Effective Dynamics of Solitons in the Presence of Rough Nonlinear Perturbations

TL;DR

This paper rigorously analyzes how solitary wave solutions of the nonlinear Schrödinger equation evolve over long times when subjected to rough nonlinear perturbations, showing they remain close to a modulated soliton governed by an effective potential.

Contribution

It provides a rigorous long-time analysis of soliton dynamics under rough nonlinear perturbations, linking the soliton's center of mass to an effective potential.

Findings

Solutions stay close to a modulated soliton over long times.

The soliton's center of mass follows dynamics dictated by an effective potential.

The analysis applies to initial states near a traveling soliton in $H^1$ norm.

Abstract

The effective long-time dynamics of solitary wave solutions of the nonlinear Schr\"odinger equation in the presence of rough nonlinear perturbations is rigorously studied. It is shown that, if the initial state is close to a slowly travelling soliton of the unperturbed NLS equation (in $H^1$ norm), then, over a long time scale, the true solution of the initial value problem will be close to a soliton whose center of mass dynamics is approximately determined by an effective potential that corresponds to the restriction of the nonlinear perturbation to the soliton manifold.