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The second Painleve equation, its hierarchy and associated special polynomials

Clarkson, Peter, Mansfield, Elizabeth L. (2003) The second Painleve equation, its hierarchy and associated special polynomials. Nonlinearity, 16 (3). R1-R26. ISSN 0951-7715. (doi:10.1088/0951-7715/16/3/201) (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) (KAR id:691)

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.
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
DOI/Identification number: 10.1088/0951-7715/16/3/201
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Elizabeth Mansfield
Date Deposited: 19 Dec 2007 18:25 UTC
Last Modified: 16 Nov 2021 09:39 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/691 (The current URI for this page, for reference purposes)

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