Stability of the sum of two solitary waves for (gDNLS) in the energy space

TL;DR

This paper proves the stability of the sum of two solitary waves with weak interactions for the generalized derivative Schrödinger equation in the energy space, extending previous work to a more challenging non-invariant setting.

Contribution

It demonstrates the stability of two solitary waves in the energy space for gDNLS without Galilean, pseudo-conformal, or gauge invariances, covering the supercritical case.

Findings

Stability of the sum of two solitary waves established.

Applicable to the energy space for gDNLS.

Addresses the supercritical case with $\sigma>1$.

Abstract

In this paper, we continue the study in \cite{MiaoTX:DNLS:Stab}. We use the perturbation argument, modulational analysis and the energy argument in \cite{MartelMT:Stab:gKdV, MartelMT:Stab:NLS} to show the stability of the sum of two solitary waves with weak interactions for the generalized derivative Schr\"{o}dinger equation (gDNLS) in the energy space. Here (gDNLS) hasn't the Galilean transformation invariance, the pseudo-conformal invariance and the gauge transformation invariance, and the case $\sigma>1$ we considered corresponds to the $L^2$-supercritical case.