Long time dynamics of highly concentrated solitary waves for the nonlinear Schrödinger equation

In this paper we study the behavior of solutions of a nonlinear Schrödinger equation in presence of an external potential, which is allowed to be singular at one point. We show that the solution behaves like a solitary wave for long time even if we start from an unstable solitary wave, and its dynamics coincides with that of a classical particle evolving according to a natural effective Hamiltonian.