Long time motion of NLS solitary waves in a confining potential

TL;DR

This paper analyzes the long-term dynamics of solitary waves in a nonlinear Schrödinger equation with a slowly varying external potential, showing their motion closely follows classical Newtonian trajectories.

Contribution

It introduces a Lyapunov-Schmidt decomposition and energy estimates to control solitary wave motion over extended periods in a confining potential.

Findings

Solitary wave centers follow Newtonian trajectories in the potential.

Motion control is achieved over long, finite time intervals.

The approach combines decomposition and energy estimates for analysis.

Abstract

We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schroedinger equations with a confining, slowly varying external potential, $V(x)$. A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval. We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential $V(x)$ over a long time interval.