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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Bibliographic Details
Main Authors: Guest, Martin A., Hertling, Claus (Author)
Format: eBook
Language:English
Published: Cham Springer International Publishing 2017, 2017
Edition:1st ed. 2017
Series:Lecture Notes in Mathematics
Subjects:
Online Access:
Collection: Springer eBooks 2005- - Collection details see MPG.ReNa
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245 0 0 |a Painlevé III: A Case Study in the Geometry of Meromorphic Connections  |h Elektronische Ressource  |c by Martin A. Guest, Claus Hertling 
250 |a 1st ed. 2017 
260 |a Cham  |b Springer International Publishing  |c 2017, 2017 
300 |a XII, 204 p. 12 illus  |b online resource 
505 0 |a 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 
653 |a Algebraic Geometry 
653 |a Functions of complex variables 
653 |a Special Functions 
653 |a Functions of a Complex Variable 
653 |a Algebraic geometry 
653 |a Ordinary Differential Equations 
653 |a Special functions 
653 |a Differential equations 
700 1 |a Hertling, Claus  |e [author] 
041 0 7 |a eng  |2 ISO 639-2 
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490 0 |a Lecture Notes in Mathematics 
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082 0 |a 515.352 
520 |a 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