Solitary wave dynamics in time-dependent potentials

TL;DR

This paper rigorously analyzes the long-term behavior of solitary waves in nonlinear Schrödinger equations with slowly varying external potentials, revealing their dynamics follow Hamiltonian equations with radiation effects, relevant for physical applications.

Contribution

It establishes well-posedness for a generalized nonautonomous nonlinear Schrödinger equation and derives soliton dynamics in the space-adiabatic regime, including physical applications.

Findings

Soliton center dynamics follow Hamilton's equations with radiation damping.

Well-posedness of the generalized nonautonomous nonlinear Schrödinger equation.

Applications to adiabatic soliton transport and Mathieu instability.

Abstract

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.