CAREER: Long-time asymptotics of completely integrable systems with connections to random matrices and partial differential equations

职业：与随机矩阵和偏微分方程相关的完全可积系统的长时间渐近

• 批准号: 1150427
• 负责人: Irina Nenciu
• 金额: $ 50万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-05-01 至 2019-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1150427&HistoricalAwards=false
• 关键词: CAREER; Long; time; asymptotics; completely

The research side of this project focuses on understanding the long-time asymptotics for a diverse set of models from analysis and mathematical physics. In particular, the principal investigator will concentrate on three projects: (1) the continued study of a discrete version of the defocusing nonlinear Schroedinger equation, in particular, its connections with orthogonal polynomials on the unit circle, Lie-Poisson algebras, and its continuum limits; (2) the study of properties (e.g., expected diagonalization times) of certain random matrix ensembles when considered as initial data for integrable flows such as the Toda lattice and the QR-algorithm; and (3) the study, using Riemann-Hilbert methods, of long-time asymptotics for integrable systems, namely, the Toda lattice and the Korteweg-deVries equation, and for certain nonintegrable perturbations of these systems. The research topics of this project fall in the wide areas of mathematical physics and partial differential equations, with a particular emphasis on questions related to such well-established mathematical areas as completely integrable systems, random matrices, and their applications to certain numerical algorithms. Partial differential equations emerge in a variety of physical contexts as mathematical models for the time-evolution of certain physical quantities. A particular class of such equations are the so-called completely integrable systems, which are characterized by the fact they satisfy a sufficiently large (in a sense which can be made precise) number of conservation laws. The theory of completely integrable systems has a long and distinguished history. Over the past thirty years, in particular, interest in the field has been fueled on the one hand by the fact that many of the equations known to be completely integrable are obtained as models of fluid dynamics, and on the other hand by the close connections that have been discovered with many fields of pure mathematics, such as Lie algebras or symplectic and Poisson geometry. One of the fundamental goals of the current project is to expand and deepen the understanding of these connections and to widen the class of models to which the techniques of completely integrable theory apply. Hence one aims to describe as fully as possible the solutions to these models and hopefully to gain thereby insights into real-life phenomena. An essential part of the proposal is its educational component, which is centered around the organization of an annual summer school for advanced graduate students and recent Ph.D.'s on various topics of current research interest at the juncture of analysis, partial differential equations, and numerical analysis. All talks in the summer school will be delivered on preassigned articles by the participants. Beyond its broad mathematical impact, the project will allow the principal investigator to expand other educational activities as well. She will support the work of graduate students through advanced courses, seminars, and student research workshops, and she will work with local chapters of the Association for Women In Mathematics (AWM) in the Chicago area to promote the advancement of women in mathematical careers.

该项目的研究重点是理解来自分析和数学物理的多种模型的长期渐近性。特别是，首席研究员将专注于三个项目：（1）继续研究散焦非线性薛定谔方程的离散版本，特别是其与单位圆上的正交多项式、李泊松代数及其连续统的联系限制； (2) 研究某些随机矩阵系综在被视为可积流（例如 Toda 格和 QR 算法）的初始数据时的属性（例如，预期对角化时间）； (3) 使用黎曼-希尔伯特方法研究可积系统（即 Toda 晶格和 Korteweg-deVries 方程）以及这些系统的某些不可积扰动的长期渐近性。该项目的研究主题属于数学物理和偏微分方程的广泛领域，特别强调与完全可积系统、随机矩阵及其在某些数值算法中的应用等成熟数学领域相关的问题。偏微分方程作为某些物理量随时间演化的数学模型出现在各种物理环境中。此类方程的一类特定类型是所谓的完全可积系统，其特征在于它们满足足够大（在某种意义上可以使其精确）的守恒定律。完全可积系统理论有着悠久而辉煌的历史。特别是在过去的三十年里，人们对该领域的兴趣一方面是由于许多已知完全可积的方程都是作为流体动力学模型获得的，另一方面是由于它们之间的密切联系已在纯数学的许多领域中发现，例如李代数或辛几何和泊松几何。当前项目的基本目标之一是扩大和加深对这些联系的理解，并扩大完全可积理论技术所适用的模型类别。因此，我们的目标是尽可能全面地描述这些模型的解决方案，并希望借此深入了解现实生活中的现象。该提案的一个重要组成部分是其教育部分，其核心是为高级研究生和最近的博士学位组织一年一度的暑期学校，讨论当前研究兴趣的各种主题，包括分析、偏微分方程，以及数值分析。暑期学校的所有演讲都将根据参与者预先指定的文章进行。除了广泛的数学影响之外，该项目还将允许首席研究员扩大其他教育活动。她将通过高级课程、研讨会和学生研究讲习班来支持研究生的工作，并将与芝加哥地区女性数学协会 (AWM) 的当地分会合作，促进女性在数学职业领域的进步。