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Solving the Sixth Painlevé Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae

TitleSolving the Sixth Painlevé Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae
Publication TypeJournal Article
Year of Publication2012
AuthorsGuzzetti, D
JournalInt Math Res Notices (2012) 2012 (6): 1352-1413
Abstract

The critical behavior of a three real parameter class of solutions of the\\r\\nsixth Painlev\\\\\\\'e equation is computed, and parametrized in terms of monodromy\\r\\ndata of the associated $2\\\\times 2$ matrix linear Fuchsian system of ODE. The\\r\\nclass may contain solutions with poles accumulating at the critical point. The\\r\\nstudy of this class closes a gap in the description of the transcendents in one\\r\\nto one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of solutions are\\r\\nmissing in the literature, so they are computed here. A computational procedure to write the full expansion of the four and three real parameter class of solutions is proposed.

URLhttp://hdl.handle.net/1963/6093
DOI10.1093/imrn/rnr071

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