Journal of Mathematical Sciences

, Volume 135, Issue 1, pp 2680–2694 | Cite as

Contact-equivalence problem for linear hyperbolic equations

  • O. I. Morozov

Abstract

We consider the local equivalence problem for the class of linear second-order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. É. Cartan’s method is used for finding the Maurer-Cartan forms for symmetry groups of equations from the class and computing structure equations and complete sets of differential invariants for these groups. The solution of the equivalence problem is formulated in terms of these differential invariants.

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References

  1. 1.
    É. Cartan, “Sur la structure des groupes infinis de transformations,” in: Œuvres Complètes, Pt. II, Vol. 2, Gauthier-Villars, Paris (1953), pp. 571–714.Google Scholar
  2. 2.
    É. Cartan, “Les sous-groupes des groupes continus de transformations,” Œuvres Complètes, Pt. II, Vol. 2, Gauthier-Villars, Paris (1953)ibid., pp. 719–856.Google Scholar
  3. 3.
    É. Cartan, “Les groupes de transformations continus, infinis, simples,” Œuvres Complètes, Pt. II, Vol. 2, Gauthier-Villars, Paris (1953)ibid., pp. 857–925.Google Scholar
  4. 4.
    É. Cartan, “La structure des groupes infinis,” Œuvres Complètes, Pt. II, Vol. 2, Gauthier-Villars, Paris (1953)ibid., pp. 1335–1384.Google Scholar
  5. 5.
    É. Cartan, “Les probl`emes d’équivalence,” Œuvres Complètes, Pt. II, Vol. 2, Gauthier-Villars, Paris (1953)ibid., pp. 1311–1334.Google Scholar
  6. 6.
    M. Fels and P. J. Olver, “Moving coframes, I. A practical algorithm,” Acta Appl. Math., 51, 161–213 (1998)CrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Fels and P. J. Olver, “Moving coframes, II. Regularization and theoretical foundations,” Acta Appl. Math., 55, 127–208 (1999).CrossRefMathSciNetGoogle Scholar
  8. 8.
    R. B. Gardner, The Method of Equivalence and Its Applications, SIAM, Philadelphia (1989).Google Scholar
  9. 9.
    N. H. Ibragimov, “Infinitesimal method in the theory of invariants of algebraic and differential equations,” Notices South African Math. Soc., 29, 61–70 (1997).Google Scholar
  10. 10.
    N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley and Sons, New York (1999).Google Scholar
  11. 11.
    N. H. Ibragimov, “Laplace type invariants for parabolic equations,” Nonlinear Dynam., 28, 125–133 (2002).MATHMathSciNetGoogle Scholar
  12. 12.
    N. H. Ibragimov, “Invariants of hyperbolic equations: Solution to Laplace’s problem,” Prikl. Mekh. Tekh. Fiz., 45, No. 2, 11–21 (2004).MATHGoogle Scholar
  13. 13.
    I. K. Johnpillai and F. M. Mahomed, “Singular invariant equation for the (1+1) Fokker-Planck equation,” J. Phys. A, 34, 11033–11051 (2001).CrossRefMathSciNetGoogle Scholar
  14. 14.
    I. K. Johnpillai, F. M. Mahomed, and C. Wafo Soh, “Basis of joint invariants for (1+1) linear hyperbolic equations,” J. Nonlinear Math. Phys., 9, Supplement 2, 49–59 (2002).MathSciNetGoogle Scholar
  15. 15.
    P. S. Laplace, “Recherches sur le calcul intégral aux différences partielles,” Mémoires de l’Académie Royale de Sciences de Paris, 341–401 (1773–1777), reprinted in Œuvres Complètes, IX, Gauthier-Villars, Paris (1893), pp. 3–68. English translation: New York (1966).Google Scholar
  16. 16.
    S. Lie, Gesammelte Abhandlungen, B. 1–6, BG Teubner, Leipzig (1922–1937).Google Scholar
  17. 17.
    I. G. Lisle and G. J. Reid, “Geometry and structure of Lie pseudogroups from infinitesimal defining equations,” J. Symbolic Comput., 26 355–379 (1998).CrossRefMathSciNetGoogle Scholar
  18. 18.
    I. G. Lisle and G. J. Reid, Symmetry Classification Using Invariant Moving Frames, http://www.apmaths.uwo.ca/∼reid (1999).
  19. 19.
    O. I. Morozov, “Moving coframes and symmetries of differential equations,” J. Phys. A, 35, 2965–2977 (2002).CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    O. I. Morozov, Contact Equivalence Problem for Linear Parabolic Equations, arXiv:math.ph/0304045 (2003).Google Scholar
  21. 21.
    O. I. Morozov, “Symmetries of differential equations and Cartan’s equivalence method,” in: Proc. Fifth Int. Conf. “Symmetry in Nonlinear Mathematical Physics,” Kyiv, Ukraine, 23–29 June 2003, Pt. 1, pp. 196–203 (2004).Google Scholar
  22. 22.
    L. V. Ovsiannikov, “Group properties of the Chaplygin equation,” Prikl. Mekh. Tekh. Fiz., No. 3, 126–145 (1960).Google Scholar
  23. 23.
    L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York (1982).Google Scholar
  24. 24.
    P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge Univ. Press, Cambridge (1995).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • O. I. Morozov
    • 1
  1. 1.Department of MathematicsMoscow State Technical University of Civil AviationMoscowRussia

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