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Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions

✍ Scribed by Raimundas Vidunas; Alexander V. Kitaev

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
286 KB
Volume
280
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

In 1991, one of the authors showed the existence of quadratic transformations between the Painlevé VI equations with local monodromy differences (1/2, a, b, ±1/2) and (a, a, b, b). In the present paper we give concise forms of these transformations. They are related to the quadratic transformations obtained by Manin and Ramani–Grammaticos–Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particularly important as it can be realized as a quadratic‐pull back transformation of isomonodromic Fuchsian equations. The new formulas are illustrated by derivation of explicit expressions for several complicated algebraic Painlevé VI functions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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