Resonance Problems for Linear and Nonlinear Waves

点击翻译按钮获取中文摘要

DMS-0412305 Title: Resonance Problems for Linear and Nonlinear WavesPI: Michael I. Weinstein Department of Applied Physics and Applied Mathematics Columbia University New York, New York Abstract:The Principal Investigator will study problems involving resonant energy transfer in Hamiltonian wave equations governing propagation in nonlinear and inhomogeneous, deterministic and random media. The proposed research will deepen our understanding of and inform analytical and computational approaches to the study of important wave phenomena in physical systems. Three theme areas are considered. (1) Nonlinear scattering and resonant energy transfer: This concerns fundamental questions for infinite dimensional dynamical systems on the interaction of bound states (solitary waves, kinks, vortices) and radiation. (2) Control of soliton-like coherent structures: Motivated by the problem of trapping optical pulses in inhomogeneous media with defects, this work concerns design of propagation media in order to achieve **controlled** energy transfer. The ideas are also of direct relevance in the study of mathematical models of Bose-Einstein condensation. (3) Linear and nonlinear photonic structures and homogenization: This work addresses determination of properties of photonic microstructures, by effective media (e.g. higher order homogenization) approaches with applications to efficient numerical study of optical microstructures and their optimal design. Technological advances have made possible the fabrication of novel optical media (photonic microstructures), which have great potential for applications ranging from transmission media and devices in communication networks to the fundamental science of optical and quantum computing.These media are material structures made up of features which are of micro- (millionth of a meter) or nano- (billionth of a meter) scales. The shape of individual microfeatures, their spatial arrangement and the material contrasts among them (e.g. refractive index variations) offer multiple degrees of freedom, which can now be tuned in order to influence light pulses sent through such media. Devices based on these novel structures can be used to shape, filter and amplify light pulses which encode the bits of information in optical communication networks. Transmission media using such structures can potentially achieve much lower losses and lower distortion of pulses than in currently used optical fiber. The number of possible designs is so huge that an approach mainly based on direct computer exploration is notfeasible. Rather, a fundamental mathematical understanding of such phenomena must be joined with computation to achieve rational and systematic design of optical media. The goal of this research is to develop fundamental mathematical insights in the fields of partial differential equations and dynamical systems and to apply them to an understanding of light propagation (linear and nonlinear) in photonic microstructures.