Skip to main content
SpringerLink
Log in

On the properties of solutions of equations in the K 2 hierarchy

  • Ordinary Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

For the sixth-order equation of the K 2 hierarchy, we establish conditions for the existence of families of polar solutions, construct polynomial Hamiltonians and Bäcklund transformations, represent meromorphic solutions via a ratio of special entire functions, and give a criterion for the existence of rational solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ince, E.L., Ordinary Differential Equations, London: Lonmans, Green & Co., 1926. Translated under the title Obyknovennye differentsial’nye uravneniya, Kharkov: GONTI, 1939.

    Google Scholar 

  2. Ablowitz, M.J., Kaup, D., Newell, A., and Segur, H., Stud. Appl. Math., 1974, vol. 53, pp. 249–315.

    MathSciNet  Google Scholar 

  3. Ablowitz, M., Ramani, A., and Segur, H., Lett. Nuovo Cim., 1978, vol. 23, pp. 333–338.

    Article  MathSciNet  Google Scholar 

  4. Flashka, H. and Newell, A.C., Comm. Math. Phys., 1980, vol. 76, pp. 65–116.

    Article  MathSciNet  Google Scholar 

  5. Airault, H., Stud. Appl. Math., 1979, vol. 61, pp. 31–53.

    MathSciNet  MATH  Google Scholar 

  6. Jimbo, M., Miwa, T., and Ueno, K., Phys. D, 1981, vol. 2, pp. 306–352.

    Article  MathSciNet  Google Scholar 

  7. Pickering, A., Teoret. Mat. Fiz., 2003, vol. 137, pp. 445–456, 1733–1742.

    MathSciNet  Google Scholar 

  8. Gordia, P.R. and Pickering, A., J. Math. Phys., 1999, vol. 40, pp. 5749–5786.

    Article  MathSciNet  Google Scholar 

  9. Clarkson, P.A., Joshi, N., and Pickering A., Inverse Problems, 1999, vol. 15, pp. 175–187.

    Article  MathSciNet  MATH  Google Scholar 

  10. Conte, R., in The Painlevé Property, CRM Ser. Math. Phys., New York, 1999, pp. 77–180.

    MATH  Google Scholar 

  11. Okamoto, K., in The Painlevé Property, CRM Ser. Math. Phys., New York, 1999, pp. 735–788.

    Google Scholar 

  12. Cosgrove, C.M., Stud. Appl. Math., 2000, vol. 104, pp. 1–65.

    Article  MathSciNet  MATH  Google Scholar 

  13. Kudryashov, N.A., Phys. Lett. A, 1997, vol. 224, pp. 353–360.

    Article  MathSciNet  Google Scholar 

  14. Noumi, M., Painlevé Equations Trough Symmetry, AMS, Providence, Rhode Island, 2004, vol. 223.

    Google Scholar 

  15. Ablowitz, M. and Clarkson, P.A., Solitons, Nonlinear Evolutions Equations and Inverse Scattering, Lect. Notes Math, vol. 149. Cambridge, 1991.

  16. Iwasaki, K., Kimura, H., Shimomura, Sh., and Yoshida, M., From Gauss to Painlevé. A Modern Theory of Special Functions, Aspects of Mathematics E16, Braunschweig, 1991.

    MATH  Google Scholar 

  17. Gromak, V.I., Laine, I., and Shimomura, S., Painlevé Differential Equations in the Complex Plane, vol. 28, Berlin; New York, 2002.

  18. Kudryashov, N.A., Teoret. Mat. Fiz., 2000, vol. 122, pp. 72–87.

    MathSciNet  MATH  Google Scholar 

  19. Kudryashov, N.A., ANZIAM J., 2002, vol. 44, pp. 149–160.

    Article  MathSciNet  MATH  Google Scholar 

  20. Hone Andrew, N.W., Phys. D, 1998, vol. 118, pp. 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kudryashov, N.A., Phys. Lett. A, 2000, vol. 273, pp. 194–202.

    Article  MathSciNet  MATH  Google Scholar 

  22. Gromak, V.I., Differ. Uravn., 1984, vol. 20, no. 12, pp. 2042–2048.

    MathSciNet  Google Scholar 

  23. Gromak, V.I., Differ. Uravn., 2006, vol. 42, no. 8, pp. 1017–1026.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Belarus State University, Minsk, Belarus

    V. I. Gromak

Authors
  1. V. I. Gromak
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © V.I. Gromak, 2008, published in Differentsial’nye Uravneniya, 2008, Vol. 44, No. 2, pp. 172–180.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gromak, V.I. On the properties of solutions of equations in the K 2 hierarchy. Diff Equat 44, 176–185 (2008). https://doi.org/10.1134/S0012266108020043

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266108020043

Keywords

Search

Navigation