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Special polynomials and rational solutions of the hierarchy of the second Painlevé equation

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Abstract

We study special polynomials used to represent rational solutions of the hierarchy of the second Painlevé equation. We find several recursion relations satisfied by these polynomials. In particular, we obtain a differential-difference relation that allows finding any polynomial recursively. This relation is an analogue of the Toda chain equations.

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References

  1. M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM Stud. Appl. Math., Vol. 4), SIAM, Philadelphia (1981).

    MATH  Google Scholar 

  2. A. Newell, Solitons in Mathematics and Physics (CBMS-NSF Regional Conf. Ser. in Appl. Math., Vol. 48), SIAM, Philadelphia (1985).

    Google Scholar 

  3. N. A. Kudryashov, Analytic Theory of Nonlinear Differential Equations [in Russian], Inst. Computer Studies, Moscow (2004).

    Google Scholar 

  4. A. I. Yablonskii, Vesti AN BSSR, Ser. fiz.-tech. nauk, 3, 30–35 (1959).

    Google Scholar 

  5. A. P. Vorob’ev, Differential Equations, 1, 58–59 (1965).

    MATH  Google Scholar 

  6. P. A. Clarkson, Phys. Lett. A, 319, 137–144 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. K. Okamoto, Math. Ann., 275, 221–255 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Umemura, Sugaki Expositions, 11, No. 1, 77–100 (1998).

    MathSciNet  Google Scholar 

  9. N. A. Kudryashov, Phys. Lett. A, 224, 353–360 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  10. P. A. Clarkson and E. L. Mansfield, Nonlinearity, 16, No. 3, R1–R26 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. M. V. Demina and N. A. Kudryashov, Chaos Solitons Fractals, 32, 526–537 (2007).

    Article  MathSciNet  Google Scholar 

  12. N. A. Kudryashov and A. Pickering, “Rational and special solutions of the P II hierarchy, ” in: SIDE III: Symmetries and Integrability of Difference Equations (CRM Proc. Lecture Notes, Vol. 25, D. Levi and O. Ragnisco, eds.), Amer. Math. Soc., Providence, R. I. (2000), pp. 245–253.

    Google Scholar 

  13. V. I. Gromak, I. Laine, and S. Shimomura, Painlevé Differential Equations in the Complex Plane (de Gruyter Stud. Math., Vol. 28), de Gruyter, Berlin (2002).

    MATH  Google Scholar 

  14. N. A. Kudryashov and M. B. Soukharev, Phys. Lett. A, 237, 206–216 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. H. Airault, Stud. Appl. Math., 61, 31–53 (1979).

    MATH  ADS  MathSciNet  Google Scholar 

  16. A. N. W. Hone, Phys. D, 118, 1–16 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  17. V. É. Adler, A. B. Shabat, and R. I. Yamilov, Theor. Math. Phys., 125, 1603–1661 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  18. V. G. Marikhin and A. B. Shabat, Theor. Math. Phys., 118, 173–182 (1999).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Moscow Engineering Physical Institute (State University), Moscow, Russia

    M. V. Demina & N. A. Kudryashov

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  1. M. V. Demina
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  2. N. A. Kudryashov
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Correspondence to M. V. Demina.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 58–67, October, 2007.

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Demina, M.V., Kudryashov, N.A. Special polynomials and rational solutions of the hierarchy of the second Painlevé equation. Theor Math Phys 153, 1398–1406 (2007). https://doi.org/10.1007/s11232-007-0123-z

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  • DOI: https://doi.org/10.1007/s11232-007-0123-z

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