Effective noise theory for the Nonlinear Schr\"odinger Equation with disorder

TL;DR

This paper investigates the effective noise in the nonlinear Schrödinger equation with disorder, confirming previous assumptions and exploring how it influences localization and subdiffusion phenomena.

Contribution

It provides a numerical study of the properties of the effective noise, verifying earlier assumptions and analyzing its dependence on localization length.

Findings

Effective noise properties are numerically characterized.

Dependence of noise on localization length is established.

A scenario for long-term breakdown of the theory is proposed.

Abstract

For the Nonlinear Shr\"odinger Equation with disorder it was found numerically that in some regime of the parameters Anderson localization is destroyed and subdiffusion takes place for a long time interval. It was argued that the nonlinear term acts as random noise. In the present work the properties of this effective noise are studied numerically. Some assumptions made in earlier work were verified, the dependence of various quantities on the localization length of the linear problem were computed. A scenario for the possible breakdown of the theory for a very long time is outlined.