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Vector Hyperbolic Equations on the Sphere Possessing Integrable Third-Order Symmetries

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Abstract

The complete lists of vector hyperbolic equations on the sphere that have integrable third-order vector isotropic and anisotropic symmetries are presented. Several new integrable hyperbolic vector models are found. By their integrability, we mean the existence of vector Bäcklund transformations depending on a parameter. For all new equations, such transformations are constructed.

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References

  1. Sokolov V.V., Shabat A.B.: Classification of integrable evolution equations. Sov. Sci. Rev. Sect. C. 4, 221–280 (1984)

    MATH  MathSciNet  Google Scholar 

  2. Mikhailov A.V., Shabat A.B., Yamilov R.I.: The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems. Russ. Math. Surv. 42(4), 1–63 (1987)

    Article  MathSciNet  Google Scholar 

  3. Mikhailov A.V., Sokolov V.V., Shabat A.B.: The symmetry approach to classification of integrable equations. In: Zakharov, V.E. (eds) What is Integrability? Springer Series in Nonlinear Dynamics, pp. 115–184. Springer, Berlin (1991)

    Google Scholar 

  4. Mikhailov A.V., Sokolov V.V.: Symmetries of differential equations and the problem of integrability. In: Mikhailov, A.V. (eds) Integrability Lecture. Notes in Physics, vol. 767, pp. 19–88. Springer, Berlin (2009)

    Google Scholar 

  5. Svinolupov S.I., Sokolov V.V.: Evolution equations with nontrivial conservation laws. Funct. Anal. Appl. 16(4), 86–87 (1982) (in Russian)

    MathSciNet  Google Scholar 

  6. Svinolupov, S.I., Sokolov, V.V.: On conservation laws for equations having a non-trivial Lie-Bäcklund algebra. In: Shabat, A.B. (ed.) Integrable Systems, pp. 53–67. BFAN SSSR, Ufa (1982) (in Russian)

  7. Meshkov A.G., Sokolov V.V.: Integrable evolution equations with the constant separant. Ufa Math. J. 4(3), 104–154 (2012) (in Russian)

    Google Scholar 

  8. Zhiber A.V., Shabat A.B.: Klein–Gordon equations with a nontrivial group. Sov. Phys. Dokl. 247(5), 1103–1107 (1979)

    MathSciNet  Google Scholar 

  9. Zhiber A.V., Shabat A.B.: Systems of equations u x = p(u, v), v y = q(u, v) possessing symmetries. Sov. Math. Dokl. 30, 23–26 (1984)

    MATH  Google Scholar 

  10. Zhiber A.V., Sokolov V.V.: Exactly integrable hyperbolic equations of Liouville type. Russ. Math. Surv. 56(1), 63–106 (2001)

    Article  MathSciNet  Google Scholar 

  11. Meshkov A.G., Sokolov V.V.: Hyperbolic equations with third-order symmetries. Theor. Math. Phys. 166(1), 43–57 (2011)

    Article  MATH  Google Scholar 

  12. Meshkov A.G., Sokolov V.V.: Integrable evolution equations on the N-dimensional sphere. Commun. Math. Phys. 232, 1–18 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Balakhnev, M.Ju., Meshkov, A.G.: Integrable anisotropic evolution equations on a sphere. SIGMA 1, paper 027, 11 (2005). http://www.imath.kiev.ua/~igma/

  14. Ibragimov N.Kh., Shabat A.B.: Infinite Lie-Bäcklund algebras. Funct. Anal. Appl. 14(4), 79–80 (1980) (in Russian)

    MATH  MathSciNet  Google Scholar 

  15. Golubchik I.Z., Sokolov V.V.: Multicomponent generalization of the hierarchy of the Landau-Lifshitz equation. Theor. Math. Phys. 124(1), 909–971 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Meshkov A.G., Sokolov V.V.: Classification of integrable divergent N-component evolution systems. Theor. Math. Phys. 139(2), 609–622 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Balakhnev M.Ju.: A class of integrable evolutionary vector equations. Theor. Math. Phys. 142(1), 8–14 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Balakhnev M.Ju., Meshkov A.G.: On a classification of integrable vectorial evolutionary equations. J. Nonlinear Math. Phys. 15(2), 212–226 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Borisov A.B., Zykov S.A.: The dressing chain of discrete symmetries and proliferation of nonlinear equations. Theor. Math. Phys. 115(2), 530–541 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zakharov V.E., Manakov S.V., Novikov C.P., Pitaevsky L.P.: Theory of Solitons: Inverse Problem Method. Nauka, Moscow (1980)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Orel State University, Orel, Russia

    Anatoly Meshkov

  2. Landau Institute for Theoretical Physics, Moscow, Russia

    Vladimir Sokolov

Authors
  1. Anatoly Meshkov
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  2. Vladimir Sokolov
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Correspondence to Anatoly Meshkov.

Additional information

This work was supported in part by the Russian Foundation for Basic Research (Grant No 14-01-00751-a), and by the Program for Supporting Leading Scientific Schools (Grant No NSh-6501.2010.2).

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Meshkov, A., Sokolov, V. Vector Hyperbolic Equations on the Sphere Possessing Integrable Third-Order Symmetries. Lett Math Phys 104, 341–360 (2014). https://doi.org/10.1007/s11005-013-0665-y

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  • DOI: https://doi.org/10.1007/s11005-013-0665-y

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