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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 20/1997

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

Abstract:

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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