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.
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## 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: | 14 May 2014 13:57 UTC |

Resource URI: | https://kar.kent.ac.uk/id/eprint/4048 (The current URI for this page, for reference purposes) |

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