## Long time motion of NLS solitary waves in a confining potential.(English)Zbl 1100.81019

Authors’ summary: We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schrödinger equations with a confining, slowly varying external potential, $$V(x)$$.
A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval.
We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential $$V(x)$$ over a long time interval.

### MSC:

 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35Q55 NLS equations (nonlinear Schrödinger equations)
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