We study the reducibility of an isomonodromic family of Fuchsian systems on the Riemann sphere which is determined by some initial Fuchsian system and its monodromy to an isomonodromic family of upper triangular Fuchsian systems. Under some conditions on the monodromy of the initial Fuchsian system, a nonlinear Schlesinger system is reduced to a system of homogeneous and inhomogeneous linear Pfaffian Jordan–Pochhammer differential equations. Such Jordan–Pochhammer systems possess hypergeometric type solutions admitting an explicit description of their integral representations.
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References
A. A. Bolibruch, Inverse Monodromy Problems of the Analytic Theory of Differential Equations, Springer, Berlin (2006).
R. R. Gontsov and I. V. V’yugin, “Solvability of linear differential systems with small exponents in Liouvillian sense,” Arnold Math J. 1, No. 4, 445–471 (2015).
S. Malek, “On the reducibility of the Schlesinger equations,” J. Dyn. Control Syst. 8, No. 4, 505–527 (2002).
R. R. Gontsov and V. P. Leksin, “On the reducibility of Schlesinger isomonodromic families,” In: Analytic Methods of Analysis and Differential Equations: AMADE pp. 21–34, Cambridge Scientific Publishers, Cambridge (2014).
V. Dragović, R. R. Gontsov, and V. Schramchenko, “Triangular Schlesinger systems and superelliptic curves,” Physica D 424, Article ID 132947 (2021).
M. Kapovich and J. Millson, “Quantization of bending deformations of polygons in 𝔼3, hypergeometric integrals and the Gassner representation,” Can. Math. Bull. 44, No. 1, 36–60 (2001).
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Translated from Problemy Matematicheskogo Analiza 123, 2023, pp. 85-90.
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Leksin, V.P. Reduction of Schlesinger Systems to Linear Jordan–Pochhammer Systems. J Math Sci 270, 714–720 (2023). https://doi.org/10.1007/s10958-023-06382-5
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DOI: https://doi.org/10.1007/s10958-023-06382-5