Painlevé III : a case study in the geometry of meromorphic connections / Martin A. Guest, Claus Hertling.

Format:

Book

Author/Creator:

Guest, Martin A., author.

Hertling, Claus, author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 2198.

Lecture notes in mathematics, 0075-8434 ; 2198

Language:

English

Subjects (All):

Painlevé equations.

Physical Description:

xii, 202 pages : illustrations ; 24 cm.

Other Title:

Painlevé 3

Place of Publication:

Cham, Switzerland : Springer, [2017]

Summary:

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 of all zeros and poles of all real solutions of PIII (0, 0, 4, -4) on R>0 is given.

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 Painlevé equations

10. Solutions of the Painlevé equation PIII (0, 0, 4, -4)

11. Comparison with the setting of Its, Novokshenov, and Niles

12. Asymptotics of all solutions near 0

13. Rank 2 TEPA bundles with a logarithmic pole

14. Symmetries of the universal family of solutions

15. Three families of solutions on R>0

16. TERP structures and P3D6-TEP bundles

17. Orbits of TERP structures and mixed TERP structures

18. Real solutions of PIII(0, 0, 4, -4) on R>0.

Notes:

Includes bibliographical references (pages 197-200) and index.

ISBN:

9783319665252

3319665251

OCLC:

994640070