Analytical Studies of Nonlinear Optics in Periodic Media

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The objective of this proposal is to analytically investigate novel phenomena of nonlinear light propagation in new types of periodic media. Specifically, bifurcation and linear stability of solitary waves in combined linear and nonlinear lattices, general parity-time symmetric lattices and general two-dimensional lattices will be analytically determined through the development of a sophisticated exponential asymptotics technique. Both the positions of solitary waves and their linear-stability eigenvalues will be explicitly calculated; hence the existence and linear-stability properties of these solitary waves in periodic media will be analytically obtained. From a broader perspective, nonlinear optics in periodic media is currently at the forefront of research in optics and applied mathematics. With appropriate engineering of microscale periodic structures, light propagation can be controlled and steered in various different ways, which point to promising applications for micro-scale optical data-processing devices, low-distortion image transmission, and potential optical chips for ultrafast computers. The study of these exciting but difficult physical problems calls for advanced mathematical techniques. In this project, we will develop a powerful exponential asymptotics method, which can yield very detailed information on the existence and stability of solitary waves in periodic media. This information will allow us to assess the potential of optical solitons for various physical applications such as data processing on microscopic periodic structures. In addition, this project will facilitate interdisciplinary training of a graduate student in this important area.