Integrable Systems and Random Matrices

该提案描述了PI计划在资助期间调查的三组问题。首先，PI将继续她对Ablowitz-Ladik （AL）方程的研究，并解决与有限系统的广义轨道和多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边界条件的AL的解和NLS方程的临界现象的研究感兴趣，这被视为AL方程的连续统极限。其次，在她对一般β -集成的矩阵模型的研究基础上，PI计划通过Adler和van Moerbeke的方法来研究这些模型的渐近性质。这将模型的各种特征值统计描述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，而一般的β -系综从未在这种情况下使用过。最后，PI计划与P. Deift共同研究小振幅/长波长条件下粗糙数据下水波问题解的长期渐近性问题。在他们的研究中，他们将把这个问题视为一个完全可积PDE的扰动，KdV方程，并使用相关的散射变换和黎曼-希尔伯特技术来控制扰动。特别地，本项目的第一步是严格处理具有Sobolev初始数据的KdV方程的长时间渐近性。本文所描述的研究涉及数学中两个最活跃领域的经典问题：随机矩阵理论和可积系统。在过去的五十年里，最引人入胜的科学发展之一是发现了各种各样的数学和物理现象都是由随机矩阵的特征值来建模的。特别是，随机矩阵理论描述了中子在大原子核上的散射，复杂平面上临界线上黎曼ζ函数零点的统计，以及“现实世界”中的问题，例如墨西哥Cavalierness市的公共汽车调度，或者高速公路上汽车之间的距离。PI提出的研究目标是描述模拟上述一些现象的某些矩阵系综的渐近性质。该提议的另一部分涉及研究两个显著的演化方程的性质：第一个是Ablowitz-Ladik （AL）方程，它是众所周知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，上述两个方程还有许多科学应用，其中最重要的应用之一是光学。PI用正交多项式理论和完全可积系统的方法来研究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平稳相位方法，研究可用于模拟海啸的水波方程。

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.