Numerical computation of soliton dynamics for NLS equations in a driving potential

TL;DR

This paper numerically investigates soliton dynamics in the nonlinear Schrödinger equation with an external potential, demonstrating that the soliton's center of mass follows a Newtonian law in the semi-classical regime and analyzing numerical errors.

Contribution

It introduces a numerical approach to study soliton dynamics in NLS equations with external potentials, linking the center of mass motion to Newtonian laws and analyzing errors in 2D harmonic potentials.

Findings

Center of mass follows Newtonian dynamics in semi-classical regime

Numerical errors are analyzed in 2D harmonic potential case

Ground state solutions are computed for elliptic equations

Abstract

We provide some numerical computations for the soliton dynamics of the nonlinear Schr\"odinger equation with an external potential. After computing the ground state solution $r$ of a related elliptic equation we show that, in the semi-classical regime, the center of mass of the solution with initial datum modelled on $r$ is driven by the solution of a Newtonian type law. Finally, we provide some examples and analyze the numerical errors in the two dimensional case when $V$ is an harmonic potential.