Painleve Equations

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ABSTRACTThe purpose of this project is to study a number of fundamental problems for the Painleve differential equations. These equations are second order nonlinear differential equations whose solutions do not have so-called movable singularities and cannot be expressed in terms of elementary or special functions. Their importance arises from the connections of the Painleve property to integrability theory, as well as from the numerous applications of the solutions, the Painleve transcendents. In this project, the principal investigator will study the following aspects of Painleve-related analysis: order of growth of single-valued meromorphic solutions; value distribution and branching of solutions to Painleve equations; andPainleve-type equations which admit movable branch points but only those whose multiplicity is bounded by a preassigned constant. Work performed under this proposal will lead to a greater understanding of and concrete results for this class of differential equations.Painleve equations are of great importance in pure and applied mathematics as well as inthe applications of mathematics to other sciences and to engineering. Within mathematics,Painleve equations are being applied in differential geometry, random matrix models, and integrability. The following are examples of areas of current activity outside mathematics in which Painleve equations have been found useful and have arisen in a natural way: the Ising model in physics, statistical mechanics in elasticity, correlation functions in an antiferromagnet model, quantum field theory and topological field theory,general relativity and cosmology, supersymmetry gauge theories in physics,resonant oscillations in shallow water, Hele-Shaw problems in viscous fluids,plasma physics, superconductivity, nonlinear optics and fiber optics,polymers, polyelectrolytes, and colloids. This list alone provides a clear indication of the empowering impact of mathematics in science and engineering, and of the value that theoretical understanding and precise problem solving in mathematics can contribute to modeling and theory building in other sciences. This project will provide additional knowledge and methods to this area of wide applicability.