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.

Additional Information

Eduard Kirr (University of Illinois at Urbana-Champaign)