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Communications in Mathematical Physics

, Volume 203, Issue 3, pp 613–633 | Cite as

On the Algebro-Geometric Integration¶of the Schlesinger Equations

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

Abstract:

A new approach to the construction of isomonodromy deformations of 2× 2 Fuchsian systems is presented. The method is based on a combination of the algebro-geometric scheme and Riemann–Hilbert approach of the theory of integrable systems. For a given number 2g+ 1, g≥ 1, of 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 the Painlevé VI equation first obtained by N. J. Hitchin [H1].

Keywords

Closed Form Integrable System Function Technique Hilbert Problem Fuchsian System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • P. Deift
    • 1
  • A. Its
    • 2
  • A. Kapaev
    • 3
  • X. Zhou
    • 4
  1. 1.Courant Institute of Mathematical Sciences, New York, NY 10003, USAUS
  2. 2.Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, Indianapolis, IN 46202-3216, USA. E-mail: itsa@math.iupui.eduUS
  3. 3.St. Petersburg Branch of Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg, 191011, RussiaRU
  4. 4.Department of Mathematics, Duke University, Durham, NC 27708-0320, USAUS

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