Sala Seminari (Dip. Matematica).
We consider an equation (or a system of equations) inspired by the Peierls-Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior potential which is either attractive or repulsive, according to the orientation of the dislocations. Collision of dislocations with opposite orientations may occur in finite time, and we study these collisions and the times of relaxation of the system. We also consider a system of stationary equations with a perturbed potential and we construct heteroclinic, homoclinic and multibump orbits, providing an example of symbolic dynamics in a fractional setting.