Asymptotic stability of the sine-Gordon kink

TL;DR

This paper proves the full asymptotic stability of the sine-Gordon kink under small perturbations, using advanced Fourier and modulation techniques to handle slow decay and resonance effects.

Contribution

It introduces a novel space-time resonances approach combined with modulation methods to establish stability of moving solitons in scalar field theories.

Findings

Proves asymptotic stability of sine-Gordon kink

Develops a systematic distorted Fourier theory for solitons

Handles slow decay due to threshold resonances

Abstract

We establish the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. Our proof consists of a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects combined with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the moving kink. Our analysis crucially relies on two remarkable null structures in the quadratic nonlinearities of the evolution equation for the radiation term and of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is not specific to the sine-Gordon model and extends to many other asymptotic stability problems for moving solitons in relativistic scalar field theories on the line.