Wave Propagation and Resonance in Complex Media

本计画系关于线性与非线性非均匀介质中波动之偏微分方程式。 考虑的问题范围从（a）波传播理论中的基本分析问题（非线性散射理论，散射共振的计算，以及相对于某些状态的寿命的微观结构的优化）到（B）光学（线性和非线性）和宏观量子系统（玻色-爱因斯坦凝聚）的应用。 不同的研究方向由以下主题统一：（i）不同模式之间的能量传递（例如，相干局域结构，如孤子、涡旋和辐射模式）和（ii）弱耦合到环境的相干结构的控制。波（光、声、流体、电子、引力等）的相互作用不均匀性在自然界和工程系统中普遍存在。 这些相互作用由物理方程控制，这些物理方程的基本形式包含了所有相关的物理效应，是难以解决的：在大多数有趣的情况下，即使使用当今最强大的计算机，它们也无法解决。 涉及简化数学模型、数学分析和科学计算的方法是解决最重要问题的关键。 本研究的目的是发展这种混合方法的类波相互作用的问题，具有潜在的应用，例如，光学器件和量子信息科学的设计。

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.