extends previous short-time results to long-time dynamics.

Abstract

In this paper, we confirm the conjecture of Wang and Zhang (J. Stat. Phys. 134 (5-6): 953--968, 2009) in a long time scale, i.e., the displacement of the wavefront for $1D$ nonlinear random Schroedinger equation is of logarithmic order in time $|t|$.