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Geometric Methods in the Analytic Theory of Differential Equations

微分方程解析论中的几何方法

基本信息

批准号：
1665115
负责人：
Alexandre Eremenko
金额：
$ 21万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665115&HistoricalAwards=false
关键词：
Geometric Methods Analytic Theory Differential

项目摘要

Theory of differential equations is a fundamental mathematical tool of physics and engineering. Few differential equations can be solved explicitly, and approximate numerical solutions do not always give essential features of the behavior of exact solutions. A class of simple differential equations that frequently occur in applications has been studied by mathematicians for centuries; their solutions are called special functions of mathematical physics, and they are widely used in science. The general goal of this project is to extend this class of well-understood equations. Building on earlier work relevant for applications to physics, control theory, and materials science, this project aims to apply a variety of recently-developed methods to study longstanding questions of intrinsic mathematical interest. The work is anticipated to further improve understanding of the qualitative features of analytic functions defined by some basic differential equations arising in mathematical physics and geometry.Most of the special functions of mathematical physics are defined by linear differential equations with at most three singularities. Solutions of the Heun equation (with four regular singularities) and the Painlevé VI equation (non-linear, with four fixed singularities and no movable singularities) lie on the boundary of the class of special functions. Because of their intrinsic mathematical interest and numerous applications in science, they have been extensively studied since the beginning of the 20th century. This research project aims to advance understanding of these important functions through the use of new geometric methods. The main topics of this project are the qualitative study of real solutions of the Painlevé VI equation, the study of Riemannian metrics of constant positive curvature with conic singularities, with the emphasis on the metrics with four singularities closely related to the Heun equation, and finally the study of the eigenvalues of some parity-time symmetric anharmonic oscillators.

微分方程理论是物理学和工程学的基本数学工具。很少有微分方程可以显式求解，近似数值解并不总是给出精确解行为的基本特征。一类在应用中经常出现的简单微分方程，几个世纪以来一直被数学家研究;它们的解被称为数学物理的特殊函数，在科学中有着广泛的应用。这个项目的总体目标是扩展这类很好理解的方程。在与物理学，控制理论和材料科学应用相关的早期工作的基础上，该项目旨在应用各种最近开发的方法来研究内在数学兴趣的长期问题。本文的工作将进一步加深对数学物理和几何中的一些基本微分方程所定义的解析函数的定性特征的理解。数学物理中的大多数特殊函数都是由至多具有三个奇点的线性微分方程所定义的。Heun方程（有四个正则奇点）和Painlevé VI方程（非线性，有四个固定奇点，没有可动奇点）的解位于特殊函数类的边界上。由于其内在的数学兴趣和在科学中的众多应用，自世纪以来，它们一直被广泛研究。该研究项目旨在通过使用新的几何方法来促进对这些重要功能的理解。该项目的主要课题是Painlevé VI方程的真实的解的定性研究，具有圆锥奇点的常正曲率Riemannian度量的研究，重点是与Heun方程密切相关的四个奇点的度量，最后是一些宇称对称非简谐振子本征值的研究。