Skip to main content
SpringerLink
Log in

Hyperbolic equations with third-order symmetries

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

Abstract

We present a complete list of nonlinear one-field hyperbolic equations with integrable third-order x and y symmetries. The list includes both equations of the sine-Gordon type and equations linearizable by differential substitutions.

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. V. Sokolov and A. B. Shabat, “Classification of integrable evolution equations,” in: Mathematical Physics Reviews (Sov. Sci. Rev., Sect. C, Vol. 4, S. P. Novikov, ed.), Harwood, Chur (1984), pp. 221–280.

    Google Scholar 

  2. A. V. Mikhailov, A. B. Shabat and R. I. Yamilov, Russ. Math. Surveys, 42, No. 4, 1–63 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  3. A. V. Mikhajlov, A. B. Shabat, and V. V. Sokolov, “The symmetry approach to the classification of integrable equations [in Russian],” in: Integrability and Kinetic Equations for Solitons (V. G. Bakhtar’yar, V. E. Zakharov, and V. M. Chernousenko, eds.), Naukova Dumka, Kiev (1990), pp. 213–279; English transl. in: What is Integrability? (V. E. Zakharov, ed.), Springer, Berlin (1991), pp. 115–184.

    Google Scholar 

  4. S. I. Svinolupov and V. V. Sokolov, Funct. Anal. Appl., 16, No. 4, 317–319 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  5. S. I. Svinolupov and V. V. Sokolov, “On conservation laws for equations having a nontrivial Lie-Bäcklund algebra [in Russian],” in: Integrable Systems (A. B. Shabat, eds.), Bashkirskii Filial Akademii Nauk SSSR, Ufa (1982), pp. 53–67.

    Google Scholar 

  6. A. V. Zhiber and A. B. Shabat, Sov. Phys. Dokl., 24, 607–609 (1979).

    ADS  Google Scholar 

  7. A. V. Zhiber and A. B. Shabat, Sov. Math. Dokl., 30, 23–26 (1984).

    MATH  Google Scholar 

  8. A. V. Zhiber and V. V. Sokolov, Russ. Math. Surveys, 56, 61–101 (2001).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. S. Ya. Startsev, Theor. Math. Phys., 120, 1009–1018 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  10. A. B. Borisov and S. A. Zykov, Theor. Math. Phys., 115, 530–541 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  11. J. A. Sanders and J. P. Wang, J. Differ. Equations, 147, 410–434 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. B. Borisov, S. A. Zykov, and M. V. Pavlov, Theor. Math. Phys., 131, 550–557 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  13. A. V. Zhiber, Russian Acad. Sci. Izv. Math., 45, 33–54 (1995).

    Article  MathSciNet  Google Scholar 

  14. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York (1955).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Orel State Technical University, Orel, Russia

    A. G. Meshkov

  2. Landau Institute for Theoretical Physics, RAS, Moscow, Russia

    V. V. Sokolov

Authors
  1. A. G. Meshkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Sokolov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Meshkov.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 166, No. 1, pp. 51–67, January, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Meshkov, A.G., Sokolov, V.V. Hyperbolic equations with third-order symmetries. Theor Math Phys 166, 43–57 (2011). https://doi.org/10.1007/s11232-011-0004-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-011-0004-3

Keywords

Search

Navigation