Skip to main content
SpringerLink
Log in

Nonlinear fourth-order differential equations with solutions in the form of transcendents

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We present two hierarchies of ordinary differential equations and give the relations between these hierarchies. We find rational and special solutions of one hierarchy. Solutions of two differential equations are shown to be essentially transcendental functions with respect to the integration constants.

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. N. A. Kudryashov,J. Phys. A,31, L129 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  4. M. J. Ablowitz, A. Ramani, and H. Segur,J. Math. Phys.,21, 715, 1006 (1980).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. M. J. Ablowitz and P. A. Clarkson,Solitons, Nonlinear Evolution Equations, and Inverse Scattering, Cambridge Univ. Press, Cambridge (1991).

    MATH  Google Scholar 

  6. P. J. Caudrey, R. K. Dodd, and J. D. Gibbon,Proc. Roy Soc. London A,351, 407 (1976).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. R. K. Dodd and J. D. Gibbon,Proc. Roy Soc. London A,358, 287 (1977).

    ADS  MathSciNet  Google Scholar 

  8. B. Kuperschmidt and G. Wilson,Invent. Math.,62, 403 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  9. J. Weiss,J. Math. Phys.,25, 13 (1984).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. J. Weiss,J. Math. Phys.,27, 1293 (1986).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. R. Conte, “The Painlevé approach to nonlinear ordinary differential equations”, in:The Painlevé Property, One Century Later (R. Conte, ed.) (Proc. Sci. School in Corgèse, 3–22 June 1966. CRM Series in Math. Phys.), Springer, New York (1999), p. 77; Preprint solv-int/9710020 (1997).

    Google Scholar 

  12. R. Conte, A. P. Fordy, and A. Pickering,Physica D,69, 33 (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. N. A. Kudryashov,J. Phys. A,27, 2457 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. N. A. Kudryashov,J. Phys. A,30, 5445 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. N. A. Kudryashov and A. Pickering,J. Phys. A (forthcoming).

  16. E. L. Ince,Ordinary Differential Equations [in Russian], GNTI Ukraine, Khar'kov (1939); English transl., Dover, New York (1956).

    Google Scholar 

  17. V. I. Gromak and N. A. Lukashevich,Analytic Properties of Solutions of the Painlevé Equations [in Russian], Universitetskoe, Minsk (1990).

    Google Scholar 

  18. V. V. Golubev,Lecture on the Analytic Theory of Differential Equations [in Russian], GITTL, Moscow (1941).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Engineering Physics Institute, Moscow, Russia

    N. A. Kudryashov

Authors
  1. N. A. Kudryashov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 72–87, January, 1999.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kudryashov, N.A. Nonlinear fourth-order differential equations with solutions in the form of transcendents. Theor Math Phys 122, 58–71 (2000). https://doi.org/10.1007/BF02551170

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02551170

Keywords

Search

Navigation