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Bäcklund transformations and solution hierarchies for the third Painlevé equation

Citation

Milne, AE and Clarkson, PA and Bassom, AP, Backlund transformations and solution hierarchies for the third Painleve equation, Studies in Applied Mathematics, 98, (2) pp. 139-194. ISSN 0022-2526 (1997) [Refereed Article]

Copyright Statement

Copyright 1997 Massachusetts Institute of Technology

DOI: doi:10.1111/1467-9590.00044

Abstract

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
,                             (1)

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.

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:108525
Year Published:1997
Web of Science® Times Cited:33
Deposited By:Mathematics and Physics
Deposited On:2016-04-21
Last Modified:2016-07-08
Downloads:0

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