Solitary wave dynamics in time-dependent potentials

Details Solitary wave dynamics in time-dependent potentials Solitary wave dynamics in time-dependent potentials

2007-07-02

arxiv.org/abs/0707.0272v2

Physics

Mathematical Physics

38 pages, some typos corrected, one reference added, one remark added

Scientific paper

10.1063/1.2837429

We rigorously study the long time dynamics of solitary wave solutions of the nonlinear Schr\"odinger equation in {\it time-dependent} external potentials. To set the stage, we first establish the well-posedness of the Cauchy problem for a generalized nonautonomous nonlinear Schr\"odinger equation. We then show that in the {\it space-adiabatic} regime where the external potential varies slowly in space compared to the size of the soliton, the dynamics of the center of the soliton is described by Hamilton's equations, plus terms due to radiation damping. We finally remark on two physical applications of our analysis. The first is adiabatic transportation of solitons, and the second is Mathieu instability of trapped solitons due to time-periodic perturbations.

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