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Essential singularity of the second-genus Painlevé function and the nonlinear stokes phenomenon

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Abstract

By the isomonodromic deformation method, the leading term of the elliptic asymptotics as x→∞ of the solution of the second Painlevé equation is constructed in the generic case. The equations for the modulus of this elliptic sine (which depends only on arg x) are given. The phase of the elliptic sine for any arg x is explicitly expressed in terms of first integrals of the Painlevé equation, i.e., in terms of the Stokes multipliers of the associated linear system. A nonlinear Stokes phenomenon typical for the asymptotic behavior of the Painlevé function is described. Bibliography: 25 titles.

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Literature Cited

  1. P. Painlevé,Bull. Soc. Math. France,28, 201–261 (1900).

    MATH  MathSciNet  Google Scholar 

  2. P. Painlevé,Acta Math.,25, 1–86 (1902).

    Google Scholar 

  3. B. Gambier,Acta Math.,33, 1–55 (1910).

    MathSciNet  Google Scholar 

  4. V. V. Golubev,Math. Sb.,28, No. 2, 323–349 (1912).

    MATH  MathSciNet  Google Scholar 

  5. P. Boutroux,Ann. Sci. Ec. Norm. Sup.,30, 255–376 (1900).

    MathSciNet  Google Scholar 

  6. H. Flaschka and A. C. Newell,Commun. Math. Phys.,76, No. 1, 65–111 (1980).

    MathSciNet  Google Scholar 

  7. M. Jimbo and T. Miwa,Physica D,2, No. 3, 407–448 (1981).

    MathSciNet  Google Scholar 

  8. R. Fuchs,Math. Ann.,63, 301–321 (1907).

    Article  MATH  MathSciNet  Google Scholar 

  9. L. Schlesinger,J. Rein. Angew. Math.,141, 96–145 (1912).

    MATH  Google Scholar 

  10. R. Garnier,Ann. Sci. Ec. Norm. Sup.,29, 1–126 (1912).

    MATH  MathSciNet  Google Scholar 

  11. A. R. It-s,Izv. Akad. Nauk SSSR, Ser. Mat.,49, No. 3, 530–565 (1985).

    MathSciNet  Google Scholar 

  12. V. Yu. Novokshenov,Funkts. Anal. Prilozh.,19, No. 3, 90–91 (1984).

    MathSciNet  Google Scholar 

  13. A. R. It-s (Its) and V. Yu. Novokshenov,Lecture Notes in Math.,1181, 1–313 (1986). A. S. Fokas and M. J. Ablowitz,Commun. Math. Phys.,91, 381 (1983).

    Google Scholar 

  14. A. S. Fokas, U. Mugan, and M. J. Ablowitz,M. J. Phys. D,30, 247–283 (1988).

    MathSciNet  Google Scholar 

  15. A. R. It-s and A. A. Kapaev,Izv. Akad. Nauk SSSR, Ser. Mat.,51, No. 4, 873–892 (1987).

    Google Scholar 

  16. V. Yu. Novokshenov,Dokl. Akad. Nauk SSSR,283, No. 5, 1161–1165 (1985).

    MATH  MathSciNet  Google Scholar 

  17. A. V. Kitaev,Teor. Math. Fiz.,64, No. 3, 347–369 (1985).

    MATH  MathSciNet  Google Scholar 

  18. B. M. McCoy and Sh. Tang,Physica D,19, 42–72 (1986);20, 187–216 (1986).

    Article  MathSciNet  Google Scholar 

  19. A. V. Kitaev,Math. Sb.,134, No. 11, 421–444 (1987).

    MATH  Google Scholar 

  20. A. A. Kapaev,Teor. Math. Fiz.,77, No. 3, 323–332 (1988).

    MATH  MathSciNet  Google Scholar 

  21. A. A. Kapaev, (Self-review of the Candidate Thesis), LGU, Leningrad (1987).

    Google Scholar 

  22. M. V. Fedoryuk,Asymptotic Methods for Ordinary Differential Equations [in Russian], Moscow (1983).

  23. H. Bateman and A. Erdelyi,Higher Transcendental Functions, Mc Graw-Hill, New York-Toronto-London (1953).

    Google Scholar 

  24. V. Yu. Novokshenov,Dokl. Akad. Nauk SSSR,311, No. 2, 288–291 (1990).

    MATH  MathSciNet  Google Scholar 

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Authors
  1. A. A. Kapaev
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Additional information

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 187, pp. 139–170, 1990.

Translated by O. A. Ivanov.

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Kapaev, A.A. Essential singularity of the second-genus Painlevé function and the nonlinear stokes phenomenon. J Math Sci 73, 500–517 (1995). https://doi.org/10.1007/BF02364572

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  • DOI: https://doi.org/10.1007/BF02364572

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