Crystal Dislocations with Different Orientations and Collisions

We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superposition of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time. More precisely, we consider the evolutionary equation(Formula Presented.) is the atom dislocation function at time t >  0 at the point (Formula Presented.),Is is an integro-differential operator of order (Formula Presented.) is a small parameter that takes into account the small periodicity scale of the crystal. We suppose that (Formula Presented.) is the superposition of N−K transition layers in the positive direction and K in the negative one (Formula Presented.), depending on the orientation of the transition layer u, which in turn solves the stationary equation (Formula Presented.). We show that our problem possesses a unique solution and that, as (Formula Presented.), it approaches the sum of Heaviside functions H with different orientations centered at points x_i(t), (Formula Presented.).The point x_i evolves in time from (Formula Presented.), being subject to the external stress and a singular potential, which may be either attractive or repulsive, according to the different orientation of the transitions; more precisely, the speed (Formula Presented.).The evolution of such a dynamical system may lead to collisions in finite time. We give a detailed description of such collisions when N = 2, 3 and we show that the solution itself keeps track of such collisions; indeed, at the collision time T_c the two opposite dislocations have the tendency to annihilate each other and make the dislocation vanish, but only outside the collision point x_c, according to the (Formula Presented.). We also study some specific cases of N dislocation layers, namely when two dislocations are initially very close and when the dislocations are alternate. To the best of our knowledge, the results obtained are new even in the model case s = 1/2.