Painlevé III: A Case Study in the Geometry of Meromorphic Connections

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.

List of contents

1. Introduction.- 2.- The Riemann-Hilbert correspondence for P3D6 bundles.- 3. (Ir)Reducibility.- 4. Isomonodromic families.- 5. Useful formulae: three 2 × 2 matrices.- 6. P3D6-TEP bundles.- 7. P3D6-TEJPA bundles and moduli spaces of their monodromy tuples.- 8. Normal forms of P3D6-TEJPA bundles and their moduli spaces.- 9. Generalities on the Painleve´ equations.- 10. Solutions of the Painleve´ equation PIII (0, 0, 4, -4).- 13. Comparison with the setting of Its, Novokshenov, and Niles.- 12. Asymptotics of all solutions near 0.- ...Bibliography. Index.

Summary

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

Product details

• Authors: Martin A. Guest, Claus Hertling
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 15.10.2017
• EAN: 9783319665252
• ISBN: 978-3-31-966525-2
• No. of pages: 204
• Dimensions: 156 mm x 236 mm x 14 mm
• Weight: 351 g
• Illustrations: XII, 204 p. 12 illus.
• Series: Lecture Notes in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Analysis, B, Algebraische Geometrie, Funktionalanalysis und Abwandlungen, Komplexe Analysis, komplexe Variablen, Funktionentheorie, Mathematics and Statistics, Algebraic Geometry, Ordinary Differential Equations, Complex analysis, complex variables, Differential equations, Special Functions, Functional analysis & transforms, Functions of a Complex Variable, Functions of complex variables