Painlevé differential equations in the complex plane
type: Item
instance of: scholarly article
title: Painlevé differential equations in the complex plane
zbMATH Open document ID: 1043.34100
author: Valeriĭ Ivanovic Gromak
author: Ilpo Laine
author: Shun Shimomura
published in: De Gruyter Studies in Mathematics
publication date: 2003-01-27T00:00:00Z
review text: Few books concentrating on topics in Painlevé equations have been published, especially in English, since the appearance of those equations in the beginning of the last century. For this reason, this book written by Gromak, Laine and Shimomura is valuable for those who wish to learn quickly the fundamental properties of Painlevé equations without the need to refer to a large number of original papers on them. In fact, this book, containing almost 300 pages and consisting of ten chapters, appendices, etc., deals exclusively with topics and materials, including some open problems, in the theory of Painlevé equations which range from past study to recent progress. In particular, it contains a detailed description of the complex analytic study of Painlevé transcendents, which is a distinguishing characteristic, because other books do not deal with this in detail. Most parts of this book are an exposition of the results obtained by or closely related to the three authors' research on Painlevé equations. The main part of this book begins with the study of the Painlevé property of the Painlevé equations (Chapter l). After the discovery of the Painlevé equations, Painlevé posed the question whether every solution of each Painlevé equation is free of movable singularities other than movable poles, i.e., whether each Painlevé equation has the so-called ``Painlevé property'' . He regarded this question as one of the most fundamental problems in the theory of Painlevé equations, and in fact, he tried to resolve affirmatively the question of the Painlevé property for the first Painlevé equation, but his proof of the Painlevé property had some gaps. The first rigorous proof of the Painlevé property for the first and sixth equations was given in 1960 by Hukuhara, who made precise estimates for the behavior of solutions. The idea of his proof is also applicable to proving the Painlevé property of the other Painlevé equations. In fact, several Japanese authors (Shimomura among them)…
reviewed by: Humihiko Watanabe
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 34-02
zbMATH DE Number: 1859263
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1859263