Asymptotic localization of stationary states in the nonlinear Schroedinger equation

TL;DR

This paper analytically investigates the asymptotic localization properties of stationary states in the nonlinear Schrödinger equation with randomness, linking it to the linear case and calculating the localization length.

Contribution

It introduces a novel analytical approach to determine the localization length of stationary states in the nonlinear Schrödinger equation with a random potential.

Findings

Derived the localization length for nonlinear Schrödinger stationary states.

Established the asymptotic growth rates of wave function moments.

Connected nonlinear localization properties to the linear case through resummation.

Abstract

The mapping of the Nonlinear Schroedinger Equation with a random potential on the Fokker-Planck equation is used to calculate the localization length of its stationary states. The asymptotic growth rates of the moments of the wave function and its derivative for the linear Schroedinger Equation in a random potential are computed analytically and resummation is used to obtain the corresponding growth rate for the nonlinear Schroedinger equation and the localization length of the stationary states.