Integrable Systems and Random Matrices

可积系统和随机矩阵

• 批准号: 0701026
• 负责人: Irina Nenciu
• 金额: $ 9.9万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701026&HistoricalAwards=false
• 关键词: Integrable; Systems; Random; Matrices

Abstract The proposal describes three sets of problems that the PI plans to investigate during the funding period. First, the PI will continue her study of the Ablowitz-Ladik (AL) equation, and address questions related to generalized orbits and multi-Hamiltonian structures for the finite system, as well as their connection to the analogous Toda lattice problems. She is also interested in finding the solution of AL with periodic boundary conditions, and in the study of critical phenomena for the NLS equation, viewed as a continuum limit of the AL equation. Second, building on her work on matrix models for general beta-ensembles, the PI plans to investigate the asymptotic properties of these models via the approach of Adler and van Moerbeke. This describes various eigenvalue statistics of the model as solutions of completely integrable systems; so far, only the cases with beta equal to 1, 2 or 4 have been studied, and general beta-ensembles have never been used in this context. Finally, the PI plans to investigate, jointly with P. Deift, the question of long-time asymptotics for solutions of the water-wave problem with rough data, in the small amplitude/long wavelength regime. In their investigation, they will treat the problem as a perturbation of a completely integrable PDE, the KdV equation, and use the associated scattering transform and Riemann-Hilbert techniques to control the perturbation. In particular, a first step in this project is the rigorous treatment of the long-time asymptotics for the KdV equation with Sobolev initial data. The research described in this proposal concerns classical problems in two of the most active fields in mathematics, random matrix theory and integrable systems. One of the most fascinating scientific developments over the last fifty years has been the discovery that a wide variety of mathematical and physical phenomena are modeled by the eigenvalues of a random matrix. In particular, random matrix theory describes the scattering of neutrons off large nuclei, the statistics of the zeros of the Riemann zeta function on the critical line in the complex plane, as well as problems in the "real world", such as the bus scheduling in the city of Cavalierness in Mexico, or distances between cars on the freeway. The goal of the PI's proposed research is to describe the asymptotic properties of certain matrix ensembles which model some of the phenomena described above. Another part of the proposal is concerned with studying the properties of two remarkable evolution equations: The first is the Ablowitz-Ladik (AL) equation, which is a discrete version of the well-known nonlinear Schroedinger equation (NLS). Beyond their theoretical interest, both of the aforementioned equations have numerous scientific applications, one of the most important of which is in optics. The PI approaches the study of the AL equation using the methods from the theories of orthogonal polynomials and completely integrable systems. Finally, the PI proposes to study the water wave equation in a regime which can be used to model tsunamis, by further developing the method of nonlinear stationary phase used in the treatment of Riemann-Hilbert problems.

摘要 该提案描述了 PI 计划在资助期间调查的三组问题。首先，PI 将继续研究 Ablowitz-Ladik (AL) 方程，并解决与有限系统的广义轨道和多重哈密顿结构相关的问题，以及它们与类似的 Toda 晶格问题的联系。她还对寻找具有周期性边界条件的 AL 的解以及研究 NLS 方程的临界现象（被视为 AL 方程的连续统极限）感兴趣。其次，在她对一般 beta 系综矩阵模型的研究基础上，PI 计划通过 Adler 和 van Moerbeke 的方法研究这些模型的渐近性质。这将模型的各种特征值统计量描述为完全可积系统的解；到目前为止，只研究了 beta 等于 1、2 或 4 的情况，而一般的 beta 系综从未在这种情况下使用过。最后，PI 计划与 P. Deift 联合研究在小振幅/长波长范围内用粗糙数据解决水波问题的长时渐近问题。在他们的研究中，他们将把问题视为完全可积偏微分方程（KdV 方程）的扰动，并使用相关的散射变换和黎曼-希尔伯特技术来控制扰动。特别是，该项目的第一步是使用 Sobolev 初始数据严格处理 KdV 方程的长期渐近性。该提案中描述的研究涉及数学中两个最活跃的领域的经典问题：随机矩阵理论和可积系统。过去五十年来最令人着迷的科学发展之一是发现各种各样的数学和物理现象都是由随机矩阵的特征值建模的。特别是，随机矩阵理论描述了中子离开大原子核的散射、复平面临界线上黎曼 zeta 函数零点的统计，以及“现实世界”中的问题，例如总线调度在墨西哥的卡瓦利内斯市，或高速公路上汽车之间的距离。 PI 提出的研究目标是描述某些矩阵系综的渐近性质，这些矩阵系综对上述一些现象进行了建模。该提案的另一部分涉及研究两个显着演化方程的性质：第一个是 Ablowitz-Ladik (AL) 方程，它是著名的非线性薛定谔方程 (NLS) 的离散版本。除了理论上的兴趣之外，上述两个方程还有许多科学应用，其中最重要的应用之一是光学。 PI 使用正交多项式和完全可积系统理论的方法来研究 AL 方程。最后，PI建议通过进一步发展用于处理黎曼-希尔伯特问题的非线性固定相方法来研究可用于模拟海啸的水波方程。