Solutions of a modified third Painlevé equation are meromorphic
type: Item
instance of: scholarly article
title: Solutions of a modified third Painlevé equation are meromorphic
zbMATH Open document ID: 1015.34082
DOI: 10.1007/BF02788086
author: Ilpo Laine
author: Aimo Hinkkanen
published in: Journal d'Analyse Mathématique
publication date: 2002-07-10T00:00:00Z
review text: It is known that all solutions to the first, second and fourth Painlevé differential equations are meromorphic in the plane. That is in general not true for the third Painlevé equation \[ w''= {(w')^2\over w}-{w' \over z}+{\alpha w^2+\beta \over z}+\gamma w^3+{\delta\over w}, \quad \alpha, \beta,\gamma, \delta\in \mathbb{C}. \] Using the transformation \(W(\zeta)= w(z)z\) and \(z=e^{\zeta/2}\) one gets the modified equation \[ W''={(W')^2\over W}+\alpha W^2+ \gamma W^3+\beta e^{\zeta}+ {\delta e^{2\zeta}\over W} \] with \((4\alpha,4 \beta, 4\gamma,4 \delta)\) instead of \((\alpha,\beta, \gamma,\delta)\). Now the authors prove that all solutions to this modified equation again are meromorphic functions in the plane.
reviewed by: Günter Frank
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 34M05
zbMATH DE Number: 1765851
zbMATH Keywords: Painlevé equations
zbMATH Keywords: meromorphic solutions
MaRDI profile type: Publication
cites work: Solutions of the first and second Painlevé equations are meromorphic
cites work: On Painlevé's equations I, II and IV
cites work: Q5665539
description: scientific article; zbMATH DE number 1765851