Mathematical Analysis for Problems in Nonlinear Optics

摘要，DMS-0505681，J Moeser，科罗拉多大学标题：非线性光学问题的数学分析本项目的目的是开发和实施分析技术，用于研究非线性光学中出现的问题。应用范围从光纤和非线性波导中的光脉冲传播到时空孤子的稳定，无论是对于确定性系统还是具有随机分量的系统。根据具体应用，相关的数学模型是非线性偏微分方程、积分方程或离散方程。核心的数学问题是渐近理论的基本理解，无论是分析和数值，适定性或爆破的建模方程，并通过变分法的直接方法寻找孤波解。本计画主要探讨非线性光学的三大应用：随机性系统、离散系统及广义色散管理系统。随机性在非线性光学中普遍存在，本项目的一个重要目标是研究在严格的概率意义上物理相关和可分析的噪声模型。这种模型已经产生了新型的基本孤子，他们的研究可能有助于工程师设计出性能实际上通过少量随机性增强的光学器件。离散方程出现在许多重要的背景下，包括非线性波导，全光开关和玻色-爱因斯坦凝聚的研究。离散模型与连续模型相比具有非常不同的性质，虽然空间变量的解的规律性不再是问题，但通常缺乏标度不变性使其分析更加困难。因此，离散的问题需要开发新的分析工具，最终将有助于解释实验观察。最后私家侦探将研究广义色散管理系统。色散管理技术在实现更高比特率的通信中起了重要作用，并为许多有趣的数学研究提供了动力。出现的近似模型具有非局部非线性，这提出了有趣的数学和数值挑战。通常，模型方程中的非定域性反映了原始物理系统中的稳定效应，一个重要的反复出现的问题是如何利用非定域结构来帮助解释观察到的稳定性。私家侦探将研究色散管理技术在光纤通信以外的环境中的应用，例如寻找时空孤子，其中P.I.希望获得有关其可能的稳定作用的新信息。非线性光学系统的基本数学理解是核心的技术，将能够支持未来互联网扩展的不断增长的需求的发展。快速、稳定的数据传输对于涉及国家利益的广泛领域至关重要，从银行、证券交易所、保险到医疗服务、交通运输和国土安全。通过拟议的研究收集的信息将有助于设计和实现新的光学系统，这将有助于满足日益增长的带宽需求。

Abstract, DMS-0505681, J Moeser, University of Colorado Title: Mathematical Analysis for Problems in Nonlinear Optics 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.