RUI: The Darboux-Halphen Equations and their Generalizations

这个建议是关于一类非线性微分方程的研究产生在这样不同的领域，如流体动力学，宇宙学，拓扑量子场论以及理论的自守函数。这些方程承认一个大家庭的解决方案，在模块化的形式，并表现出新颖的奇点的形式，在复杂的域中的可移动的自然边界。本项目的主要目标是通过分析这些非线性系统在复平面上的通解、对称性和奇异性结构来刻画它们的复杂动力学。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有着重要的应用。本文的目的是研究一类非线性微分方程的通解，从而进一步揭示这些方程所描述的物理系统的性质。该项目还将探索这些非线性方程与自守函数理论之间的联系，自守函数理论传统上出现在数论等纯数学领域。拟议的研究活动将在一个主要的本科院校进行。强烈预期，拟议的研究将导致在几个领域，包括非线性波，动力系统，特殊功能和保角映射，所有这些都有广泛的物理应用的本科项目。

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.