Global solutions and stable ground states on nonlinear Schroedinger equations

I present a new method to obtain sufficient conditions for the global existence of solutions of nonlinear Schroedinger equations with focusing nonlinearities. It turns out that the regime of global solutions is essentially determined by the ground state solutions of these equations. In particular, I can show that the orbital stability of certain ground states implies the global existence of all solutions. Furthermore sharp conditions for the global existence of sufficiently small solutions are derived. (orig.)