On the nonlinear Schrödinger equation outside an obstacle

In this talk, we consider the focusing nonlinear Schrödinger equation (NLS) in the exterior of a compact and strictly convex obstacle, with Dirichlet boundary conditions. We study the asymptotic behavior of the solution for large times and finite time. We prove the existence of these types of solutions: solitary wave solutions (solitons), blow-up solutions (solutions with finite time of existence) and scattering solutions (global and behaving asymptotically as linear solutions), for the NLS equation in the exterior of a convex obstacle.