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

该合作研究项目的目标是开发用于分析、建模和计算非线性光学中产生的随机偏微分方程组的方法。这些系统中噪声的研究受几个因素的影响而变得复杂。特别是：(I)许多物理效应相互作用造成的复杂性，(Ii)存在多个长度和时间尺度，以及(Iii)代表系统故障的关键事件在设计上极其罕见的事实。我们将通过以下方式解决这些挑战：(A)创建一套分析方法，以获得从控制系统行为的随机偏微分方程组到一组低维随机微分方程的约化；(B)开发跨多个时间和空间尺度的研究噪声的有效方法；以及(C)实施混合分析-计算方法，可以有效地针对感兴趣的罕见事件并最终估计这些系统的可靠性。光波系统有广泛的应用，特别是作为激光和光通信线路，它们的出现在过去50年里对我们的生活产生了巨大的影响。激光在与扫描和通信相关的应用中使用功率较低，而在较高功率时用于外科和工业应用。由于噪声是光学系统中的一个主要限制因素，进一步的技术进步将关键取决于对噪声的影响的了解和能否控制由噪声引起的系统故障。这项研究将提供新的分析和计算工具，支持下一代光学设备的分析和设计。

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.