Wave Propagation and Resonance in Complex Media

This project concerns partial differential equations governing wave propagation in linear and nonlinear inhomogeneous media. The problems considered range from (a) fundamental analytical ones in wave propagation theory (nonlinear scattering theory, calculation of scattering resonances, and optimization of microstructures with respect to the lifetime of certain states) to (b) applications to optics (linear and nonlinear) and macroscopic quantum systems (Bose-Einstein condensation). The different research directions are unified by the themes of (i) energy transfer among different modes (e.g., coherent localized structures, such as solitons, vortices, and radiation modes) and (ii) control of coherent structures that are weakly coupled to an environment. Interactions of waves (light, acoustic, fluid, electronic, gravitational, etc.) with inhomogeneities are ubiquitous in nature as well as in engineered systems. These interactions are governed by equations of physics, which, in the fundamental forms that incorporate all relevant physical effects, are intractable: In most interesting cases they cannot be solved, even with today's most powerful computers. Methods involving simplified mathematical models, mathematical analysis, and scientific computation working in tandem are essential to progress on the most important problems. This research is aimed at the development of such hybrid approaches to classes of wave interaction problems, with potential applications to, for example, design of optical devices and quantum information science.