Dynamics of Linear and Nonlinear Waves in Complex Media

相干结构是波场中可识别的局域特征，在许多物理系统中扮演着能量、信息等载体的核心角色。几个例子是：水/空气界面上的孤立水波、空气动力冲击波和通信网络中使用的光束或脉冲。本研究涉及光学、电磁学、流体力学和量子物理中控制波动现象的非线性偏微分方程组，特别是相干结构的研究。偏微分方程组的相干结构解往往是由于非线性和色散效应的共同作用而产生的。非线性倾向于集中能量，而色散倾向于分散能量；它们的平衡通常导致持久的或长寿命的相干结构。我们将研究相干结构的动力学--它们的稳定性和散射相互作用。我们还将考虑这种相干结构与空间不均匀介质的相互作用。非均匀性(偏微分方程中的空间变化系数)引入了另一类局域态，称为“缺陷模”。我们将研究具有缺陷模和辐射模的相干结构之间的非线性动力学和能量交换。最后，我们将把我们以前关于能量守恒偏微分方程组中离散(束束态)和连续(辐射)模之间的能量传递的工作应用于波传播的控制问题，例如设计一种光子结构来最大化指定光学态的寿命。我们的数学方法从严格的解析方法到形式渐近方法，再到偏微分方程组的数值模拟和优化。微纳结构介质的设计和制造的进步推动了数学研究，以确定它们的有效特性和波在此类结构中的传播特性。这种微结构是当前和未来通信和信息处理技术的重要组成部分。例如，它们能够以类似于多年来在固态计算机芯片中操纵电(电子)的方式来操纵光(光子)。优势来自于在光子介质中更快的速度和几乎无耗散的传播。由于这些问题的规模、多尺度特性和复杂性，对于器件设计问题，依靠控制偏微分方程组的全计算机模拟是不现实的。本项目中探讨的数学问题旨在开发系统的方法来表征这种微结构介质，并了解波在其中传播的特性。这些将被用来开发混合分析/计算方法，以解决需要控制相干结构的问题。

Coherent structures are identifiable localized characteristics in a wave field, which play a central role as carriers of energy, information etc in many physical systems. Several examples are: solitary water waves at the water / air interface, aerodynamic shock waves and optical beams or pulses used in communication networks. This research concerns the study of nonlinear partial differential equations (PDEs) governing wave phenomena, in particular coherent structures, in optics, electromagnetics, hydrodynamics and quantum physics. Coherent structure solutions of PDEs often arise due to a combination of nonlinear and dispersive effects. Nonlinearity tends to concentrate energy and dispersion tends to spread it out; their balance often results in persistent or long-lived coherent structures. We will study the dynamics of coherent structures - their stability and scattering interactions. We shall also consider the interaction of such coherent structures with spatially inhomogeneous media. Inhomogeneities (spatially varying coefficients in the PDEs) introduce another class of localized states called "defect modes". We will study the nonlinear dynamics and energy exchange between coherent structures with defect modes and radiation modes. Finally, we will apply our previous work on energy transfer between discrete (bound state) and continuum (radiation) modes in energy conserving PDEs to the question of control of wave propagation, e.g. designing a photonic structure to maximize the lifetime of a designated optical state. Our mathematical approaches range from rigorous analytical methods to formal asymptotic methods to numerical simulation of PDEs and optimization.Advances in the design and fabrication of micro- and nano-structured media are driving mathematical research to determine their effective properties and those of waves propagating in such structures. Such microstructures are important components in current and future communication and information processing technologies. For example, they enable the manipulation of light (photons) in a manner analogous to the way electricity (electrons) has been manipulated in solid state computer chips for many years. Advantages come through greater speed and virtually dissipation free propagation in photonic media. Due to the size, multi-scale character and complexity of these problems, relying on full computer simulations of the governing partial differential equations is not practical for the problem of device design. The mathematical problems explored in this project are aimed at the development of systematic approaches to characterization of such microstructured media and an understanding the properties of waves traveling through them. These will be used to develop hybrid analytical / computational approaches to the problems requiring the control of coherent structures.