Long-time asymptotics for evolutionary crystal dislocation models

Abstract

We consider a family of evolution equations that generalize the Peierls-Nabarro model for crystal dislocations. They can be seen as semilinear parabolic reaction-diffusion equations in which the diffusion is regulated by a fractional Laplace operator of order $2s\in (0,2)$ acting in one space dimension and the reaction is determined by a $1$-periodic multi-well potential. We construct solutions of these equations that represent the typical propagation of $N\ge 2$ equally oriented dislocations of size $1$. For large times, the dislocations occur around points that evolve according to a repulsive dynamical system. When $s\in (1/2,1)$, these solutions are shown to be asymptotically stable with respect to odd perturbations.