Painleve Equations

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PI: Aimo Hinkkanen, University of Illinois at Urbana-ChampaignDMS-0200752Abstract:The six Painleve differential equations are prototypes of second order differential equations which do not have movable singularities. In this project, the principal investigator will study the followingfundamental problems for the Painleve equations and Painleve-related analysis. Firstly, the task of obtaining rigorous proofs for the Painleve property of the solutions to Painleve equations in the remaining open cases. Secondly, the question of finding sharp upper and lower bounds for the usual order of growth of single-valued meromorphic solutions, and for other natural measures of growth for multi-valued solutions, as well as the determination of the cases where less than maximal growth occurs. Thirdly, applications motivated particularly by differential geometry lead to the question of classifying Painleve-type differential equations which are allowed tohave movable branch points subject to restrictions on their multiplicity. The principal investigator will work towards such a classification and to determine whether certain particular equations arising in applications have this property.Over the last ten years, after a quiet period of many decades, an enormous amount of literature has appeared on the Painleve equations and their generalizations, due to a great number of interdisciplinary connections and applications that have been found. On the theoretical side, the Painleve property is related to the concept of integrability for non-linear ordinary and partial differential equations. Other applications in pure mathematics include differential geometry and random matrix models. On the interdisciplinary side, the applications of Painleveequations include the following: the Ising and antiferromagnet models in physics,statistical mechanics in elasticity, quantum field theory and topological field theory,general relativity and cosmology, supersymmetric gauge theories in physics,resonant oscillations in shallow water, Hele-Shaw problems in viscous fluids,plasma physics, superconductivity, nonlinear optics and fiber optics,polymers, and polyelectrolytes. This provides a clear manifestation of the enablingpower of mathematics in science and engineering, and of the value that theoretical understanding and precise problem solving in mathematics can add to modeling and theory buildingelsewhere. Work performed under this proposal will lead to a greater understanding of and concrete results for this class of differential equations which is being used in numerousapplications in other areas of mathematics as well as in other sciences and engineering.