Studies on the Painlevé equations. III: Second and fourth Painlevé equations, \(P_{II}\) and \(P_{IV}\)
type: Item
instance of: scholarly article
title: Studies on the Painlevé equations. III: Second and fourth Painlevé equations, \(P_{II}\) and \(P_{IV}\)
zbMATH Open document ID: 0589.58008
DOI: 10.1007/BF01458459
author: Kazuo Okamoto
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164145
review text: [Part II is to appear in Jap. J. Math., Part I in Ann. Mat. Pura Appl., IV. Ser.] In this paper, which is the third part of the series of papers ''Studies on the Painlevé equations'', we study the second and the fourth Painlevé equations by means of the method of birational canonical transformations. We associate with each equation the nonautonomous Hamiltonian system (H), called the Painlevé system. The group of birational canonical transformations of (H) is investigated by the use of the notion of the affine Weyl group; we attach the root system (R) to each (H). We consider also the families of particular solutions of the Painlevé systems, written in terms of the Airy functions or the Hermite functions. In particular, the rational solutions of (H) are studied in detail. The \(\tau\)-functions related to (H) is the other main object of this article; it is shown that the sequence \(\{\tau_ n\); \(n\in {\mathbb{Z}}\}\) of \(\tau\)-functions of (H) satisfies the Toda equation: \(\delta^ 2\log \tau_ n=\tau_{n-1}\tau_{n+1}/\tau^ 2_ n\).
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 3947258
zbMATH Keywords: Painlevé equations
zbMATH Keywords: birational canonical transformations
zbMATH Keywords: Hamiltonian system
MaRDI profile type: Publication
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OpenAlex ID: W2998014864
description: scientific article; zbMATH DE number 3947258