RUI: The Darboux-Halphen Equations and their Generalizations

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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.