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Application of uniform asymptotics to the second Painlevé transcendent

Citation

Bassom, AP and Clarkson, PA and Law, CK and McLeod, JB, Application of uniform asymptotics to the second Painlevé transcendent, Archive for Rational Mechanics and Analysis, 143, (3) pp. 241-271. ISSN 0003-9527 (1998) [Refereed Article]

Copyright Statement

Copyright 1998 Springer-Verlag

DOI: doi:10.1007/s002050050105

Abstract

In this work we propose a new method for investigating connection problems for the class of nonlinear second‐order differential equations known as the Painlevé equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along different directions in the complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the ideas of isomonodromic deformation and the matching of WKB solutions. However, the implementation of these methods often tends to be heuristic in nature and so the task of rigorising the process is complicated. The method we propose here develops uniform approximations to solutions. This removes the need to match solutions, is rigorous, and can lead to the solution of connection problems with minimal computational effort. Our method relies on finding uniform approximations of differ ential equations of the generic form d2Φ/dη2 = −ξ2F(η, ξ)Φ as the complex-valued parameter ξ → ∞. The details of the treatment rely heavily on the locations of the zeros of the function F in this limit. If they are isolated, then a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. On the other hand, if two of the zeros of F coalesce as |ξ| → ∞, then an approximation can be derived in terms of parabolic cylinder functions. In this paper we discuss both cases, but illustrate our technique in action by applying the parabolic cylinder case to the "classical" connection problem associated with the second Painlevé transcendent. Future papers will show how the technique can be applied with very little change to the other Painlevé equations, and to the wider problem of the asymptotic behavio ur of the general solution to any of these equations.

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:107982
Year Published:1998
Web of Science® Times Cited:18
Deposited By:Mathematics and Physics
Deposited On:2016-04-01
Last Modified:2016-07-08
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