Orbital Stability Of A Sum Of Solitons And Breathers Of The Modified Korteweg-de Vries Equation

TL;DR

This paper proves the orbital stability of a combination of solitons and breathers in the modified Korteweg-de Vries equation (mKdV), showing solutions remain close over time if initially close in H^2 norm.

Contribution

It establishes the orbital stability of multi-breathers of mKdV and derives a uniqueness result using inverse scattering methods.

Findings

Orbital stability of multi-solitons and breathers in mKdV in H^2.

Solutions stay close to initial configurations if initially close.

Uniqueness of multi-breather solutions is demonstrated.

Abstract

In this article, we prove that a sum of solitons and breathers of the modified Korteweg-de Vries equation (mKdV) is orbitally stable. The orbital stability is shown in H^2. More precisely, we will show that if a solution of (mKdV) is close enough to a sum of solitons and breathers with distinct velocities at t=0 in the H^2 sense, then it stays close to this sum of solitons and breathers, up to space translations for solitons and space or phase translations for breathers. From this, we deduce the orbital stability of a multi-breather of (mKdV), constructed in [49]. As an application of the orbital stability and the formula for multi-breathers obtained by inverse scattering method [51], we deduce a result about uniqueness of multi-breathers.