Nonlinear Waves in Uniform and Periodic Media

Duke University, Durham NC

Investigators

Abstract

The main aims of this project are in three directions. (a) To advance existing approaches and to develop new methods for the rigorous asymptotic solution of integrable nonlinear wave equations, primarily the nonlinear Schroedinger equation, in the small dispersion limit; also to understand how the structure of the solution, critically affected by the modulational instability, depends on the degree of smoothness of the initial data. (b) To develop methods within the theory of integrable and near-integrable wave equations to deal with wave reflection from boundaries and time dependent forcing. (c) To develop numerical and analytical tools for the study of electromagnetic resonant fields and surface wave phenomena in periodic and random photonic crystal slabs with as well as without the presence of nonlinearity. INTEGRABLE WAVE PROPAGATION is often a very good approximation to physical wave propagation in real materials (e.g. the nonlinear Schroedinger equation and its perturbations model transmission of light in fiber optics), and related mathematical aspects like the modulational instability play a crucial role in the optimal design of communications systems. PHOTONIC CRYSTALS, first developed in the eighties, allow light to be trapped in or guided through regions of dimensions comparable to its wavelength. They are already in use in microwave antennas, low threshold lasers and other devices. In most of their current applications they function as linear devices. We investigate nonlinear behavior and therefore pay particular attention to resonant behavior, which generates large fields for which the nonlinearity of the medium plays a central role.