Dynamics of wave packets for the nonlinear Schroedinger equation with a random potential

TL;DR

This paper investigates the evolution of localized wave packets in a disordered nonlinear Schrödinger system, revealing subdiffusive behavior governed by a fractional Fokker-Planck equation through analytical and probabilistic methods.

Contribution

It provides an analytical perturbative solution for the initial dynamics and introduces a probabilistic framework describing long-term subdiffusive behavior.

Findings

Localized wave functions remain localized initially

Wave packet dynamics exhibit subdiffusion at large times

Fractional Fokker-Planck equation models the probability distribution

Abstract

The dynamics of an initially localized Anderson mode is studied in the framework of the nonlinear Schroedinger equation in the presence of disorder. It is shown that the dynamics can be described in the framework of the Liouville operator. An analytical expression for a wave function of the initial time dynamics is found by a perturbation approach. As follows from a perturbative solution the initially localized wave function remains localized. At asymptotically large times the dynamics can be described qualitatively in the framework of a phenomenological probabilistic approach by means of a probability distribution function. It is shown that the probability distribution function may be governed by the fractional Fokker-Planck equation and corresponds to subdiffusion.