Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms

TL;DR

This paper proves the asymptotic stability of sine-Gordon kinks under small weighted Sobolev norm perturbations, using Bäcklund transforms and inverse scattering, and provides an asymptotic formula for the perturbations.

Contribution

It establishes the asymptotic stability of sine-Gordon kinks with new techniques and derives an explicit asymptotic formula for perturbations, extending previous local stability results.

Findings

Proved asymptotic stability of sine-Gordon kinks in weighted Sobolev norms.

Derived an explicit asymptotic formula for perturbations.

Applied inverse scattering and wave packet methods for decay analysis.

Abstract

We study the asymptotic stability of the sine-Gordon kinks under small perturbations in weighted Sobolev norms. Our main tool is the B\"acklund transform which reduces the study of the asymptotic stability of the kinks to the study of the asymptotic decay of solutions near zero. Our results consist of two parts. First, we prove an asymptotic stability result similar to the local results in arXiv:2003.09358 and arXiv:2009.04260. Our assumptions are the same as those in the local result in arXiv:2009.04260. In its proof, we apply a result obtained by the inverse scattering method on the local decay of the solutions with sufficiently small and localized initial data. Moreover, we derive an asymptotic formula for the perturbations, i.e. the difference between solutions and kinks. This result is similar to that in arXiv:2106.09605 and the full asymptotic stability result in arXiv:2009.04260. In its proof, we apply a result obtained by the method of testing by wave packets on the pointwise decay of the solutions with small and localized data.