eCite Digital Repository

New exact solutions of the discrete fourth Painlevé equation

Citation

Bassom, AP and Clarkson, PA, New exact solutions of the discrete fourth Painlevé equation, Physics Letters A, 194, (5-6) pp. 358-370. ISSN 0375-9601 (1994) [Refereed Article]

Copyright Statement

Copyright 1994 Elsevier B.V.

DOI: doi:10.1016/0375-9601(94)91294-7

Abstract

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

                   
𝒳𝑛+1𝒳𝑛−1 + 𝒳𝑛(𝒳𝑛+1 + 𝒳𝑛−1) = 
−2𝓏𝑛𝒳3𝑛 + (𝜂 − 3𝛿−2 − 𝓏2𝑛)𝑥2𝑛 + 𝜇2
(𝑥𝑛 + 𝓏𝑛 + 𝛾) (𝑥𝑛 + 𝓏𝑛 − 𝛾)
,                    (1)

where 𝑧𝑛 = 𝑛𝛿 and 𝜂, 𝛿, 𝜇 and 𝛾 are constants. In an appropriate limit (1) reduces to the fourth Painlevé (PIV) equation

                             
d2𝓌
d𝓏2
 = 
1
2𝓌
(
d𝓌
d𝓏
)
2
 + 
3
2
𝓌3
+
4𝓌2
+
2(𝓏2 - 𝛼)𝓌
+
𝛽
𝓌
,                             (2)

where α and β are constants and (1) is commonly referred to as the discretised fourth Painlevé equation. A suitable factorisation of (1) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable 𝑧𝑛. Limits of these solutions yield rational solutions of PIV (2). It is also known that there exist exact solutions of PIV (2) that are expressible in terms of the complementary error function and in this article we show that a discrete analogue of this function can be obtained by analysis of (1).

Item Details

Item Type:Refereed Article
Research Division:Mathematical Sciences
Research Group:Applied Mathematics
Research Field:Theoretical and Applied Mechanics
Objective Division:Expanding Knowledge
Objective Group:Expanding Knowledge
Objective Field:Expanding Knowledge in the Mathematical Sciences
Author:Bassom, AP (Professor Andrew Bassom)
ID Code:107885
Year Published:1994
Web of Science® Times Cited:10
Deposited By:Mathematics and Physics
Deposited On:2016-03-30
Last Modified:2016-11-25
Downloads:0

Repository Staff Only: item control page