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.
- 0707850 The PI investigated linear and nonlinear wave problems arising in a number of areas of fundamental and applied physics and engineering, e.g. self-focusing of beams nonlinear optics, guided modes in linear and nonlinear photonic waveguides, geophysical magma dynamics. The first set of advances concern conditions for stability and instability of coherent structures (e.g. solitary waves) of nonlinear dispersive waves with applications. Coherent structures are dominant carriers of energy in many physical systems. Examples range form hydrodynamic waves at the water-air interface of the ocean to ultra-short optical pulses in laser physics. A coherent structure is stability if small or moderate perturbations of it do not significantly effect its properties. This notion underlies the observability and controllability of many physical phenomena. We advanced the theory of metastability of coherent structures. This concerns a theory of coherent structures which are very long-lived but eventually decay or ``break apart’’ due to coupling to an environment. We investigate this phenomenon in energy-conserving systems where the mechanism of eventual decay is scattering loss. Such coupling to an environment in nonlinear or parametrically forced conservative systems was understood by the PI and collaborators in terms of a generalization of ``Fermi’s Golden Rule’’, first developed in atomic physics in the study of spontaneous emission. Using the framework we developed, we proved an asymptotic energy equipartition law and obtained results on the control of energy flow among multiple competing bound states in systems characterized by nonlinear waves and inhomeneities. Finally, we have studied many problems related to scattering resonances in high contrast, rapidly varying media – which are the key to understanding the flow and storage of energy in micro- and nano-structures in many engineering device applications.
