On the Dynamics of solitons in the nonlinear Schroedinger equation

TL;DR

This paper investigates the behavior of soliton solutions in a nonlinear Schrödinger equation with singular nonlinearities, demonstrating that as a parameter approaches zero, the soliton dynamics resemble classical particle motion.

Contribution

It introduces a novel analysis of soliton dynamics under strong nonlinearities with singular limits, linking quantum soliton behavior to classical particle trajectories.

Findings

Solitons exist and maintain shape under certain conditions.

As the nonlinearity parameter approaches zero, soliton orbits converge to classical particle paths.

The study provides stability results for solitons in the singular limit.

Abstract

We study the behavior of the soliton solutions of the equation i((\partial{\psi})/(\partialt))=-(1/(2m)){\Delta}{\psi}+(1/2)W_{{\epsilon}}'({\psi})+V(x){\psi} where W_{{\epsilon}}' is a suitable nonlinear term which is singular for {\epsilon}=0. We use the "strong" nonlinearity to obtain results on existence, shape, stability and dynamics of the soliton. The main result of this paper (Theorem 1) shows that for {\epsilon}\to0 the orbit of our soliton approaches the orbit of a classical particle in a potential V(x).