Nonlinear stability of 2-solitons of the Sine-Gordon equation in the energy space

TL;DR

This paper proves the orbital stability of 2-soliton solutions of the sine-Gordon equation in the energy space, using Bäcklund transformations and extending previous methods to a vector-valued context.

Contribution

It provides the first rigorous proof of the stability of 2-solitons of the sine-Gordon equation in the energy space, employing Bäcklund transformations and addressing vector-valued challenges.

Findings

2-soliton solutions are orbitally stable in the energy space

Bäcklund transformations reduce stability analysis to the vacuum case

Extends stability results to vector-valued sine-Gordon solutions

Abstract

In this article we prove that 2-soliton solutions of the sine-Gordon equation (SG) are orbitally stable in the natural energy space of the problem. The solutions that we study are the {\it 2-kink, kink-antikink and breather} of SG. In order to prove this result, we will use B\"acklund transformations implemented by the Implicit Function Theorem. These transformations will allow us to reduce the stability of the three solutions to the case of the vacuum solution, in the spirit of previous results by Alejo and the first author, which was done for the case of the scalar modified Korteweg-de Vries equation. However, we will see that SG presents several difficulties because of its vector valued character. Our results improve those in Alejo et al., and give a first rigorous proof of the stability in the energy space of SG 2-solitons.