Waves in Complex Media and Applications

这个项目的目的是对波，特别是光波和电子波如何在新颖和复杂的介质中传播进行数学和计算研究。这项工作与超材料领域基础、应用和工程物理界最近的紧张活动直接相关。超材料是一种复合介质，它的宏观特性(折射率、导电性等)非常不寻常，在复合材料的单个微观成分中是不可能实现的。例如，装满负折射率流体的玻璃杯中的吸管将出现向后弯曲。最近设计的折射率为负的超材料可能会有广泛的应用，从超级透镜到隐形覆盖物，使隐形斗篷覆盖的对象。能够以极高的精度制造这种微结构和纳米结构的介质，激励了理论界帮助预测这类新材料的性能。首席研究人员的目标是，在他之前线性和非线性波动现象工作的基础上，进一步发展混合数学和计算分析，预测复杂微结构的特性并对其进行优化。在对复杂介质和超材料的研究中，主要研究人员将集中于应用偏微分方程组、数学分析、谱理论、渐近分析和最优化的数学问题：(A)蜂窝和其他新型结构中的线性和非线性波(B)受局域缺陷和随机性扰动的周期结构中的波，以及从局域缺陷模到谱隙的分叉(C)离散和连续非线性波系统中相干结构的动力学和(D)具有微结构和超材料的波现象的控制

This project is aimed at a mathematical and computational investigation of how waves, particularly optical and electronic waves, propagate through novel and complex media. This work connects directly with recent intense activity in the fundamental and applied and engineering physics communities in the field of metamaterials. Metamaterials are composite media whose emergent macroscopic properties (refractive index, conductivity, and others) are very unusual and are not achievable in individual microscopic constituents of the composite. For example, a straw in a glass filled with a negative refractive index fluid will appear bending backwards. The recently designed metamaterials with negative refractive index may have a broad variety of applications from super-lenses to cloaking covers that make objects covered by the cloak "invisible." The ability to fabricate such micro- and nano-structured media with great precision has inspired the theoretical community to help predict the properties of this new class of materials. The goal of the principal investigator is, building on his previous work in linear and nonlinear wave phenomena, to further develop hybrid mathematical and computational analysis that will predict the properties of and give insight into optimization of complex microstructures. In his study of complex media and metamaterials, the principal investigator will focus on the following mathematical problems at the nexus of applied Partial Differential Equations, Mathematical Analysis, Spectral Theory, Asymptotic Analysis and Optimization:(A) Linear and nonlinear waves in honeycomb and other novel structures(B) Waves in periodic structures perturbed by localized defects and randomness, and bifurcation from the continuum of localized defect modes into spectral gaps(C) Dynamics of coherent structures in discrete and continuous nonlinear wave systems and (D) Control of wave phenomena with microstructure and metamaterials