Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

Yang Shi; Nalini Joshi

doi:10.1098/rspa.2011.0167

Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

Yang Shi, Nalini Joshi

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

In this paper, we present a new method of deducing infinite sequences of exact solutions of q-discrete Painleve equations by using their associated linear problems. The specific equation we consider in this paper is a q-discrete version of the second Painleve equation (q-PII) with affine Weyl group symmetry of type (A2 + A1)⁽¹⁾. We show, for the first time, how to use the q-discrete linear problem associated with q-PII to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q-PII here, is also applicable to other discrete Painlevé equations.

Original language	English
Pages (from-to)	3443-3468
Number of pages	26
Journal	PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume	467
Issue number	2136
DOIs	https://doi.org/10.1098/rspa.2011.0167
Publication status	Published - 8 Dec 2011
Externally published	Yes

Keywords

q-discrete
Painlevé equations
discrete equations
Iso-monodeomy deformation
Painlevé
Equations
Special solutions

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title = "Exact solutions of a q-discrete second Painlev{\'e} equation from its iso-monodromy deformation problem: I. Rational solutions",

abstract = "In this paper, we present a new method of deducing infinite sequences of exact solutions of q-discrete Painleve equations by using their associated linear problems. The specific equation we consider in this paper is a q-discrete version of the second Painleve equation (q-PII) with affine Weyl group symmetry of type (A2 + A1)(1). We show, for the first time, how to use the q-discrete linear problem associated with q-PII to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q-PII here, is also applicable to other discrete Painlev{\'e} equations.",

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author = "Yang Shi and Nalini Joshi",

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doi = "10.1098/rspa.2011.0167",

language = "English",

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Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem : I. Rational solutions. / Shi, Yang; Joshi, Nalini.

In: PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol. 467, No. 2136, 08.12.2011, p. 3443-3468.

Research output: Contribution to journal › Article › peer-review

TY - JOUR

T1 - Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem

T2 - I. Rational solutions

AU - Shi, Yang

AU - Joshi, Nalini

PY - 2011/12/8

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N2 - In this paper, we present a new method of deducing infinite sequences of exact solutions of q-discrete Painleve equations by using their associated linear problems. The specific equation we consider in this paper is a q-discrete version of the second Painleve equation (q-PII) with affine Weyl group symmetry of type (A2 + A1)(1). We show, for the first time, how to use the q-discrete linear problem associated with q-PII to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q-PII here, is also applicable to other discrete Painlevé equations.

AB - In this paper, we present a new method of deducing infinite sequences of exact solutions of q-discrete Painleve equations by using their associated linear problems. The specific equation we consider in this paper is a q-discrete version of the second Painleve equation (q-PII) with affine Weyl group symmetry of type (A2 + A1)(1). We show, for the first time, how to use the q-discrete linear problem associated with q-PII to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q-PII here, is also applicable to other discrete Painlevé equations.

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KW - Painlevé equations

KW - discrete equations

KW - Iso-monodeomy deformation

KW - Painlevé

KW - Equations

KW - Special solutions

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DO - 10.1098/rspa.2011.0167

M3 - Article

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SP - 3443

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JO - PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES

JF - PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES

SN - 1364-5021

IS - 2136

ER -

Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

Abstract

Keywords

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