Solitary wave dynamics in time-dependent potentials

Abstract

We rigorously study the long time dynamics of solitary wave solutions of the nonlinear Schrödinger 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 Schrödinger 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.

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

References in corpus (4)

• Solitary Wave Dynamics in an External Potential
• Asymptotic stability of N-soliton states of NLS
• Long time motion of NLS solitary waves in a confining potential
• Slow soliton interaction with delta impurities

Cited by in corpus (8)

• Colliding solitons for the nonlinear Schrodinger equation
• On the Dynamics of solitons in the nonlinear Schroedinger equation
• Solitary waves for the Hartree equation with a slowly varying potential
• Soliton interaction with slowly varying potentials
• Soliton dynamics for the nonlinear Schrödinger equation with magnetic field
• An Adiabatic Theorem for the Gross-Pitaevskii Equation
• Stochastic acceleration of solitons for the nonlinear Schrödinger equation
• Soliton dynamics of NLS with singular potentials