Waves in Complex Media and Applications

该项目旨在对波，特别是光波和电子波如何通过新颖复杂的介质传播进行数学和计算研究。这项工作与超材料领域的基础物理、应用物理和工程物理界最近的激烈活动直接相关。超材料是一种复合介质，其出现的宏观性质（折射率，导电性等）是非常不寻常的，并且在复合材料的单个微观成分中无法实现。例如，一根吸管在充满负折射率流体的玻璃杯中会出现向后弯曲。 最近设计的具有负折射率的超材料可能有各种各样的应用，从超级透镜到使斗篷覆盖的物体“隐形”的斗篷覆盖物。“以极高的精度制造这种微米和纳米结构介质的能力激发了理论界帮助预测这类新材料的特性。首席研究员的目标是，在他以前在线性和非线性波动现象方面的工作的基础上，进一步发展混合数学和计算分析，以预测复杂微结构的特性并深入了解其优化。在复杂介质和超材料的研究中，首席研究员将专注于应用偏微分方程，数学分析，谱理论，渐近分析和优化的关系中的以下数学问题：（A）蜂窝结构和其它新颖结构中的线性和非线性波（B）受局部缺陷和随机性扰动的周期性结构中的波，（C）离散和连续非线性波系统中相干结构的动力学和（D）用微结构和超材料控制波动现象

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