Boutroux's method vs. re-scaling. Lower estimates for the orders of growth of the second and fourth Painlevé transcendents
type: Item
instance of: scholarly article
title: Boutroux's method vs. re-scaling. Lower estimates for the orders of growth of the second and fourth Painlevé transcendents
zbMATH Open document ID: 1080.34074
author: Norbert Steinmetz
published in: Portugaliae Mathematica. Nova Série
publication date: 2005-03-30T00:00:00Z
full work available at URL: https://eudml.org/doc/52043
review text: This paper gives lower estimates on the growth of transcendental solutions of the second and fourth Painlevé equations \[ \begin{aligned} &w''=2w^3+zw+\alpha, \tag{II} \\ &2ww''=(w')^2+3w^4+8zw^3+4(z^2-\alpha)w^2+2\beta\quad\quad (\,'\,=d/dz). \tag{IV} \end{aligned} \] For every transcendental solution \(w\) of (II), the re-scaling process \(y_n(\mathfrak {z})=p_n^{-1/2}w(p_n+p_n^{-1/2}\mathfrak {z})\) associated with a subsequence of poles [\textit{N. Steinmetz}, Isr. J. Math. 128, 29--52 (2002; Zbl 1016.34091)] leads to the equation \[ \mathfrak{y}''=2\mathfrak{y}^3+\mathfrak{y}\quad\quad (\,'\,=d/d\mathfrak{z}). \] This coincides with the limit of the equation \[ \Theta''=2\Theta^3+\Theta-\frac{\Theta'}{\xi}+\frac{2\alpha} {3\xi}+\frac{\Theta}{9\xi^2}\quad\quad (\,'\,=d/d\xi) \] as \(\xi\to\infty,\) which is derived from (II) by Boutroux's transformation \(\xi=(2/3)z^{3/2},\) \(\Theta=z^{-1/2}w.\) Using these facts, the author constructs a sequence of different poles \(\tilde{p}_n\) of \(w\) such that \(| \tilde{p}_{n+1}| \leq | \tilde{p}_n| +O(| \tilde{p}_n| ^{-1/2}),\) which yields the lower estimate on the growth \(\varrho(w) \geq 3/2.\) For every transcendental solution of (IV), the estimate \(\varrho(w)\geq 2\) is obtained by almost the same argument.
reviewed by: Shun Shimomura
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 34M10
zbMATH DE Number: 2149497
zbMATH Keywords: Painlevé differential equations
zbMATH Keywords: order of growth
zbMATH Keywords: re-scaling
zbMATH Keywords: Boutroux's method
MaRDI profile type: Publication
Recommended article: Growth results for Painlevé transcendents
Recommended article: Q3425003
Recommended article: Q3606307
Recommended article: A unified approach to the Painlevé transcendents
Recommended article: Q4350200
Recommended article: Lower Estimates for the Growth of Painlevé Transcendents
Recommended article: LOWER ESTIMATES FOR THE GROWTH OF THE FOURTH AND THE SECOND PAINLEVÉ TRANSCENDENTS
Recommended article: An old new class of meromorphic functions
Recommended article: Sub-normal solutions to Painlevé's second differential equation
Recommended article: Q4676347
description: scientific article; zbMATH DE number 2149497