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Theoretical and Mathematical Physics

, Volume 141, Issue 2, pp 1509–1527 | Cite as

One Class of Liouville-Type Systems

  • D. K. Demskoi
Article

Abstract

We prove that a class of systems with the Lagrangian of the form \(L = {{\left[ {g_{\{ ij\} } (u)u_x^i u_t^j } \right]} \mathord{\left/ {\vphantom {{\left[ {g_{\{ ij\} } (u)u_x^i u_t^j } \right]} {2 + f(u)}}} \right. \kern-\nulldelimiterspace} {2 + f(u)}}\) is of the Liouville type. We construct new integrable Hamiltonian systems related to the symmetries of the hyperbolic systems under consideration by substitutions of the Miura transformation type. For one of the systems obtained, we construct the second-order recursion operator.

system of Liouville type higher pseudoconstants recursion operator 

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REFERENCES

  1. 1.
    A. V. Zhiber and A. B. Shabat, Sov.Phys.Dokl., 224, 607–609 (1979); R. K. Dodd and R. K. Bullough, Proc. Roy.Soc.London A, 351, 499-523 (1976).Google Scholar
  2. 2.
    A. V. Zhiber and V. V. Sokolov, Russ.Math.Surveys, 56, 61–101 (2001).Google Scholar
  3. 3.
    A. N. Leznov and M. V. Saveliev, Group-Theoretical Methods or Integration o Nonlinear Dynamical Systems [in Russian], Nauka, Moscow (1985); English transl., Birkhäuser, Basel (1992).Google Scholar
  4. 4.
    I. M. Anderson and N. Kamran, Duke Math.J., 87, 265–319 (1997); M. Juras and I. M. Anderson, Duke Math.J, 89, 351-375 (1997).Google Scholar
  5. 5.
    E. Goursat, Lecons sur l 'intégration deséquations aux dérivées partielles du second ordre à deux variables indépendantes, Vols. 1, 2, Hermann, Paris (1896, 1898).Google Scholar
  6. 6.
    A. V. Zhiber, Russ.Acad.Sci.Izv.Math., 45, 33–54 (1995).Google Scholar
  7. 7.
    A. V. Zhiber and V. V. Sokolov, Theor.Math.Phys., 120, 834–839 (1999).Google Scholar
  8. 8.
    A. B. Shabat and R. I. Yamilov, "Exponential systems of type I and Cartan matrices [in Russian]," Preprint, Bashkirian Branch, USSR Acad. Sci., Ufa (1981).Google Scholar
  9. 9.
    A. N. Leznov, V. G. Smirnov, and A. B. Shabat, Theor.Math.Phys., 51, 322–330 (1982).Google Scholar
  10. 10.
    V. V. Sokolov, Russ.Math.Surveys, 43, No. 5, 165–204 (1988).Google Scholar
  11. 11.
    D. K. Demskoi and A. G. Meshkov, "New integrable string-like fields in 1+1 dimensions," in: Proc.2nd Intl. Conf.Quantum Field Theory and Gravity (July 28-August 2, 1997, Tomsk, Russia, I. L. Bukhbinder and K. E. Osetrin, eds.), Tomsk Pedagogical Univ. Press, Tomsk (1998), pp. 282–285.Google Scholar
  12. 12.
    D. K. Demskoi and A. G. Meshkov, Theor.Math.Phys., 134, 351–364 (2003); Inverse Problems, 19, 563-571 (2003).Google Scholar
  13. 13.
    S. Ya. Startsev, Theor.Math.Phys., 116, 1001–1010 (1998).Google Scholar
  14. 14.
    A. V. Zhiber and S. Ya. Startsev, Math.Notes, 74, 803–811 (2003).Google Scholar
  15. 15.
    P. Olver, Applications o Lie Groups to Differential Equations, Springer, New York (1986).Google Scholar
  16. 16.
    N. Yajima and M. Oikawa, Progr.Theoret.Phys., 56, 1719–1739 (1976).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • D. K. Demskoi
    • 1
  1. 1.Orel State UniversityOrelRussia

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