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Special Polynomials Associated with Rational Solutions of the Painlevé Equations and Applications to Soliton Equations

Clarkson, Peter (2006) Special Polynomials Associated with Rational Solutions of the Painlevé Equations and Applications to Soliton Equations. Computational Methods and Function Theory, 6 (2). pp. 329-401. ISSN 1617-9447. (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)

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. (Contact us about this Publication)

Abstract

Rational solutions of the second, third and fourth Painlev´e equations can be expressed in terms of

special polynomials defined through 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, as well as the special

polynomials associated with algebraic solutions of the third and fifth Painlev´e equations and equations in the

PII hierarchy, are studied. It is shown that the roots of these polynomials have an intriguing, highly symmetric

and regular structure in the complex plane. Further, using the Hamiltonian theory for the Painlev´e equations,

other properties of these special polynomials are studied. Soliton equations, which are solvable by the inverse

scattering method, are known to have symmetry reductions which reduce them to Painlev´e equations. Using this

relationship, rational solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations and rational

and rational-oscillatory solutions of the non-linear Schr¨odinger equation are expressed in terms of these special polynomials.

Item Type: Article
Uncontrolled keywords: Hamiltonians, Painlevé equations, rational 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: 01 Sep 2008 13:27 UTC
Last Modified: 01 Aug 2019 10:30 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/4048 (The current URI for this page, for reference purposes)
Clarkson, Peter: https://orcid.org/0000-0002-8777-5284
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