04095nam 22006375 450 99646652220331620200706093823.03-319-66526-X10.1007/978-3-319-66526-9(CKB)4100000000882282(DE-He213)978-3-319-66526-9(MiAaPQ)EBC5591096(PPN)220120994(EXLCZ)99410000000088228220171015d2017 u| 0engurnn|008mamaatxtrdacontentcrdamediacrrdacarrierPainlevé III: A Case Study in the Geometry of Meromorphic Connections[electronic resource] /by Martin A. Guest, Claus Hertling1st ed. 2017.Cham :Springer International Publishing :Imprint: Springer,2017.1 online resource (XII, 204 p. 12 illus.) Lecture Notes in Mathematics,0075-8434 ;21983-319-66525-1 Includes bibliographical references and index.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.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.Lecture Notes in Mathematics,0075-8434 ;2198Differential equationsAlgebraic geometrySpecial functionsFunctions of complex variablesOrdinary Differential Equationshttps://scigraph.springernature.com/ontologies/product-market-codes/M12147Algebraic Geometryhttps://scigraph.springernature.com/ontologies/product-market-codes/M11019Special Functionshttps://scigraph.springernature.com/ontologies/product-market-codes/M1221XFunctions of a Complex Variablehttps://scigraph.springernature.com/ontologies/product-market-codes/M12074Differential equations.Algebraic geometry.Special functions.Functions of complex variables.Ordinary Differential Equations.Algebraic Geometry.Special Functions.Functions of a Complex Variable.515.352Guest Martin Aauthttp://id.loc.gov/vocabulary/relators/aut67285Hertling Clausauthttp://id.loc.gov/vocabulary/relators/autMiAaPQMiAaPQMiAaPQBOOK996466522203316Painlevé III: A Case Study in the Geometry of Meromorphic Connections1964434UNISA