Freezing of energy of a soliton in an external potential

TL;DR

This paper analyzes the dynamics of solitons in a generalized nonlinear Schrödinger equation with a small external potential, showing that their energy remains nearly conserved over long times and their motion closely follows an effective mechanical system.

Contribution

It establishes an effective mechanical description for soliton dynamics and proves near conservation of energy over long timescales in the presence of a small external potential.

Findings

Energy of the soliton is almost conserved up to times of order psilon^{-r}

Soliton orbit remains close to the mechanical system's trajectory in rotationally invariant cases

Provides a rigorous link between soliton dynamics and classical mechanics in perturbed systems

Abstract

In this paper we study the dynamics of a soliton in the generalized NLS with a small external potential $\epsilon V$ of Schwartz class. We prove that there exists an effective mechanical system describing the dynamics of the soliton and that, for any positive integer $r$, the energy of such a mechanical system is almost conserved up to times of order $\epsilon^{-r}$. In the rotational invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order $\epsilon^{-r}$.