Math @ Duke

Publications [#244269] of Xin Zhou
Papers Published
 with Deift, P; Its, A; Kapaev, A; Zhou, X, On the algebrogeometric integration of the Schlesinger equations,
Communications in Mathematical Physics, vol. 203 no. 3
(1999),
pp. 613633 [MR2000f:34183]
(last updated on 2018/10/23)
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 algebrogeometric scheme and RiemannHilbert approach of the theory of integrable systems. For a given number 2g + 1, g ≥ 1, of finite (regular) singularities, the method produces a 2gparameter submanifold of the Fuchsian monodromy data for which the relevant RiemannHilbert problem can be solved in closed form via the BakerAkhiezer function technique. This in turn leads to a 2gparameter 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].


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