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

摘要DMS-0354373 FRG：可积非线性波动理论中的半经典渐近问题的合作研究应用数学的目标是以严格的数学方法研究科学、工程或工业上感兴趣的方程，理解这样的方程常常需要求出某些参数趋于零的极限。虽然其中一些问题在极限内以可预测的方式表现，但其他有趣且重要的问题涉及不稳定的行为，随着感兴趣的参数变得越来越小，这种行为变得越来越不可预测。 这一概念的一个最好的例证是当粘性阻力很小时流体流动的行为，流体的行为是湍流和混沌的方式。该模型的小参数极限对应于超短脉冲传输的极限，该极限被称为半经典极限，本项目深入研究了几个偏微分方程在半经典极限下的初值问题。现有的正式理论失败，因为它们导致模型问题是不适定的，因此没有预测在合理的初始条件。 这个项目的目标是发展不基于任何特定的假设的渐近理论，并且不需要初始数据的非物理条件。 其中，对一般数据的聚焦非线性Schroedinger方程进行严格的半经典分析是可积系统领域中最重要的公开问题之一。 该项目的具体目标包括发展新的无ansatz渐近分析方法--用于谱理论和反向散射理论的Riemann-Hilbert问题--对初始数据的分析性质不敏感，并具有“非线性Riemann-Lebesgue”特征，直接利用分析变形不可能的振荡引起的抵消。 这些技术，一旦开发，也将有重要的影响领域，只是切相关，例如，理论的正交多项式的大的次数和统计分析的flagerandommatrix.The主要目标的拟议工作是开发预测工具，适用于极端不稳定的系统时，初始条件是粗糙或噪音。这种系统的一个例子是控制超短数据脉冲在某些光纤中传播的系统，因此，对这种系统的任何新的认识都将在电信领域产生影响。我们提出要研究的一些问题需要详细分析，因为它们是理想化的，在某种意义上是可以精确求解的;然而，我们的工作也有望导致适用于不太理想化系统的通用方法。这些技术也将在仅仅是切线相关的物理领域中具有重要意义，例如，大次正交多项式理论和大随机矩阵的统计分析，该项目也具有重要的教育意义。我们计划让研究生和博士后研究人员参与这项工作，对下一代研究人员的高级培训是这项建议的重要组成部分。我们还计划举办一个跨学科的研讨会，以进一步传播这项工作的成果，超越数学界的界限。

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.