Solitary wave dynamics in time-dependent potentials

We rigorously study the long time dynamics of solitary wave solutions of the nonlinear Schrödinger equation in time-dependent external potentials. To set the stage, we first establish the well-posedness of the Cauchy problem for a generalized nonautonomous nonlinear Schrödinger equation. We then show that when 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.