Trapping of Waves by Solitons

We will discuss several situations in which small amplitude waves propagate in a time-dependent potential that is induced by an excitation in a self-consistent nonlinear field. Although the small amplitude waves do not influence the nonlinear field, they are modulated by its presence. This modulation can lead to scattering, resonant amplification, or under certain circumstances, "trapping" or localization of wave energy. The trapping phenomenon is associated with a kind of integrability of the coupled system consisting of the nonlinear field and the modulated linear field. With the help of this integrability, a generalized transform method will be presented for solving the general initial-value problem for the modulated linear waves. Perturbation theory for nearly integrable couplings will be presented, and numerical simulations will be used to illustrate the scattering and resonance effects that are present far from integrability. Applications range from planar waveguide optics to wave propagation in molecular chains.

This talk summarizes joint work with N. N. Akhmediev, J. A. Besley, P. L. Christiansen, S. R. Clarke, A. Soffer, and M. I. Weinstein.