Clarkson, Peter (2008) The fourth Painleve equation. In: Guo, Li and Sit, William Y., eds. Differential Algebra and Related Topics. World Scientific, Singapore. ISBN 9789812833716. (Full text available)
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Abstract
The six Painleve equations (PI–PVI) were first discovered about a hundred years ago by Painleve and his colleagues in an investigation of nonlinear secondorder 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 oneparameter 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.
Item Type:  Book section 

Subjects:  Q Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations Q Science > QA Mathematics (inc Computing science) > QA351 Special functions 
Divisions:  Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Applied Mathematics 
Depositing User:  Peter A Clarkson 
Date Deposited:  26 Feb 2010 13:43 
Last Modified:  18 Jul 2014 08:03 
Resource URI:  http://kar.kent.ac.uk/id/eprint/23090 (The current URI for this page, for reference purposes) 
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