The Nonlinear Schroedinger equation: solitons dynamics

TL;DR

This paper studies the behavior of solitons in the nonlinear Schrödinger equation as a parameter approaches zero, showing they follow particle-like dynamics with diminishing error.

Contribution

It proves that solitons approximately follow a point particle dynamics in the small parameter limit, under certain assumptions.

Findings

Solitons follow a Newtonian-like equation of motion with small error term.

The error term H_h(t) tends to zero uniformly as h approaches zero.

The analysis provides a rigorous link between soliton dynamics and classical particle motion.

Abstract

In this paper we investigate the dynamics of solitons occurring in the nonlinear Schroedinger equation when a parameter h->0. We prove that under suitable assumptions, the the soliton approximately follows the dynamics of a point particle, namely, the motion of its barycenter $q_h(t)$ satisfies the equation $ddot{q}_h(t)+\nabla V(q_h(t))=H_h(t)$ where $\sup_{t\in R} |H_h(t)| -> 0$ as h->0.