2015/06/20 by Vladimir Dragović, Vasilisa Shramchenko, Dragovic, Vladimir +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematics and Applications #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1506.06301
openalex publication_date 2015/06/20 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
A new method to construct algebro-geometric solutions of rank two Schlesinger\nsystems is presented. For an elliptic curve represented as a ramified double\ncovering of CP1, a meromorphic differential is constructed with the following\nproperty: the common projection of its two zeros on the base of the covering,\nregarded as a function of the only moving branch point of the covering, is a\nsolution of a Painleve VI equation. This differential provides an invariant\nformulation of a classical Okamoto transformation for the Painleve VI\nequations. A generalization of this differential to hyperelliptic curves is\nalso constructed. In this case, positions of zeros of the differential provide\npart of a solution of the multidimensional Garnier system. The corresponding\nsolutions of the rank two Schlesinger systems associated with elliptic and\nhyperelliptic curves are constructed in terms of this differential. The initial\ndata for construction of the meromorphic differential include a point in the\nJacobian of the curve, under the assumption that this point has nonvariable\ncoordinates with respect to the lattice of the Jacobian while the branch points\nvary. It appears that the cases where the coordinates of the point are rational\ncorrespond to periodic trajectories of the billiard ordered games associated\nwith g confocal quadrics in (g+1)-dimensional space. This is a generalization\nof a situation studied by Hitchin, who related algebraic solutions of a\nPainleve VI equation with the Poncelet polygons.\n