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On solutions of the Schlesinger equation in the neigborhood of the Malgrange \(\Theta \)-divisor. (English. Russian original) Zbl 1169.34340
Math. Notes 83, No. 5, 707-711 (2008); translation from Mat. Zametki 83, No. 5, 779-782 (2008).
The author specifies the structure of the solution of the Schlesinger equation
\[ dB_{i}(a)=-\sum_{j=1,\, j\neq i}^{n}\frac{[B_{i}(a),\, B_{j}(a)]}{a_{i}-a_{j}}d(a_{i}-a_{j}) \]
in the neighborhood of the \(\Theta\)-divisor. The singularities of the solution \(B(a)\) are determined by means of a method proposed by Bolibrukh for calculating the local \(\tau\)-function [see A. A. Bolibrukh, Math. Notes, 74, 184–191 (2003; Zbl 1068.34082)].

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
Full Text: DOI
[1] B. Malgrange, ”Sur les déformations isomonodromiques. I: Singularités régulières,” in Mathematics and Physics, Progr. Math., Paris, 1979/1982 (Birkhäuser Boston, Boston, MA, 1983), Vol. 37, pp. 401–426.
[2] M. Jimbo and T. Miwa, ”Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II,” Phys. D 2(3), 407–448 (1981). · Zbl 1194.34166
[3] A. A. Bolibrukh, ”On the tau function for the Schlesinger equation of isomonodromic deformations,” Mat. Zametki, 74(2), 177–184 (2003) [Math. Notes, 74 (1–2), 184–191 (2003)]. · Zbl 1068.34082
[4] A. A. Bolibrukh, ”On Isomonodromic Confluences of Fuchsian Singularities,” Trudy Mat. Inst. Steklov 221, 127–142 (1998) [Proc. Steklov Inst. Math. 221, 117–132 (1998)]. · Zbl 0951.34069
[5] A. A. Bolibrukh, Fuchsian Differential Equations and Holomorphic Fibering, in Modern Courses (MTsNMO, Moscow, 2000) [in Russian].
[6] I. V. V’yugin and R. R. Gontsov, ”Additional parameters in inverse monodromy problems,” Mat. Sb. 197(12), 43–64 (2006) [Russian Acad. Sci. Sb. Math. 197 (12), 1753–1773 (2006)].
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