Asymptotic forms of solutions to the third Painlevé equation
type: Item
instance of: scholarly article
title: Asymptotic forms of solutions to the third Painlevé equation
zbMATH Open document ID: 1193.34175
DOI: 10.1134/S1064562408050335
published in: Doklady Mathematics
publication date: 2010-01-12T00:00:00Z
review text: The authors consider the implicit differential equation \[ -xyy''+ xy^{\prime 2}- yy'+ ay^3+ by+ cxy^4+ dx= 0 \] which can be obtained from the third Painlevé equation by multiplying it by \(xy\). It is assumed that all complex parameters \(a\), \(b\), \(c\) and \(d\) are different from zero. By means of two- and three-dimensional power geometry [\textit{A. D. Bruno}, Power geometry in algebraic and differential equations. Transl. from the Russian. Corrected and enlarged ed. North-Holland Mathematical Library. 57. Amsterdam: North-Holland (2000; Zbl 0955.34002)], the authors determine near the singularities \(x= 0\) and \(x=\infty\) the asymptotic forms of the solutions.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 5656097
author: A. D. Bryuno
author: Irina Goryuchkina
MaRDI profile type: Publication
cites work: Painlevé differential equations in the complex plane
cites work: Asymptotic behaviour and expansions of solutions of an ordinary differential equation
cites work: Q5820721
cites work: Q4389339
full work available at URL: s1064562408050335
OpenAlex ID: W4240934854
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description: scientific article; zbMATH DE number 5656097