Integrable Systems and Random Matrices

该提案描述了投资促进局计划在筹资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有关的问题，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的解以及研究NLS方程的临界现象感兴趣，NLS方程被视为AL方程的连续统极限。其次，在她对一般贝塔系综矩阵模型工作的基础上，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，而一般的贝塔系综从未在此背景下被使用过。最后，PI计划与P.Deift一起研究在小幅度/长波长条件下，具有粗糙数据的水波问题的解的长期渐近问题。在他们的研究中，他们将把这个问题视为一个完全可积的偏微分方程组的扰动，KdV方程，并使用相关的散射变换和Riemann-Hilbert技术来控制扰动。特别是，这个项目的第一步是严格处理具有Soblev初值的KdV方程的长时间渐近性。这项建议中描述的研究涉及数学中最活跃的两个领域--随机矩阵理论和可积系统中的经典问题。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是由随机矩阵的特征值来模拟的。特别是，随机矩阵理论描述了大核子的散射，复平面上临界线上Riemann Zeta函数的零点的统计，以及“现实世界”中的问题，如墨西哥卡瓦列斯市的公交调度，或高速公路上汽车之间的距离。PI提出的研究的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的一些现象。该提案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，上述两个方程都有许多科学应用，其中最重要的是在光学方面。PI利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Riemann-Hilbert问题的非线性定相方法来研究可用于模拟海啸的区域中的水波方程。

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.