Solitonization of the Anderson Localization

TL;DR

This paper explores the relationship between bright solitons and Anderson localization in a disordered nonlinear system, deriving explicit expressions and analyzing stability to deepen understanding of disorder-induced localization phenomena.

Contribution

It introduces explicit formulas for nonlinear eigenvalues and localization lengths in disordered nonlinear Schrödinger systems, combining perturbation theory, variational methods, and numerical stability analysis.

Findings

Localization profiles follow a nonlocal nonlinear Schrödinger equation

Explicit expressions for eigenvalues and localization lengths are derived

Numerical stability analysis reveals superlocalizations

Abstract

We study the affinities between the shape of the bright soliton of the one-dimensional nonlinear Schroedinger equation and that of the disorder induced localization in the presence of a Gaussian random potential. With emphasis on the focusing nonlinearity, we consider the bound states of the nonlinear Schroedinger equation with a random potential; for the state exhibiting the highest degree of localization, we derive explicit expressions for the nonlinear eigenvalue and for the localization length by using perturbation theory and a variational approach following the methods of statistical mechanics of disordered systems. We numerically investigate the linear stability and "superlocalizations". The profile of the disorder averaged Anderson localization is found to obey a nonlocal nonlinear Schroedinger equation