Asymptotic stability for odd perturbations of the the stationary kink in the variable-speed $\phi^4$ model

TL;DR

This paper extends the asymptotic stability of the kink solution in the $\, ext{phi}^4$ model to cases with variable propagation speeds close to constant, using spectral analysis and a perturbative approach.

Contribution

It demonstrates the existence of kink solutions for variable-speed $\, ext{phi}^4$ models and extends stability results through a perturbative spectral analysis.

Findings

Existence of kink solutions for variable-speed $\, ext{phi}^4$ models.

Extension of asymptotic stability to non-constant speeds.

Spectral analysis of the linearized operator around the variable-speed kink.

Abstract

We consider the $\phi^4$ model in one space dimension with propagation speeds that are small deviations from a constant function. In the constant-speed case, a stationary solution called the kink is known explicitly, and the recent work of Kowalczyk, Martel, and Mu\~noz established the asymptotic stability of the kink with respect to odd perturbations in the natural energy space. We show that a stationary kink solution exists also for our class of non-constant propagation speeds, and extend the asymptotic stability result by taking a perturbative approach to the method of Kowalczyk, Martel, and Mu\~noz. This requires an understanding of the spectrum of the linearization around the variable-speed kink.