The fourth Painleve equation

The six Painleve equations (PI–PVI) were first discovered about a hundred years ago by Painleve and his colleagues in an investigation of nonlinear second-order ordinary differential equations. During the past 30 years there has been considerable interest in the Painleve equations primarily due to the fact that they arise as symmetry reductions of the soliton equations which are solvable by inverse scattering. Although first discovered from pure mathematical considerations, the Painleve equations have arisen in a variety of important physical applications.

The Painleve equations may be thought of as nonlinear analogues of the classical special functions. They have a Hamiltonian structure and associated isomonodromy problems, which express the Painleve equations as the compatibility condition of two linear systems. The Painleve equations also admit symmetries under affine Weyl groups which are related to the associated B¨acklund transformations. These can be used to generate hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further solutions of the Painleve equations have some interesting asymptotics which are use in applications. In this paper I discuss some of the remarkable properties which the Painleve equations possess using the fourth Painleve equation (PIV) as an illustrative example.