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...

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Main Author: Guest, Martin A.
Corporate Author: SpringerLink (Online service)
Other Authors: Hertling, Claus
Format: Electronic
Language: English
Published: Cham Springer International Publishing 2017, 2017
Series: Lecture Notes in Mathematics
Subjects:
Online Access: http://dx.doi.org/10.1007/978-3-319-66526-9?nosfx=y
Collection: Springer eBooks 2005- - Collection details see MPG.ReNa
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 of0 is given
Physical Description: XII, 204 p. 12 illus online resource
ISBN: 9783319665269