Movable poles of the solutions of Painlevé's equation of the third kind and their relation with Mathieu functions
type: Item
instance of: scholarly article
title: Movable poles of the solutions of Painlevé's equation of the third kind and their relation with Mathieu functions
zbMATH Open document ID: 0616.34022
DOI: 10.1007/BF01077265
publication date: 1986-01-01T00:00:00Z
review text: The article deals with the Painlevé equation \(u''+(1/x)u'=4shu\), \(x>0\). This equation has a solution \(u(x)=r\ell nx+s+O(x^{2-| r|})\) where \(x\to 0\), \(| r| <2\) with the asymptotic property \[ u(x)=- 2\ell n(x-2)-\frac{x-a}{a}-\frac{b(x-a)^ 2}{a^ 2}+O(x-a)^ 3,\quad x\to a \] near a singular point. The purpose of this work is to find a formula connecting the parameters (a,b) with the initial data (r,s). This problem is solved by the method of monodromy-deformation [\textit{H. Flaschka} and \textit{A. Newell}, Comm. Math. Phys. 76, 65-116 (1980; Zbl 0439.34005)]. Theorems 5, 6 give functional equations for the parameters (a,b), (r,s) containing the solution of the Mathieu equation.
Mathematics Subject Classification ID: 34B30
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34A99
Mathematics Subject Classification ID: 33E10
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 3998254
zbMATH Keywords: second order differential equations
zbMATH Keywords: Painlevé equation
zbMATH Keywords: method of monodromy-deformation
zbMATH Keywords: Mathieu equation
author: V. Yu. Novokshenov
MaRDI profile type: Publication
cites work: Painlevé functions of the third kind
cites work: Monodromy- and spectrum-preserving deformations. I
cites work: Q3717531
cites work: Q3345143
cites work: Stokes multipliers for a class of ordinary differential equations
cites work: Q5723358
cites work: Q3324254
full work available at URL: https://doi.org/10.1007/bf01077265
OpenAlex ID: W1989340385
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3998254