Multi-soliton dynamics in the nonlinear Schr\"{o}dinger equation

TL;DR

This paper analyzes the long-term behavior of multiple solitons in a nonlinear Schrödinger equation with potential, showing they maintain shape and follow particle-like dynamics under certain conditions.

Contribution

It provides a rigorous proof of multi-soliton stability and describes their dynamics as particle motion influenced by the potential, extending understanding of soliton interactions in variable potentials.

Findings

All solitons maintain their shape over large times.

Soliton dynamics approximate particle motion in the potential.

Solitons' barycenters do not collide during evolution.

Abstract

In this paper, we study the Cauchy problem of the nonlinear Schr\"{o}dinger equation with a nontrival potential $V_\varepsilon(x)$. In particular, we consider the case where the initial data is close to a superposition of $k$ solitons with prescribed phase and location, and investigate the evolution of the Schr\"{o}dinger system. We prove that over a large time interval with the maximum time tending to infinity, all $k$ solitons will maintain the shape, and the solitons dynamics can be regarded as an approximation of $k$ particles moving in $\mathbb{R}^N$ with their accelerations dominated by $\nabla V_\varepsilon$, provided the barycenters of these solitons do not coincide.