Petersburg Department of Steklov Institute of Mathematics

PREPRINT 20/1997

P. Deift, A. Its, A. Kapaev, X. Zhou

A note on the algebro-geometric integration of the Schlesinger equations and on the elliptic solutions of the Painlev\'e VI equation

This preprint was accepted October, 1997.
Contact: A.Kapaev

A new approach to the construction of isomonodromy deformations of the $2\times2$ Fuchsian systems is presented. The method is based on a combination of the algebrogeometric scheme and Riemann-Hilbert approach of the theory of integrable systems. For a given number $2g+1$, $g\geq 1$, of the finite (regular) singularities, the method produces a $2g$- parameter submanifold of the Fuchsian monodromy data for which the relevant Riemann-Hilbert problem can be solved in closed form via the Baker-Akhiezer function technique. This in turn leads to a $2g$-parameter family of solutions of the corresponding Schlesinger equations, explicitly described in terms of Riemann theta functions of genus $g$. In the case $g=1$ the solution found coincides with the general elliptic solution of the particular case of Painlev\'e VI equation first discovered by N. J. Hitchin [H1].

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