New exact solutions of the discrete fourth Painlevé equation

Bassom, Andrew P.

doi:10.1016/0375-9601(94)90324-7

Cited by 4 publications

(4 citation statements)

References 20 publications

(49 reference statements)

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“…As for rational solutions of P IV (4), it is known that there are three hierarchies of unique rational solutions [7](the name of the hierarchies are due to Ref. [10]), " − 1 z hierarchy": w = P n−1 (z) Q n (z) , (α, β) = (±k, −2(1 + 2l + k) 2 ), k, l ∈ Z, l ≤ −1, k ≤ −2l ,…”

Section: Introductionmentioning

confidence: 99%

Determinant structure of the rational solutions for the Painlevé IV equation

Kajiwara

Ohta

1998

J. Phys. A: Math. Gen.

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Section: Introductionmentioning

confidence: 99%

Determinant structure of the rational solutions for the Painlevé IV equation

Kajiwara

Ohta

1998

J. Phys. A: Math. Gen.

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“…As the theory of Bäcklund transformations for the discrete Painlevé equations is nowhere near as well developed as in the continuous case we are unable to apply simple transformations to deduce new solutions of d-P III from known ones. However, we can reduce the task of finding new discrete solutions to one of relatively simple algebra in a way akin to that discussed by Bassom and Clarkson [64] who were concerned with exact solutions of d-P IV .…”

Section: Introductionmentioning

confidence: 99%

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

1997

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In this article our concern is with the third Painlevé equation d2y/dx2= (1/y)(dy/dx)2− (1/x)(dy/dx) + (αy2+β)/x+γy3+δ/y where α, β, γ, and δ are arbitrary constants. It is well known that this equation admits a variety of types of solution and here we classify and characterize many of these. Depending on the values of the parameters the third Painlevé equation can admit solutions that may be either expressed as the ratio of two polynomials in either x or x1/3 or related to certain Bessel functions. It is thought that all exact solutions of (1) can be categorized into one or other of these hierarchies. We show how, given a few initial solutions, it is possible to use the underlying structures of these hierarchies to obtain many other solutions. In addition, we show how this knowledge concerning the continuous third Painlevé equation (1) can be adapted and used to derive exact solutions of a suitable discretized counterpart of (1). Both the continuous and discrete solutions we find are of potential importance as it is known that the third Painlevé equation has a large number of physically significant applications.

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“…An specially important subject related to the study of discrete equations is that concerning the discretization of Painlevé equations (cf. [2,16,27,30]). Recently there has been substantial interest in the discrete Painlevé equations (dP1-dPVI), which in a variety of physical applications and indeed dPI and dPII were first discovered in physical situations [5,17,28].…”

Section: Introductionmentioning

confidence: 99%

Discrete equations and the singular manifold method

Estévez¹,

Clarkson²

2000

SIDE III—Symmetries and Integrability Of Difference Equations

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The Painlevé expansion for the second Painlevé equation (PII) and fourth Painlevé equation (PIV) have two branches. The singular manifold method therefore requires two singular manifolds. The double singular manifold method is used to derive Miura transformations from PII and PIV to modified Painlevé type equations for which auto-Bäcklund transformations are obtained. These auto-Bäcklund transformations can be used to obtain discrete equations.

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New exact solutions of the discrete fourth Painlevé equation

Abstract: In this paper we derive a number of exact solutions of the discrete equation

Cited by 4 publications

References 20 publications

Determinant structure of the rational solutions for the Painlevé IV equation

Determinant structure of the rational solutions for the Painlevé IV equation

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

Discrete equations and the singular manifold method

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