The purpose of this project is to develop and implement analytical techniques for the study of problems arising in nonlinear optics. The array of applications ranges from light pulse propagation in optical fibers and nonlinear waveguides to stabilization of spatio-temporal solitons, both for deterministic systems and systems with a stochastic component. Depending on the particular application, the relevant mathematical models are nonlinear partial differential equations, integral equations, or discrete equations. The core mathematical issues are fundamental understanding of asymptotic theories, both analytically and numerically, well posedness or blow up for the modeling equations, and the search for solitary wave solutions via direct methods of calculus of variations. This project focuses on three major applications in nonlinear optics: systems with randomness, discrete systems, and generalized dispersion managed systems. Randomness arises ubiquitously in nonlinear optics and an important goal of this project is to study noise models that are physically relevant and analyzable in a rigorous probabilistic sense. Such models have already yielded new types of fundamental solitons and their study may help engineers to design optical devices whose performance is actually enhanced by small amounts of randomness. Discrete equations arise in many important contexts, including the study of nonlinear waveguides, all-optical switches, and Bose-Einstein condensates. Discrete models have very different properties than those of their continuous counterparts, and though regularity of solutions in the spatial variable is no longer an issue, the lack of scaling invariances in general makes their analysis more difficult. Thus discrete problems invite the development of new analytical tools which ultimately will help explain experimental observations. Finally, the P.I. will study generalized dispersion managed systems. The technique of dispersion management has been instrumental in enabling higher bit rate communications and has provided the impetus for many interesting mathematical investigations. The approximate models that arise have a nonlocal nonlinearity which presents interesting mathematical and numerical challenges. Often the nonlocality in the modeling equation reflects a stabilizing effect in the original physical system, and an important recurring question is how to exploit the nonlocal structure in order to help explain the observed stabilization. The P.I. will examine the application of dispersion management technology in contexts other than fiber optic communications, such as the search for spatio-temporal solitons, where the P.I. expects to obtain new information on it's possible stabilizing effects.
Fundamental mathematical understanding of nonlinear optical systems is central to the development of technologies that will be able to support the ever increasing demands of future Internet expansion. Fast, stable data transmission is critically important to a wide array of sectors of national interest, ranging from banking, the stock exchange, and insurance to health services, transportation, and homeland security. The information gleaned through the proposed research will aid in the design and implementation of novel optical systems that will help meet the growing need for bandwidth.
