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Uspekhi Mat. Nauk, 2009, Volume 64, Issue 1(385), Pages 51–134 (Mi umn9261)  

This article is cited in 10 scientific papers (total in 10 papers)

On canonical parametrization of the phase spaces of equations of isomonodromic deformations of Fuchsian systems of dimension $2\times 2$. Derivation of the Painlevé VI equation

M. V. Babich

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: This survey considers the factorization, by linear changes of the sought vector-function, of the manifold of $2\times 2$ matrix linear differential equations of first order with simple poles on the right-hand side. It is shown how under a parametrization of such quotient manifolds there naturally appear the Garnier–Painlevé VI equations, as well as algebro-geometric constructions related to them: the Okamoto surface and a rational atlas of the Darboux coordinates on it.

DOI: https://doi.org/10.4213/rm9261

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English version:
Russian Mathematical Surveys, 2009, 64:1, 45–127

Bibliographic databases:

UDC: 517.912
MSC: Primary 34-02, 34M55; Secondary 33E17, 34A26, 34A30, 34M35, 37J05
Received: 20.10.2008

Citation: M. V. Babich, “On canonical parametrization of the phase spaces of equations of isomonodromic deformations of Fuchsian systems of dimension $2\times 2$. Derivation of the Painlevé VI equation”, Uspekhi Mat. Nauk, 64:1(385) (2009), 51–134; Russian Math. Surveys, 64:1 (2009), 45–127

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. V. Brezhnev, “A $\tau$-function solution of the sixth Painlevé transcendent”, Theoret. and Math. Phys., 161:3 (2009), 1616–1633  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Zabrodin A., Zotov A., “Quantum Painlevé-Calogero correspondence for Painlevé. VI”, J. Math. Phys., 53:7 (2012), 073508, 19 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Babich M.V., “On birational Darboux coordinates of isomonodromic deformation equations phase space”, Painlevé equations and related topics, eds. Bruno A., Batkhin A., Walter de Gruyter, Berlin, 2012, 91–94  mathscinet  isi
    4. S. Yu. Slavyanov, “Derivation of Painlevé equations by antiquantization”, Painlevé equations and related topics, eds. Bruno A., Batkhin A., Walter de Gruyter, Berlin, 2012, 253–256  mathscinet  isi
    5. S. Y. Slavyanov, “Relations between linear equations and Painlevé's equations”, Constr. Approx., 39:1 (2014), 75–83  crossref  mathscinet  zmath  isi  scopus
    6. S. Yu. Slavyanov, “Polynomial degree reduction of a Fuchsian $2{\times}2$ system”, Theoret. and Math. Phys., 182:2 (2015), 182–188  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. J. Math. Sci. (N. Y.), 209:6 (2015), 910–921  mathnet  crossref
    8. D. P. Novikov, B. I. Suleimanov, ““Quantization” of an isomonodromic Hamiltonian Garnier system with two degrees of freedom”, Theoret. and Math. Phys., 187:1 (2016), 479–496  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. Its A.R., Prokhorov A., “On Some Hamiltonian Properties of the Isomonodromic Tau Functions”, Rev. Math. Phys., 30:7, SI (2018), 1840008  crossref  mathscinet  isi  scopus
    10. M. V. Babich, S. Yu. Slavyanov, “Links from second-order Fuchsian equations to first-order linear systems”, J. Math. Sci. (N. Y.), 240:5 (2019), 646–650  mathnet  crossref
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