Ground state (in-)stability and long-time behavior in multi-dimensional Schrödinger equations

Christof Sparber (Université de l’Illinois à Chicago)

We consider Schrödinger equations with competing nonlinearities in spatial dimensions up to three, for which global existence holds (i.e. for which no finite-time blow-up occurs). A typical example is the case of the (focusing-defocusing) cubic-quintic nonlinear Schrödinger equation. We recall the notions of energy minimizing and of action minimizing ground states and show that, in general, they are nonequivalent. The question of long-time behavior of solutions, in particular the problem of ground-state (in-)stability will be discussed using analytical results and numerical simulations.