Generalization of Okamoto's equation to arbitrary 2x2 Schlesinger systems

Abstract : The 2x 2 Schlesinger system for the case of four regular singularities is equivalent to the Painleve VI equation. The Painleve VI equation can in turn be rewritten in the symmetric form of Okamoto's equation; the dependent variable in Okamoto's form of the PVI equation is the (slightly transformed) logarithmic derivative of the Jimbo-Miwa tau-function of the Schlesinger system. The goal of this note is twofold. First, we find a symmetric uniform formulation of an arbitrary Schlesinger system with regular singularities in terms of appropriately defined Virasoro generators. Second, we find analogues of Okamoto's equation for the case of the 2 x2 Schlesinger system with an arbitrary number of poles. A new set of scalar equations for the logarithmic derivatives of the Jimbo-Miwa tau-function is derived in terms of generators of the Virasoro algebra; these generators are expressed in terms of derivatives with respect to singularities of the Schlesinger system.
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Advances in Mathematical Physics, Hindawi Publishing Corporation, 2009, 2009 (2009), pp.461860. 〈10.1155/2009/461860〉
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Dmitrii Korotkin, Henning Samtleben. Generalization of Okamoto's equation to arbitrary 2x2 Schlesinger systems. Advances in Mathematical Physics, Hindawi Publishing Corporation, 2009, 2009 (2009), pp.461860. 〈10.1155/2009/461860〉. 〈ensl-00401421〉

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