Mathematical Sciences: Spectral Problems and Inverse Spectral Problems

数学科学：谱问题和逆谱问题

• 批准号: 9203771
• 负责人: Percy Deift
• 金额: $ 11.5万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203771&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Spectral; Problems; Inverse

This project will focus on two fundamental areas of nonlinear mathematical analysis involving integrable systems of differential equations. Particular emphasis is placed on the relationship between eigenvalue algorithms and Hamiltonian mechanics. The first concerns the asymptotics of the oscillatory Riemann-Hilbert problems of the kind that arise in the theory of integrable nonlinear wave equations. Work will be done exploiting a newly developed steepest descent method to study the zero dispersion limit of the Korteweg de Vries equation, integrable statistical models such as the transverse Ising chain at critical magnetic field and spatially discrete problems, such as the Toda lattice. Work will also continue on the eigenvalue algorithms in the context of integrable systems. It was shown earlier how the basic diagonalization of finite dimensional matrices can be realized as flows on manifolds. A framework has been developed in which many well known systems of physical and mathematical interest can be embedded. One focus of this investigation will be to place the recent theory of Moser and Veselov describing a class of variational problems that can be solved by a generalized QR factorization into the framework of Hamiltonian flows. The principal object of this research is the analysis of systems of differential equations arising from models of the physical world and the application of that analysis to studies of finite dimensional matrix theory; in particular the development of new, computationally effective means for diagonalization of matrices. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions.

该项目将重点关注涉及微分方程可积系统的非线性数学分析的两个基本领域。 特别强调特征值算法和哈密顿力学之间的关系。第一个涉及可积非线性波动方程理论中出现的振荡黎曼-希尔伯特问题的渐近问题。 工作将利用新开发的最速下降法来研究 Korteweg de Vries 方程的零色散极限、可积统计模型（例如临界磁场下的横向伊辛链）和空间离散问题（例如 Toda 晶格）。 还将继续研究可积系统背景下的特征值算法。 前面已经展示了如何将有限维矩阵的基本对角化实现为流形上的流。 已经开发了一个框架，可以在其中嵌入许多众所周知的物理和数学系统。 这项研究的一个重点是将 Moser 和 Veselov 的最新理论放置到哈密顿流的框架中，该理论描述了一类可以通过广义 QR 分解来解决的变分问题。这项研究的主要目标是分析由物理世界模型产生的微分方程组，并将该分析应用于有限维矩阵理论的研究；特别是开发新的、计算有效的矩阵对角化方法。 偏微分方程构成了物理科学中数学建模的支柱。 涉及连续变化的现象，例如在运动、材料和能量中看到的现象，已知遵守某些一般定律，这些定律可以用偏导数之间的相互作用和关系来表达。 数学的关键作用不是陈述关系，而是从中提取定性和定量的意义，并验证表达解决方案的方法。