The Localization Length of Stationary States in the Nonlinear Schreodinger Equation

TL;DR

This paper analytically demonstrates that in the nonlinear Schrödinger equation with disorder, the localization length of stationary states remains unaffected by nonlinearity and matches the linear case, using stochastic analysis methods.

Contribution

It provides an analytical proof that the localization length in nonlinear Schrödinger equations with disorder is independent of nonlinearity strength, extending understanding of localization phenomena.

Findings

Localization length is independent of nonlinearity strength.

Analytical expression for eigenstate growth in disordered NLSE.

Main conclusions are robust across different conditions.

Abstract

For the nonlinear Schreodinger equation (NLSE), in presence of disorder, exponentially localized stationary states are found. In the present Letter it is demonstrated analytically that the localization length is typically independent of the strength of the nonlinearity and is identical to the one found for the corresponding linear equation. The analysis makes use of the correspondence between the stationary NLSE and the Langevin equation as well as of the resulting Fokker-Planck equation. The calculations are performed for the ``white noise'' random potential and an exact expression for the exponential growth of the eigenstates is obtained analytically. It is argued that the main conclusions are robust.