Stochastic acceleration of solitons for the nonlinear Schr\"odinger equation

TL;DR

This paper rigorously analyzes the stochastic acceleration of solitons in the nonlinear Schrödinger equation with a random potential, showing their dynamics approximate classical Hamiltonian motion and converge to diffusion or Brownian motion under certain limits.

Contribution

It provides a rigorous derivation of soliton dynamics in random potentials, including limit theorems showing convergence to stochastic processes in higher dimensions.

Findings

Soliton center dynamics follow Hamilton's equations with radiation damping.

Center of mass momentum converges to a diffusion process.

Trajectory converges to spatial Brownian motion in higher dimensions.

Abstract

The effective dynamics of solitons for the generalized nonlinear Schr\"odinger equation in a random potential is rigorously studied. It is shown that when the external potential varies slowly in space compared to the size of the soliton, the dynamics of the center of the soliton is almost surely described by Hamilton's equations for a classical particle in the random potential, plus error terms due to radiation damping. Furthermore, a limit theorem for the dynamics of the center of mass of the soliton in the weak-coupling and space-adiabatic limit is proven in two and higher dimensions: Under certain mixing hypotheses for the potential, the momentum of the center of mass of the soliton converges in law to a diffusion process on a sphere of constant momentum. Moreover, in three and higher dimensions, the trajectory of the center of mass of the soliton converges to a spatial Brownian motion.