Painlevé III: A Case Study in the Geometry of Meromorphic Connections
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Beschreibung
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.
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
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
Autor*in
Martin A. Guest
Themen in »Painlevé III: A Case Study in the Geometry of Meromorphic Connections«
Painlevé III movable poles isomonodromic connections Riemann-Hilbert map monodromy data TERP-structures ordinary differential equations