Painlevé equations - Nonlinear special functions

Abstract

The six Painlevé equations (PI-PVI) were first discovered around the. beginning of the twentieth century by Painlevé, Gambier and their colleagues in an investigation of nonlinear second-order ordinary differential equations. Recently there has been considerable interest in the Painlev'e equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering. Consequently the Painlev'e equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations, despite being nonlinear equations. Although first discovered from strictly mathematical considerations, the Painlev'e equations have arisen in a variety of important physical applications including statistical mechanics random matrices, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics. The Painlevé equations may be thought of as nonlinear analogues of the classical special functions. They possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Furthermore the Painlevé equations admit symmetries under affine Weyl groups which are related to the associated Bäcklund transformations. In this paper many of the remarkable properties which the Painlevé equations possess are reviewed including connection formulae, Bäcklund transformations associated discrete equations, and hierarchies of exact solutions. In particular, the second Painlevé equation PII is primarily used to illustrate these properties and some of the applications of PII are also discussed.