Applied Mathematics Seminar

The six Painleve equations (PI-PVI) were first derived around the turn of the century in an investigation by Painleve and his 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 physically applications including statistical mechanics, 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 and some exact solutions of the Painleve equations can be written in terms of special functions. For example, there exist solutions of PII-PVI that are expressed in terms of Airy, Bessel, parabolic cylinder, Whittaker and hypergeometric functions, respectively.

Recently there has also been considerable interest in integrable mappings and discrete systems, including discrete analogues of the Painleve equations which are nonlinear difference equations.

In this talk I shall describe some of plethora of remarkable properties which the discrete Painleve equations possess including nonlinear recurrence relations (commonly referred to as a Bäcklund transformations in the context of soliton equations) and hierarchies of exact solutions. In particular, I shall compare and contrast results for the discrete Painleve equations with those for the (continuous) Painleve equations and the classical special functions.