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In this paper we study the behavior of solutions of a nonlinear Schroedingerequation in presence of an external potential, which is allowed to be singularat one point. We show that the solution behaves like a solitary wave for longtime even if we start from a unstable solitary wave, and its dynamics coincidewith that of a classical particle evolving according to a natural effectiveHamiltonian.

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