FRG: Collaborative Research in Semiclassical Asymptotic Questions in Integrable Nonlinear Wave Theory

FRG：可积非线性波理论中半经典渐近问题的合作研究

• 批准号: 0451495
• 负责人: Kenneth T-R McLaughlin
• 金额: --
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0451495&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Semiclassical; Asymptotic

Abstract DMS-0354373 FRG: Collaborative Research in Semiclassical Asymptotic Questions in Integrable Nonlinear Wave Theory Peter Miller, Kenneth McLaughlin and Jared BronskiThe goal of applied mathematics is the study of equations of scientific,engineering or industrial interest in a mathematically rigorous way.Understanding such equations often requires consideringa limit in which certain parameters approach zero. Whilesome of these problems behave in a predictable manner in thelimit, other interesting and important problems involve unstablebehavior that becomes less and less predictable the smaller theparameter of interest becomes. One of the best illustrations of thisconcept is the behavior of fluid flows when the viscous drag is small,and the fluid behaves in a turbulent and chaotic way.The primary model under study in this project is the nonlinearSchrodinger equation, which is a fundamental model for the study ofpulses in optical fibers. The small parameter limit for thismodel corresponds to the limit of ultra-short pulse propagation, whichis expected to find many applications to high-speed telecommunications.This limit is known as the semiclassical limit.This project constitutes an in-depth study of initial-value problemsfor several partial differential equations (PDEs) in the semiclassicallimit. Existing formal theories fail because they lead to model problems that areill-posed and thus make no prediction at all for reasonable initialconditions. The goal of this project is to develop asymptotictheories that are not based on any particular ansatz and do notrequire unphysical conditions on the initial data. Among the problemsunder attack is the rigorous semiclassical analysis of the focusingnonlinear Schroedinger equation for general data, a problem that isgenerally considered to be one of the most important open problems inthe field of integrable systems. The specific aims of the projectinclude the development of new ansatz-free methods of asymptoticanalysis --- for spectral theory and for Riemann-Hilbert problems ofinverse-scattering theory --- that are insensitive to analyticityproperties of the initial data, and have a "nonlinearRiemann-Lebesgue" character, directly exploiting cancellation due tooscillations where analytic deformations are impossible. Thesetechniques, once developed, will also have important repercussions infields that are only tangentially related, for example, the theory oforthogonal polynomials of large degree and the statistical analysis oflarge random matrices.The main goal of the proposed work is to develop predictive tools thatapply to extremely unstable systems when the initial conditionsare rough or noisy. An example of such a system is the one governing thepropagation of ultrashort data pulses in certain optical fibers, andtherefor any new insights into such systems will have repercussionsin the field of telecommunications. Some of the problems we propose tostudy admit a detailed analysis because they are idealized and can in somesense be solved exactly; however our work is also expected to lead to generalmethods that are applicable to less idealized systems.These techniques will also have important implications in mathematicalfields that are only tangentially related, for example, the theory oforthogonal polynomials of large degree and the statistical analysis oflarge random matrices.This project also has an important educationalaspect. We plan to involve both graduate students and postdoctoral researchers inthis work, and the advanced training of the next generation of researchersis an important component of this proposal. We also plan aninterdisciplinary workshop to further disseminate the results of this work beyond theboundaries of the mathematical community.

Peter Miller， Kenneth McLaughlin和Jared bronski应用数学的目标是用严谨的数学方法研究科学、工程或工业的方程。理解这类方程通常需要考虑某些参数趋近于零的极限。虽然这些问题中的一些在极限情况下以可预测的方式表现，但其他有趣和重要的问题涉及不稳定的行为，这些行为随着感兴趣的参数变得越来越小而变得越来越不可预测。这一概念最好的例证之一是当粘性阻力很小时流体流动的行为，流体表现为湍流和混沌的方式。本课题研究的主要模型是非线性薛定谔方程，它是研究光纤脉冲的基本模型。该模型的小参数极限对应于超短脉冲传播的极限，有望在高速通信中找到许多应用。这个极限被称为半经典极限。本课题对半经典极限下若干偏微分方程的初值问题进行了深入的研究。现有的形式理论之所以失败，是因为它们导致了不适定的模型问题，因此根本无法对合理的初始条件做出预测。该项目的目标是开发不基于任何特定分析的渐近理论，并且不需要初始数据的非物理条件。在受到攻击的问题是对一般数据的聚焦非线性薛定谔方程的严格半经典分析，这个问题通常被认为是可积系统领域中最重要的开放问题之一。该项目的具体目标包括开发新的无分析渐近分析方法-用于光谱理论和宇宙散射理论的黎曼-希尔伯特问题-对初始数据的分析特性不敏感，并且具有“非线性黎曼-勒贝格”特征，直接利用由于振荡而产生的抵消，其中解析变形是不可能的。这些技术，一旦发展起来，也将在一些只是切线相关的领域产生重要的影响，例如，大程度的正交多项式理论和大型随机矩阵的统计分析。提出的工作的主要目标是开发预测工具，适用于初始条件粗糙或有噪声的极端不稳定系统。这种系统的一个例子是在某些光纤中控制超短数据脉冲传播的系统，因此，对这种系统的任何新见解都将在电信领域产生影响。我们提出要研究的一些问题，有必要进行详细的分析，因为它们是理想化的，在某种意义上是可以精确解决的；然而，我们的工作也有望导致适用于不太理想的系统的一般方法。这些技术也将在仅仅是切线相关的数学领域中产生重要的影响，例如，大程度的正交多项式理论和大型随机矩阵的统计分析。这个项目还有一个重要的教育方面。我们计划让研究生和博士后研究人员都参与这项工作，下一代研究人员的高级培训是这项计划的重要组成部分。我们还计划举办一个跨学科研讨会，进一步将这项工作的结果传播到数学界之外。