Integrable Systems and Random Matrices

可积系统和随机矩阵

• 批准号: 0962703
• 负责人: Irina Nenciu
• 金额: $ 5.09万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-03-02 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0962703&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将继续她的研究的Ablowitz-Ladik（AL）方程，并解决有关的问题，广义轨道和多哈密顿结构的有限系统，以及它们的连接到类似的户田晶格问题。她还感兴趣的是寻找解决方案的AL与周期性边界条件，并在研究的关键现象的NLS方程，被视为连续极限的AL方程。第二，建立在她的工作矩阵模型的一般β-合奏，PI计划通过阿德勒和货车Moerbeke的方法来研究这些模型的渐近性质。这描述了模型的各种本征值统计作为完全可积系统的解;到目前为止，只研究了β等于1，2或4的情况，并且在这种情况下从未使用过一般的β系综。最后，PI计划与P. Deift一起研究在小振幅/长波长范围内，具有粗糙数据的水波问题的长时间渐近解问题。在他们的研究中，他们将把这个问题作为一个完全可积的PDE，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，在这个项目中的第一步是严格的KdV方程的Sobolev初始数据的长时间渐近处理。在这个建议中所描述的研究涉及两个最活跃的数学领域，随机矩阵理论和可积系统的经典问题。在过去的50年里，最令人着迷的科学发展之一是发现了各种各样的数学和物理现象可以用随机矩阵的特征值来建模。特别是，随机矩阵理论描述了大原子核的中子散射，复平面中临界线上黎曼zeta函数零点的统计，以及“真实的世界”中的问题，如墨西哥Cavalierness市的公共汽车调度，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集合的渐近性质，这些矩阵集合模拟了上述一些现象。该计划的另一部分是关于研究两个显着的演化方程的性质：第一个是Ablowitz-Ladik（AL）方程，这是著名的非线性薛定谔方程（NLS）的离散版本。除了它们的理论兴趣之外，上述两个方程都有许多科学应用，其中最重要的一个是在光学中。PI采用正交多项式理论和完全可积系统理论的方法来研究AL方程。最后，PI建议研究水波方程的制度，可用于模拟海啸，通过进一步发展的方法，非线性固定相用于处理黎曼-希尔伯特问题。