Clarkson, P.A. and Joshi, N. and Mazzocco, M. (2006) The Lax pair for the mKdV hierarchy. In: Delabaere, E. and Loday-Richaud, M., eds. Théories Asymptotiques et Equations de Painlevé. Séminaires et Congrès (14). Sociètè Mathèematique de France, Paris, France, pp. 53-64. ISBN 978-2-85629-229-7.
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Abstract
Rational solutions of the second, third and fourth Painlevé equations (-) can be expressed in terms of logarithmic derivatives of special polynomials that are defined through coupled second order, bilinear differential-difference equations which are equivalent to the Toda equation. In this paper the structure of the roots of these special polynomials, and the special polynomials associated with algebraic solutions of the third and fifth Painlevé equations, is studied and it is shown that these have an intriguing, highly symmetric and regular structure. Further, using the Hamiltonian theory for -, it is shown that all these special polynomials, which are defined by differential-difference equations, also satisfy fourth order, bilinear ordinary differential equations.
| Item Type: | Book section |
|---|---|
| Uncontrolled keywords: | Hamiltonians, Painlevé equations, rational solutions |
| Subjects: | Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus |
| Divisions: | Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Applied Mathematics |
| Depositing User: | Peter A Clarkson |
| Date Deposited: | 07 Jun 2008 11:54 |
| Last Modified: | 18 Jul 2012 08:32 |
| Resource URI: | http://kar.kent.ac.uk/id/eprint/3223 (The current URI for this page, for reference purposes) |
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