×

zbMATH — the first resource for mathematics

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)].

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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)].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.