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 TMF, 2000, Volume 123, Number 3, Pages 395–406 (Mi tmf611)

Isomonodromic deformations of Heun and Painlevé equations

S. Yu. Slavyanov

Saint-Petersburg State University

Abstract: Continuing the study of the relationship between the Heun and the Painlevé classes of equations reported in two previous papers, we formulate and prove the main theorem expressing this relationship. We give a Hamiltonian interpretation of the isomonodromic deformation condition and propose an alternative classification of the Painlevé equations, which includes ten equations.

DOI: https://doi.org/10.4213/tmf611

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English version:
Theoretical and Mathematical Physics, 2000, 123:3, 744–753

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Citation: S. Yu. Slavyanov, “Isomonodromic deformations of Heun and Painlevé equations”, TMF, 123:3 (2000), 395–406; Theoret. and Math. Phys., 123:3 (2000), 744–753

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf/v123/i3/p395

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Tarasov, VF, “The Heun-Schrodinger radial equation with two auxiliary parameters for H-like atoms”, Modern Physics Letters B, 16:25 (2002), 937
2. Slavyanov S.Y., “Kovalevskaya's dynamics and Schrodinger equations of Heun class”, Operator Methods in Ordinary and Partial Differential Equations, Operator Theory : Advances and Applications, 132, 2002, 395–402
3. S. Yu. Slavyanov, F. R. Vukailovich, “Isomonodromic deformations and “antiquantization” for the simplest ordinary differential equations”, Theoret. and Math. Phys., 150:1 (2007), 123–131
4. V. V. Tsegel'nik, “Hamiltonians associated with the sixth Painlevé equation”, Theoret. and Math. Phys., 151:1 (2007), 482–491
5. B. I. Suleimanov, ““Quantizations” of the second Painlevé equation and the problem of the equivalence of its $L$$A$ pairs”, Theoret. and Math. Phys., 156:3 (2008), 1280–1291
6. 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”, Russian Math. Surveys, 64:1 (2009), 45–127
7. V. V. Tsegel'nik, “Hamiltonians associated with the third and fifth Painlevé equations”, Theoret. and Math. Phys., 162:1 (2010), 57–62
8. Slavyanov S., “Antiquantization of Deformed Equations of Heun Class”, Proceedings of the International Conference Days on Diffraction 2015, IEEE, 2015, 310–312
9. Rumanov I., “Beta Ensembles, Quantum Painlevé Equations and Isomonodromy Systems”, Algebraic and Analytic Aspects of Integrable Systems and Painlev? Equations, Contemporary Mathematics, 651, ed. Dzhamay A. Maruno K. Ormerod C., Amer Mathematical Soc, 2015, 125–155
10. S. Yu. Slavyanov, O. L. Stesik, “Antiquantization of deformed Heun-class equations”, Theoret. and Math. Phys., 186:1 (2016), 118–125
11. Combot T., “Integrability of the One Dimensional Schrodinger Equation”, J. Math. Phys., 59:2 (2018), 022105
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