A family of solutions of a nonlinear ordinary differential equation and its application to Painlevé equations (III), (V) and (VI)
type: Item
instance of: scholarly article
title: A family of solutions of a nonlinear ordinary differential equation and its application to Painlevé equations (III), (V) and (VI)
zbMATH Open document ID: 0638.34004
author: Shun Shimomura
published in: Journal of the Mathematical Society of Japan
publication date: 1987-01-01T00:00:00Z
review text: We deal with an equation of the form \[ (E)\quad x(xu')'=F_ 0(x,e^{- u},xe\quad u)+F_ 1(x,e^{-u},xe\quad u)(xu')+F_ 2(x,e^{- u\quad},xe\quad u)(xu')^ 2\quad ('=d/dx). \] Here \(F_ j(x,\xi,\eta)'s\) are holomorphic and bounded in the domain \(| x| <r_ 0\), \(| \xi | <r_ 1\), \(| \eta | <r_ 1\) \((r_ 0,r_ 1>0)\) and satisfy \(F_ j(0,0,0)=0\). In the neighbourhood of \(x=0\), we construct a two-parameter family of solutions of equation (E), and using the result, we give some analytic representations of solutions of Painlevé equations (III), (V), and (VI) near the singular point of the regular type. In some special cases, our representations of solutions of Painlevé equations are valid in the domain where the solution have infinitely many movable poles and zeroes.
reviewed by: Shun Shimomura
Mathematics Subject Classification ID: 34M55
zbMATH DE Number: 4040318
zbMATH Keywords: singular point
zbMATH Keywords: Painlevé equations
zbMATH Keywords: movable poles
zbMATH Keywords: zeroes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03940649
OpenAlex ID: W2045381276
Wikidata QID: Q115225690
DOI: 10.2969/JMSJ/03940649
description: scientific article; zbMATH DE number 4040318