Modul: MAT675 PDE and Mathematical Physics

## Chaotic orbits for nonlocal equations and applications to atom dislocation dynamics in crystals

Enrico Valdinoci talk

Speaker invited by: Prof. Dr. Xavier Ros-Oton

**Date:** 11.10.18 **Time:** 15.00 - 16.00 **Room:** Y27H46

In this talk we consider a nonlocal equation driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinic, homoclinic and chaotic trajectories are constructed. This result regarding symbolic dynamics in a fractional framework is part of a study of Peierls-Nabarro model for crystal dislocations. The associated evolution equation can be studied in the mesoscopic and macroscopic limits. Namely, the dislocation function has 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 can be either repulsive or attractive, depending on the relative orientations of the dislocations. For opposite orientations, collisions occur, after which the system relaxes exponentially fast.