## Long time dynamics of highly concentrated solitary waves for the nonlinear Schrödinger equation.(English)Zbl 1304.35632

Summary: 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.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations 35C08 Soliton solutions
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### References:

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