Application of uniform asymptotics to the second Painlevé transcendent
type: Item
instance of: scholarly article
title: Application of uniform asymptotics to the second Painlevé transcendent
zbMATH Open document ID: 0912.34007
author: Andrew P. Bassom
author: Peter A. Clarkson
author: Chun-Kong Law
author: J. B. McLeod
published in: Archive for Rational Mechanics and Analysis
publication date: 1998-11-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/solv-int/9609005
review text: Connection problems of Painlevé differential equations of the form \(d^2\phi/d\eta^2= -\xi^2F(\eta,\xi)\phi\) are studied. These problems involve finding uniform approximations to solutions to this equation when the independent variable passes towards infinity along different directions in the complex plane. By the method used the need to match solutions is avoided. The treatment depends on the locations of the zeros of the function \(F\) in the limit. If they are isolated a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. If two of the zeros of \(F\) coalesce as \(|\xi|\to \infty\) then an approximation can be derived in terms of parabolic cylinder functions.
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34A45
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 34M40
zbMATH DE Number: 1221505
zbMATH Keywords: transformations
zbMATH Keywords: Painlevé differential equations
zbMATH Keywords: uniform approximations to solutions
zbMATH Keywords: Airy functions
zbMATH Keywords: parabolic cylinder functions
MaRDI profile type: Publication
OpenAlex ID: W2069154027
arXiv ID: solv-int/9609005
DOI: 10.1007/S002050050105
description: scientific article