Solitary Wave Dynamics in an External Potential

TL;DR

This paper investigates how solitary waves in generalized nonlinear Schrödinger equations behave under external potentials, showing their motion closely follows classical Newtonian dynamics with minor radiation effects.

Contribution

It constructs solutions near solitary waves in external potentials and proves their center of mass follows Newton's laws with small radiation damping effects.

Findings

Solitary wave solutions approximate classical particle trajectories.

External potential influences the solitary wave's center of mass motion.

Radiation damping causes small deviations from ideal Newtonian motion.

Abstract

We study the behavior of solitary-wave solutions of some generalized nonlinear Schr\"odinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We construct solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton's equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.