The second Painleve equation, its hierarchy and associated special polynomials

Clarkson, P.A. and Mansfield, E.L. (2003) The second Painleve equation, its hierarchy and associated special polynomials. Nonlinearity, 16 (3). R1-R26. ISSN 0951-7715. (The full text of this publication is not available from this repository)

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Official URL
http://dx.doi.org/10.1088/0951-7715/16/3/201

Abstract

In this paper we are concerned with hierarchies of rational solutions and associated polynomials for the second Painleve equation (P-II) and the equations in the P-II hierarchy which is derived from the modified Korteweg-de Vries hierarchy. These rational solutions of P-II are expressible as the logarithmic derivative of special polynomials, the Yablonskii-Vorob'ev polynomials. The structure of the roots of these Yablonskii-Vorob'ev polynomials is studied and it is shown that these have a highly regular triangular structure. Further, the properties of the Yablonskii-Vorob'ev polynomials are compared and contrasted with those of classical orthogonal polynomials. We derive the special polynomials for the second and third equations of the P-II hierarchy and give a representation of the associated rational solutions in the form of determinants through Schur functions. Additionally the analogous special polynomials associated with rational solutions and representation in the form of determinants are conjectured for higher equations in the P-II hierarchy. The roots of these special polynomials associated with rational solutions for the equations of the P-II hierarchy also have a highly regular structure.

Item Type: Article
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Applied Mathematics
Depositing User: Judith Broom
Date Deposited: 19 Dec 2007 18:25
Last Modified: 14 Jan 2010 13:59
Resource URI: http://kar.kent.ac.uk/id/eprint/691 (The current URI for this page, for reference purposes)
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