Colliding solitons for the nonlinear Schrodinger equation

TL;DR

This paper investigates the collision dynamics of two fast-moving solitons in the nonlinear Schrödinger equation with an external potential, showing they largely retain their shape shortly after collision and their motion is influenced by the potential.

Contribution

It provides a detailed analysis of soliton collisions in the nonlinear Schrödinger equation with external potential, including estimates on shape preservation and center of mass dynamics.

Findings

Solitons preserve their shape up to logarithmic time scales after collision.

The centers of mass follow dynamics influenced by the external potential.

Error terms include radiation damping and soliton extension effects.

Abstract

We study the collision of two fast solitons for the nonlinear Schr\"odinger equation in the presence of a spatially adiabatic external potential. For a high initial relative speed $\|v\|$ of the solitons, we show that, up to times of order $\log\|v\|$ after the collision, the solitons preserve their shape (in $L^2$-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.