## 2008 DG-MP-PDE Seminar-08

- Date: 03/18/2008

University of British Columbia

Asymptotic stability of ground states in 3D subcritical nonlinear Schroedinger equation

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.

3:30pm, WMAX 110