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Isomonodromic deformations of Heun and Painlevé equations

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An Erratum to this article was published on 01 October 2000

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.

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References

  1. K. Heun,Math. Annal.,33, 161 (1889).

    Article  Google Scholar 

  2. A. Seeger and W. Lay, eds.,Centennial Workshop on Heun's Equation: Theory, and Applications MPI/MF, Stuttgart (1990).

    Google Scholar 

  3. A. Decarreau, M. C. Dumont-Lepage, P. Maroni, A. Robert and A. Ronveaux,Ann. Soc. Sc. Bruxelles,92, 53 (1978).

    Google Scholar 

  4. A. Decarreau, P. Maroni and A. Robert,Ann. Soc. Sc. Bruxelles,92, 151 (1978).

    Google Scholar 

  5. S. Yu. Slavyanov, W. Lay, and A. Seeger,Classification of Heun's Differential Equations, Oxford Univ. Press, Oxford (1995).

    MATH  Google Scholar 

  6. A. Seeger, W. Lay and S. Yu. Slavyanov,Theor. Math. Phys.,104, 950 (1995).

    Article  Google Scholar 

  7. P. Painlevé,Bull. C. R. Acad. Sci. Paris,126, 1697 (1898);Bull. Soc. Math. France,28, 201 (1900);Acta Math.,25, 1 (1902).

    Google Scholar 

  8. B. Gambier,Acta. Math.,33, 1 (1909).

    Article  MathSciNet  Google Scholar 

  9. R. Garnier,Ann. Sci. École Norm. Sup.,29, 1 (1912).

    Article  MathSciNet  Google Scholar 

  10. R. Fuchs,Math. Annal.,63, 301 (1907).

    Article  Google Scholar 

  11. L. Schlesinger,J. Reine Angew. Math,141, 96 (1912).

    MathSciNet  Google Scholar 

  12. M. Jimbo, T. Miwa and K. Ueno,Physica D,2, 306 (1981); M. Jimbo and T. Miwa,Physica D,2, 407 (1981); M. Jimbo and T. Miwa,Physica D,4, 26 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  13. H. Flashka and A. C. Newell,Commun. Math. Phys.,76, 65 (1980).

    Article  ADS  Google Scholar 

  14. K. Iwasaki, H. Kimura, S. Shimomura, and M. Ioshida,From Gauss to Painlevé: A Modern Theory of Special Functions, Vieweg, Braunschweig (1991).

    Book  Google Scholar 

  15. S. Yu. Slavyanov,J. Phys. A,29, 7329 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  16. J. Malmquist,Ark. Mat. Astr. Fys.,17, 1 (1922-1923).

    Google Scholar 

  17. K. Okamoto,Ann. Math. Pure Appl.,146, 337 (1986).

    Article  Google Scholar 

  18. S. Yu. Slavyanov,Theor. Math. Phys.,119, 393 (1999).

    Article  MathSciNet  Google Scholar 

  19. A. R. Its and V. Yu. Novokshenov,The Isomonodromic Deformation Method in the Theory of Painlevé Equations (Lect. Notes Math., Vol. 1191), Springer, Berlin (1986).

    MATH  Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    S. Yu. Slavyanov

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  1. S. Yu. Slavyanov
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Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 395–406, June, 2000.

A correction to this article is available at http://dx.doi.org/10.1007/BF02551047

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Slavyanov, S.Y. Isomonodromic deformations of Heun and Painlevé equations. Theor Math Phys 123, 744–753 (2000). https://doi.org/10.1007/BF02551029

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