## Freezing of energy of a soliton in an external potential.(English)Zbl 1342.35320

Summary: In this paper we study the dynamics of a soliton in the generalized NLS with a small external potential $$\epsilon V$$ of Schwartz class. We prove that there exists an effective mechanical system describing the dynamics of the soliton and that, for any positive integer $$r$$, the energy of such a mechanical system is almost conserved up to times of order $$\epsilon^{-r}$$. In the rotational invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order $$\epsilon^{-r}$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35C08 Soliton solutions

### Keywords:

generalized nonlinear Schrödinger equation; soliton
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### References:

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