2008 DG-MP-PDE Seminar-08

The talk will start with an overview of asymptotic stability results for NLS. By definition, the ground states are stable if a solution starting nearby decomposes into a part convergent to a ground state and a part radiating away. For many years the theory covered only the supercritical (large power) nonlinearities mainly because a large power of a small perturbation around a ground state is very small and easily dispersed by the Schroedinger type operator given by the linearization at the ground state. Recently results for critical nonlinearities have been obtained. The second part of the talk will focus on a new apprach which gives asymptotic stability results even for subcritical (low power) nonlinearities. It relies on linearizing the equation along a one parameter family of ground states. By continuously adapting the linearization to the actual evolution of the solution we are able to capture the correct effective potential induced by the nonlinearity into a time dependent Schroedinger type operator. The dispersive estimates we prove for this operator allow us to control the remaining nonlinear terms and obtain the asymptotic stability results.