This research focuses on the study of nonlinear waves and their applications in nonlinear optical systems. Using a combination of modern perturbation methods, rigorous analysis, and scientific computation, a wide variety of optical models will be explored and developed. In conjunction with significant modeling efforts, the mathematical objectives will be to further develop and extend the methods utilized for quantifying and understanding the wave dynamics of nonlinear, dispersive partial differential equations which arise in the study of optical soliton transmission, modelocked fiber lasers, and optical fiber network devices and applications. In particular, a variety of analytic methods will be pursued with the goal of reducing the governing equations in Hamiltonian and integrable systems to more readily handled systems of partial differential equation that still retain the fundamental characteristics of the full governing equations. The resulting dynamics will be completely classified with particular attention given to parameter regimes for which periodic or chaotic dynamics may arise in the system. Comparison between the reduced models, numerical simulations of the governing equations, and experiments will be performed whenever possible.
The goal of this research is to identify, classify, and model physical systems of current interest in the optics community. Particular interest will be given to applications of greatest interest to industrial partners at Bell Laboratories (Lucent Technologies), Tyco Submarine Systems International, the 3M corporation, Hughes Research Labs, and Honeywell Inc. With the increased importance of optical fiber technology, fiber optic applications have broadened from mere high data-rate communications (soliton transmission) to inexpensive optical fiber laser sources, local area network design and applications, and all-optical network devices and components. The proposal covers all areas of current research efforts in these areas. In particular, the close working relations with industrial partners ensures timely research in the most important areas of application. And with the readily available experimental studies provided by these industrial partners, accurate mathematical models can be constructed and developed to capture the fundamental aspects of each optical system: thus providing the potential for the mathematical models to have a significant impact on design and implementation of realistic and physically realizable systems.
