Asymptotic stability of the sine-Gordon kink under odd perturbations

TL;DR

This paper proves the asymptotic stability of the sine-Gordon kink under small odd perturbations using a perturbative approach that does not depend on the model's complete integrability, with implications for other non-integrable models.

Contribution

It introduces a novel perturbative method for stability analysis that leverages a specific linearized operator factorization and a quadratic normal form, applicable beyond integrable systems.

Findings

Proves asymptotic stability of the sine-Gordon kink under odd perturbations.

Develops a perturbative approach not relying on integrability.

Extends techniques to other non-integrable models like φ^4 and Klein-Gordon equations.

Abstract

We establish the asymptotic stability of the sine-Gordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key elements of our proof are a specific factorization property of the linearized operator around the sine-Gordon kink, a remarkable non-resonance property exhibited by the quadratic nonlinearity in the Klein-Gordon equation for the perturbation, and a variable coefficient quadratic normal form introduced in [53]. We emphasize that the restriction to odd perturbations does not bypass the effects of the odd threshold resonance of the linearized operator. Our techniques have applications to soliton stability questions for several well-known non-integrable models, for instance, to the asymptotic stability problem for the kink of the $\phi^4$ model as well as to the conditional asymptotic stability problem for the solitons of the focusing quadratic and cubic Klein-Gordon equations in one space dimension.