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On the application of solutions of the fourth Painlevé equation to various physically motivated nonlinear partial differential equations

Citation

Bassom, AP and Clarkson, PA and Hicks, AC, On the application of solutions of the fourth Painleve equation to various physically motivated nonlinear partial differential equations, Advances in Differential Equations, 1, (2) pp. 175-198. ISSN 1079-9389 (1996) [Refereed Article]

Abstract

Recently significant developments have been made in the understanding of the theory and solutions of the fourth Painlevé equation given by

ww" = 12 (w')2 + 32 w4 + 4zw3 + 2(z2 − α)w2 + β,

where α and β are arbitrary constants. All exact solutions of (1) are thought to belong to one of three solution hierarchies. In two of these hierarchies solutions may be determined in terms of the complementary error and parabolic cylinder functions whilst the third hierarchy contains rational solutions which may be expressed as ratios of polynomials in z. It is known that the fourth Painlevé equation (1) can arise as a symmetry reduction of a number of significant partial differential equations and here we consider the application of solutions of (1) to the nonlinear Schrödinger, potential nonlinear Schrödinger, spherical Boussinesq and 2+1-dimensional dispersive long wave equations. Consequently we generate a number of new exact solutions of these equations. The fourth Painlevé equation (1) is also related to problems that arise in the consideration of two-dimensional quantum gravity and this aspect is also discussed.

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:107970
Year Published:1996
Deposited By:Mathematics and Physics
Deposited On:2016-04-01
Last Modified:2016-07-08
Downloads:0

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