A sufficient condition for asymptotic stability of kinks in general (1+1)-scalar field models

TL;DR

This paper establishes a simple, explicit criterion on the potential function W that guarantees the asymptotic stability of kinks in (1+1)-dimensional scalar field models, applicable to various theories without symmetry restrictions.

Contribution

It provides the first explicit sufficient condition for asymptotic stability of kinks in general scalar field models, extending stability analysis beyond symmetric cases.

Findings

Derived a simple criterion for asymptotic stability of kinks.

Applied the criterion to P(φ)_2 and double sine-Gordon theories.

Confirmed the criterion's applicability to static and moving kinks.

Abstract

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models \begin{equation*} \partial_t^2\phi -\partial_x^2\phi + W'(\phi) = 0, \quad (t,x)\in\mathbb{R}\times\mathbb{R}. \end{equation*} The orbital stability of kinks under general assumptions on the potential $W$ is a consequence of energy arguments. Our main result is the derivation of a simple and explicit sufficient condition on the potential $W$ for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Last, motivated by the Physics literature, we present applications of the criterion to the $P(\phi)_2$ theories and the double sine-Gordon theory.