Collaborative Research: Mathematical and computational methods for stochastic systems in nonlinear optics

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The objective of this collaborative research project is to develop methods for analysis, modeling, and computation for stochastic partial differential equations arising in nonlinear optics. The study of noise in these systems is complicated by several factors. In particular:(i) the complexity due to the interplay of many physical effects,(ii) the presence of multiple length and time scales, and(iii) the fact that the critical events that represent system failures are, by design, extremely rare. We will address these challenges by (a) creating a set of analytical methods to obtain reductions from the stochastic partial differential equations that govern the system's behavior to a set of low-dimensional stochastic differential equations;(b) developing efficient methods for the study of noise across multiple time and space scales; and (c) implementing hybrid analytical-computational methods that can efficiently target the rare events of interest and ultimately estimate the reliability of these systems.Lightwave systems have a wide range of applications, in particular as lasers and optical communication lines, and their emergence has had a dramatic impact on our lives over the last fifty years. Lasers are used at low powers in applications related to scanning and communications; at higher powers they are employed in surgical and industrial applications. As noise is a major limiting factor in optical systems, further technological progress will depend critically on understanding the impact of noise and on being able to control noise-induced system failures. This research will provide new analytical and computational tools that support the analysis and design of the next generation of optical devices.