Resonance Problems for Linear and Nonlinear Waves

题目：线性和非线性波的共振问题（ei: Michael I. Weinstein）哥伦比亚大学应用物理与应用数学系摘要：首席研究员将研究非线性和非齐次、确定性和随机介质中哈密顿波动方程中涉及共振能量传递的问题。提出的研究将加深我们对物理系统中重要波现象的理解，并为分析和计算方法的研究提供信息。考虑了三个主题领域。(1)非线性散射和共振能量传递：这涉及无限维动力系统在束缚态（孤波、扭结、漩涡）和辐射相互作用下的基本问题。(2)类孤子相干结构的控制：受非均匀缺陷介质捕获光脉冲问题的启发，本工作涉及传播介质的设计，以实现**控制的**能量传递。这些想法也与研究玻色-爱因斯坦凝聚的数学模型直接相关。(3)线性和非线性光子结构和均质化：本研究通过有效介质（如高阶均质化）方法解决光子微结构性质的确定，并将其应用于光学微结构的有效数值研究及其优化设计。技术的进步使新型光介质（光子微结构）的制造成为可能，它具有巨大的应用潜力，从通信网络中的传输介质和设备到光学和量子计算的基础科学。这些介质是由微（百万分之一米）或纳米（十亿分之一米）尺度的特征组成的物质结构。单个微特征的形状、它们的空间排列和它们之间的材料对比（例如折射率变化）提供了多个自由度，现在可以对其进行调整，以影响通过这种介质发送的光脉冲。基于这些新型结构的器件可用于形成、过滤和放大光脉冲，从而在光通信网络中对信息位进行编码。与目前使用的光纤相比，使用这种结构的传输介质可以潜在地实现更低的损耗和更低的脉冲失真。可能的设计数量是如此之大，以至于主要基于直接计算机探索的方法是不可行的。相反，对这种现象的基本数学理解必须与计算相结合，以实现光学介质的合理和系统设计。本研究的目标是发展偏微分方程和动力系统领域的基本数学见解，并将其应用于光子微结构中光传播（线性和非线性）的理解。

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.