Overview
- The first monograph on Painlevé equations to treat both classical local aspects and modern global aspects simultaneously
- Introduces a new method in the study of Painlevé equations, combining local analysis and global topology
- Gives a new classification of real solutions of the Third Painlevé equation in terms of their zeros and poles
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2198)
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Table of contents (18 chapters)
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Front Matter
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Back Matter
Keywords
About this book
The purpose of this monograph is two-fold: it introduces a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation of type PIII (D6), called PIII (0, 0, 4, −4), describing its relation to isomonodromic families of vector bundles on P1 with meromorphic connections. This equation is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears widely in geometry and physics. It is used here as a very concrete and classical illustration of the modern theory of vector bundles with meromorphic connections.
Complex multi-valued solutions on C* are the natural context for most of the monograph, but in the last four chapters real solutions on R>0 (with or without singularities) are addressed. These provide examples of variations of TERP structures, which are related to tt∗ geometry and harmonic bundles.
As an application, a new global picture o0 is given.
Authors and Affiliations
Bibliographic Information
Book Title: Painlevé III: A Case Study in the Geometry of Meromorphic Connections
Authors: Martin A. Guest, Claus Hertling
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-319-66526-9
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2017
Softcover ISBN: 978-3-319-66525-2Published: 15 October 2017
eBook ISBN: 978-3-319-66526-9Published: 14 October 2017
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: XII, 204
Number of Illustrations: 12 b/w illustrations
Topics: Ordinary Differential Equations, Algebraic Geometry, Special Functions, Functions of a Complex Variable