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

Nonfiction, Science & Nature, Mathematics, Differential Equations, Geometry
Cover of the book Painlevé III: A Case Study in the Geometry of Meromorphic Connections by Claus Hertling, Martin A. Guest, Springer International Publishing
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Author: Claus Hertling, Martin A. Guest ISBN: 9783319665269
Publisher: Springer International Publishing Publication: October 14, 2017
Imprint: Springer Language: English
Author: Claus Hertling, Martin A. Guest
ISBN: 9783319665269
Publisher: Springer International Publishing
Publication: October 14, 2017
Imprint: Springer
Language: English

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.

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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.

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