Instability of Multi-Solitons for Derivative Nonlinear Schr{\"o}dinger Equations

TL;DR

This paper demonstrates that the instability of a single soliton leads to the instability of multi-solitons in derivative nonlinear Schrödinger equations, extending previous results from classical NLS equations using gauge transformations.

Contribution

It proves the prediction that a single unstable soliton causes multi-soliton instability specifically for derivative nonlinear Schrödinger equations, employing the Côté-Le Coz method with gauge transformation.

Findings

Multi-solitons are unstable if one constituent soliton is unstable.

The method extends stability analysis techniques to derivative NLS equations.

The proof adapts the Côté-Le Coz approach with gauge transformation.

Abstract

In [19] and [26], the authors proved the stability of multi-solitons for derivative nonlinear Schr{\"o}dinger equations. Roughly speaking, sum of finite stable solitons is stable. We predict that if there is one unstable solition then multi-soliton is unstable. This prediction is proved in [7] for classical nonlinear Schr{\"o}dinger equations. In this paper, we proved this prediction for derivative nonlinear Schr{\"o}dinger equations by using the method of C{\^o}te-Le Coz [7] with the help of Gauge transformation.