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On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

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Abstract

We give a review of the modern theory of isomonodromic deformations of Fuchsian systems discussing both classical and modern results, such as a general form of the isomonodromic deformations of Fuchsian systems, their differences from the classical Schlesinger deformations, the Fuchsian system moduli space structure and the geometric meaning of new degrees of freedom appeared in a non-Schlesinger case. Using this we illustrate some general relations between such concepts as integrability, isomonodromy and Painlevé property.

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References

  1. Boalch, P.: From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. Lond. Math. Soc. 90(3), 167–208 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Boalch, P.: Six results on Painlevé VI. In: SMF, Sèminaires et Congrés, vol. 14

  3. Bolibruch, A.: Fuchsian differential equations and holomorphic vector bundle II, MCCME (2008, to appear)

  4. Bolibruch, A.: On isomonodromic confluences of Fuchsian singularities. Proc. Steklov Inst. Math. 221, 127–142 (1998)

    Google Scholar 

  5. Fuchs, L.: Zur Theorie der linearen Differentialgleichungen mit veränderlichen Koeffizienten. J. Math. 68, 354–385 (1868)

    Google Scholar 

  6. Its, A., Novokshenov, V.: The isomonodromic deformation method in the theory of Painlevé equations. In: Lecture Notes in Mathematics, vol. 1191. Springer, New York (1986)

    Google Scholar 

  7. Jimbo, M., Miwa, T.: Monodromy preserving deformations of linear differential equations with rational coefficients II. Physica D 2, 407–448 (1981)

    Article  MathSciNet  Google Scholar 

  8. Levin, A., Olshanetsky, M., Zotov, A.: Painlevé VI, rigid tops and reflection equation. Commun. Math. Phys. 268, 67–103 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Malgrange, B.: Sur les déformations isomonodromiques I. In: Progr. Math., vol. 37. Birkhäuser, Boston (1983)

    Google Scholar 

  10. Novikov, D., Yakovenko, S.: Lectures on meromorphic flat connections. In: Ilyashenko, Y., Rousseau, C. (eds.) Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, pp. 387–430. Kluwer, Dordrecht (2004)

    Google Scholar 

  11. Poberezhny, V.: On the moduli space of Fuchsian systems. Math. Notes (2008, to appear)

  12. Poberezhny, V.: General linear problem for isomonodromic deformations of Fuchsian systems. Math. Notes 81(4), 529–542 (2007)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. ITEP, B. Cheremushkinskaya, 25, Moscow, 117259, Russia

    Vladimir Poberezhny

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  1. Vladimir Poberezhny
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Correspondence to Vladimir Poberezhny.

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The work is supported by N.Sh.-6849.2006.1 and RFBR 07-01-00526 grants.

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Poberezhny, V. On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems. Acta Appl Math 101, 255–263 (2008). https://doi.org/10.1007/s10440-008-9189-3

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  • DOI: https://doi.org/10.1007/s10440-008-9189-3

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