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Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation
Author(s) -
Milne Alice E.,
Clarkson Peter A.,
Bassom Andrew P.
Publication year - 1997
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00044
Subject(s) - mathematics , discretization , variety (cybernetics) , bessel function , pure mathematics , exact solutions in general relativity , mathematical analysis , mathematical physics , statistics
In this article our concern is with the third Painlevé equation d 2 y /d x 2 = (1/ y )(d y /d x ) 2 − (1/ x )(d y /d x ) + ( αy 2 + β )/ x + γy 3 + 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 x 1/3 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.