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Algebro-Geometric Approach to an Okamoto Transformation, the Painlevé VI and Schlesinger Equations

  • Vladimir DragovićEmail author
  • Vasilisa Shramchenko
Article
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Abstract

A new method of constructing algebro-geometric solutions of rank two four-point Schlesinger system is presented. For an elliptic curve represented as a ramified double covering of \(\mathbb {CP}^1\), a meromorphic differential is constructed with the following property: The common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of a Painlevé VI equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painlevé VI equations. The corresponding solution of the rank two Schlesinger system associated with a family of elliptic curves is constructed in terms of this differential. The initial data for construction of the meromorphic differential include a point in the Jacobian of the curve, under the assumption that this point has non-variable coordinates with respect to the lattice of the Jacobian while the branch points vary. It appears that the cases where the coordinates of the point are rational correspond to the Poncelet polygons inscribed and circumscribed in a pair of conics. Thus, this is a generalization of a situation studied by Hitchin, who related algebraic solutions of a Painlevé VI equation with the Poncelet polygons.

Mathematics Subject Classification

34M55 34M56 (33E05, 14H70) 

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Notes

Acknowledgements

The authors thank D. Korotkin for useful discussions and the referee for suggestions which improved the presentation. The research has been partially supported by the NSF Grant 1444147. The research of the first author has been partially supported by the Grant 174020 “Geometry and topology of manifolds, classical mechanics, and integrable dynamical systems” of the Ministry of Education and Sciences of Serbia and by the University of Texas at Dallas. The second author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada, Fonds de recherche du Québec Nature et Technologies (grant in the program “Établissement de nouveaux chercheurs universitaires”) and the University of Sherbrooke

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Texas at DallasRichardsonUSA
  2. 2.Mathematical Institute SANUBelgradeSerbia
  3. 3.Department of MathematicsUniversity of SherbrookeSherbrookeCanada

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