Waves in Complex Media and Applications

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