03152nam a2200421 i 4500991003628049707536m o d cr cnu|||unuuu190321s2017 sz a ob 001 0 eng d9783319665269331966526X10.1007/978-3-319-66526-9b14362491-39ule_instBibl. Dip.le Aggr. Matematica e Fisica - Sez. Matematicaeng515.35523AMS 34-02AMS 32G20AMS 34M55Guest, Martin A.67285Painlevé III :a case study in the geometry of Meromorphic connections[e-book] /Martin A. Guest, Claus HertlingCham :Springer,20171 online resource (xii, 204 pages) :illustrationstexttxtrdacontentcomputercrdamediaonline resourcecrrdacarrierLecture notes in mathematics,0075-8434 ;2198Includes bibliographical references and index1. 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 Painlevé equations -- 10. Solutions of the Painlevé equation PIII (0, 0, 4, −4) -- 13. Comparison with the setting of Its, Novokshenov, and Niles -- 12.  Asymptotics of all solutions near 0 -- ...Bibliography. IndexThe 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 of 0 is givenPainlevé equationsHertling, Clausauthorhttp://id.loc.gov/vocabulary/relators/aut66890Printed edition:9783319665252An electronic book accessible through the World Wide Webhttps://link.springer.com/book/10.1007/978-3-319-66526-9.b1436249103-03-2221-03-19991003628049707536Painlevé III1749801UNISALENTOle01321-03-19m@ -engsz 00