RUI: The Darboux-Halphen Equations and their Generalizations

RUI：Darboux-Halphen 方程及其概括

• 批准号: 0307181
• 负责人: Sarbarish Chakravarty
• 金额: $ 7.54万
• 依托单位: University of Colorado at Colorado Springs
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0307181&HistoricalAwards=false
• 关键词: RUI; Darboux; Halphen; Equations; their

This proposal is concerned with the study of a class of nonlinear differential equations arising in such diverse areas as fluid dynamics, cosmology, topological quantum field theory as well as theory of automorphic functions. These equations admit a large family of solutions in terms of modular forms and exhibit novel singularites in the form of movable natural boundaries in the complex domain. The main objective of this project is to characterize the complex dynamics of these nonlinear systems by analyzing their general solutions, symmetries and singularity structuresin the complex plane. Various algebraic and geometric methodsakin to the theory of completely integrable systems will be employed to carry out the proposed studies.Nonlinear differential equations are well known to have important applications in the modeling of complex physical phenomena arising in many areas of science and technology. This proposal is aimed to investigate the general solutions to a class of nonlinear differential equations, thereby shedding more light on the properties of the underlying physical systems described by these equations. The project will also explore the connection between these nonlinear equations and the theory of automorphic functions which traditionally arises in areas of pure mathematics such as number theory. The proposed research activities will be carried out in a primarily undergraduate institution. It is strongly anticipated that the proposed research will lead to undergraduate projects in several areas including nonlinear waves, dynamical systems, special functions and conformal mapping, all of which have wide ranging physical applications.

本文主要研究流体力学、宇宙学、拓扑量子场论以及自同构函数理论等领域中出现的一类非线性微分方程。这些方程以模形式存在大量的解，并在复域上以可移动的自然边界形式表现出新颖的奇点。本课题的主要目的是通过分析这些非线性系统在复平面上的通解、对称性和奇点结构来表征它们的复杂动力学特性。各种代数和几何的方法，类似于理论的完全可积系统将被用来进行提出的研究。众所周知，非线性微分方程在许多科学和技术领域的复杂物理现象的建模中具有重要的应用。本提案旨在研究一类非线性微分方程的一般解，从而更多地揭示这些方程所描述的潜在物理系统的性质。该项目还将探索这些非线性方程与自同构函数理论之间的联系，自同构函数理论传统上出现在纯数学领域，如数论。拟议的研究活动将主要在本科院校进行。我们强烈期望所提出的研究将导致包括非线性波、动力系统、特殊函数和保角映射在内的几个领域的本科项目，这些领域都具有广泛的物理应用。