The Painleve equations - nonlinear special functions

Peter A. Clarkson
Institute of Mathematics and Statistics
University of Kent, Canterbury, CT2 7NF, UK.

To be held on Wednesday, July 17, 2002 at 2:00pm, Western Science
Centre, Room 156.  Coffee and Refreshments available.


The six Painleve equations (PI-PVI) were first derived around the
beginning of the twentieth century by investigation by Painleve, Gambier

and their colleagues in a study of nonlinear second-order ordinary
differential equations. There has been considerable interest in Painleve

equations over the last few years primarily due to the fact that they
arise as reductions of soliton equations solvable by inverse scattering.

Further, the Painleve equations are regarded as completely integrable
equations and possess solutions which can be expressed in terms of the
solutions linear integral equations.  Although first discovered from
strictly mathematical considerations, the Painleve equations have
appeared in various of several important applications including
statistical mechanics, random matrices, plasma physics, nonlinear waves,    
quantum gravity, quantum field theory, general relativity, nonlinear
optics  and fibre optics.

The Painleve equations may also be thought of as nonlinear analogues of
the classical special functions such as Bessel functions. Their general
solutions are transcendental in the sense that they cannot be expressed
in terms of previously known functions. However, for special values of
the parameters, PII-PVI possess rational solutions and  solutions
expressible in terms of special functions.  For example,  there exist
special solutions of PII-PVI that are expressed in terms of Airy,
Bessel, parabolic cylinder, Whittaker and hypergeometric functions,
respectively. Further the Painleve equations admit symmetries under
affine Weyl groups which are related to the associated Backlund

In this talk I shall give an overview of some of plethora of remarkable
properties which the Painleve equations possess (including connection
formulae, Backlund transformations, associated discrete equations and
hierarchies of exact solutions) and some of their applications.