Asymptotic stability of multi-soliton solutions for nonlinear Schroedinger eqations

TL;DR

This paper proves that solutions to the nonlinear Schrödinger equation with initial data near multiple solitons asymptotically resemble a sum of slightly modified solitons plus a small dispersive component over large times.

Contribution

It establishes the asymptotic stability of multi-soliton solutions under spectral assumptions, extending understanding of long-term behavior in nonlinear Schrödinger equations.

Findings

Solutions approach a sum of modified solitons asymptotically

Dispersive term remains small over time

Spectral conditions ensure stability of multi-soliton configurations

Abstract

We consider the Cauchy problem for the nonlinear Schroedinger eqiation with initial data close to a sum of N decoupled solitons. Under some suitable assumptions on the spectral structure of the one soliton linearizations we prove that for large time the asymptotics of the solution is given by a sum of solitons with slightly modified parameters and a small dispersive term.