Skip to main content
SpringerLink
Log in

Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

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

Abstract

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known twodimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges.

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. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl., 8, 226–235 (1974).

    Article  MATH  Google Scholar 

  2. E. A. Kuznetsov and A. V. Mikhailov, Theor. Math. Phys., 30, 193–200 (1977).

    Article  Google Scholar 

  3. V. E. Zakharov and A. V. Mikhailov, Sov. Phys. JETP, 47, 1017–1027 (1978).

    ADS  Google Scholar 

  4. V. E. Zakharov and L. A. Takhtadzhyan, Theor. Math. Phys., 38, 17–23 (1979).

    Article  MathSciNet  Google Scholar 

  5. B. G. Konopelchenko, Phys. Lett. A, 92, 323–327 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  6. B. G. Konopelchenko and V. G. Dubrovsky, Phys. Lett. A, 95, 457–462 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  7. B. G. Konopelchenko and V. G. Dubrovsky, Ann. Phys., 156, 265–302 (1984).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980); English transl., Plenum, New York (1984).

    MATH  Google Scholar 

  9. L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl.: L. D. Faddeev and L. A. Takhtadzhyan, The Hamiltonian Methods in the Theory of Solitons, Berlin, Springer (1987).

    Google Scholar 

  10. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (London Math. Soc. Lect. Note Ser., Vol. 149), Cambridge Univ. Press, Cambridge (1991).

    MATH  Google Scholar 

  11. B. G. Konopelchenko, Nonlinear Integrable Equations: Recursion Operators, Group-Theoretical and Hamiltonian Structures of Soliton Equations (Lect. Notes Phys., Vol. 270), Springer, Berlin (1987).

    MATH  Google Scholar 

  12. B. G. Konopelchenko, Introduction to Multidimensional Integrable Equations: The Inverse Spectral Transform in 2+1 Dimensions, Plenum, New York (1992).

    MATH  Google Scholar 

  13. B. G. Konopelchenko, Solitons in Multidimensions: Inverse Spectral Transform Method, World Scientific, Singapore (1993).

    MATH  Google Scholar 

  14. B. G. Konopelchenko and C. Rogers, “Bäcklund and reciprocal transformations: Gauge connections,” in: Nonlinear Equations in the Applied Sciences (Math. Sci. Engrg., Vol. 185, W. F. Ames and C. Rogers, eds.), Acad. Press, New York (1992), pp. 317–362.

    Chapter  Google Scholar 

  15. V. S. Dryuma, JETP Lett., 19, 387–388 (1974).

    ADS  Google Scholar 

  16. B. G. Konopelchenko and V. G. Dubrovsky, Phys. Lett. A, 102, 15–17 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  17. V. G. Dubrovsky and A. V. Gramolin, J. Phys. A, 41, 275208 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  18. M. Boiti, J. J. P. Leon, and F. Pempinelli, Inverse Problems, 3, 371–387 (1987).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. A. Davey and K. Stewartson, Proc. Roy. Soc. London Ser. A, 338, 101–110 (1974).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. S. V. Manakov, Russ. Math. Surveys, 31, 245–246 (1976).

    MATH  MathSciNet  Google Scholar 

  21. A. R. Forsyth, Theory of Differential Equations, Vol. 6, Cambridge Univ. Press, Cambridge (1906).

    Google Scholar 

  22. L. J. F. Broer, Appl. Sci. Res., 31, 377–395 (1975).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. B. G. Konopelchenko, Inverse Problems, 4, 151–163 (1988).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. L. P. Nizhnik, Sov. Phys. Dokl., 25, 706–708 (1980).

    MATH  ADS  Google Scholar 

  25. A. P. Veselov and S. P. Novikov, Sov. Math. Dokl., 30, 588–591 (1984).

    MATH  Google Scholar 

  26. B. G. Konopelchenko, Rev. Math. Phys., 2, 399–440 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  27. L. V. Bogdanov, Theor. Math. Phys., 70, 219–223 (1987).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novosibirsk State Technical University, Novosibirsk, Russia

    V. G. Dubrovsky & A. V. Gramolin

Authors
  1. V. G. Dubrovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Gramolin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. G. Dubrovsky.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 35–48, July, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dubrovsky, V.G., Gramolin, A.V. Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations. Theor Math Phys 160, 905–916 (2009). https://doi.org/10.1007/s11232-009-0080-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-009-0080-9

Keywords

Search

Navigation