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

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.