Painlevé equations—nonlinear special functions

Clarkson, Peter (2003) Painlevé equations—nonlinear special functions. Journal of Computational and Applied Mathematics, 153 (1-2). pp. 127-140. ISSN 0377-0427 . (doi: ) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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The six Painlevé equations (PI–PVI) were first discovered about a hundred years ago by Painlevéand his colleagues in an investigation of nonlinear second-order ordinary differential equations. Recently, there has been considerable interest in the Painlevé equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering. Consequently, the Painlevé 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 Painlevé equations have arisen in a variety of important physical applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics. The Painlevé equations may be thought of a 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. Further the Painlevé equations admit symmetries under affine Weyl groups which are related to the associated Bäcklund transformations. In this paper, I discuss some of the remarkable properties which the Painlevé equations possess including connection formulae, Bäcklund transformations associated discrete equations, and hierarchies of exact solutions. In particular, the second Painlevé equation PII is used to illustrate these properties and some of the applications of PII are also discussed.

Item Type: Article
Uncontrolled keywords: Painleve equations; Backlund transformations; connection formulae; exact solutions
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics
Depositing User: Peter A Clarkson
Date Deposited: 06 Jun 2008 11:17 UTC
Last Modified: 28 Apr 2014 15:07 UTC
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