Acta Applicandae Mathematicae

, Volume 99, Issue 3, pp 309–319 | Cite as

Coverings of Differential Equations and Cartan’s Structure Theory of Lie Pseudo-Groups

Article

Abstract

We establish relations between Maurer–Cartan forms of symmetry pseudo-groups and coverings of differential equations. Examples include Liouville’s equation, the Khokhlov–Zabolotskaya equation, and the Boyer–Finley equation.

Keywords

Lie pseudo-groups Maurer–Cartan forms Symmetries of differential equations Coverings of differential equations 

Mathematics Subject Classification (2000)

58H05 58J70 35A30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989) MATHGoogle Scholar
  2. 2.
    Boyer, C.P., Finley, J.D. III: Killing vectors in self-dual, Euclidean Einstein spaces. J. Math. Phys. 23, 1126–1130 (1982) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bryant, R., Griffiths, P., Hsu, L.: Toward a geometry of differential equations. In: Geometry, Topology and Physics. Conference Proceedings and Lecture Notes in Geometry and Topology, vol. 6, pp. 1–76 (1995) Google Scholar
  4. 4.
    Bryant, R.L., Griffiths, P.A.: Characteristic cohomology of differential systems (II): conservation laws for a class of parabolic equations. Duke Math. J. 78, 531–676 (1995) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bryant, R., Griffiths, P., Hsu, L.: Hyperbolic exterior differential systems and their conservation laws. I, II. Selecta Math., New Ser. 1, 21–112 (1995), 265–323 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cartan, É.: Sur la structure des groupes infinis de transformations. In: Œuvres Complètes, Part II, vol. 2, pp. 571–714. Gauthier-Villars, Paris (1953) Google Scholar
  7. 7.
    Cartan, É.: Les sous-groupes des groupes continus de transformations. In: Œuvres Complètes, Part II, vol. 2, pp. 719–856. Gauthier-Villars, Paris (1953) Google Scholar
  8. 8.
    Cartan, É.: La structure des groupes infinis. In: Œuvres Complètes, Part II, vol. 2, pp. 1335–1384. Gauthier-Villars, Paris (1953) Google Scholar
  9. 9.
    Cartan, É.: Les problèmes d’équivalence. In: Œuvres Complètes, Part II, vol. 2, pp. 1311–1334. Gauthier-Villars, Paris (1953) Google Scholar
  10. 10.
    Cheh, J., Olver, P.J., Pohjanpelto, J.: Algorithms for differential invariants of symmetry groups of differential equations. Found. Comput. Math. (to appear) Google Scholar
  11. 11.
    Cheh, J., Olver, P.J., Pohjanpelto, J.: Maurer–Cartan equations for Lie symmetry pseudo-groups of differential equations. J. Math. Phys. 46, 023504 (2005) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Clairin, J.: Sur les transformations de Baecklund. Ann. Sci. École Norm. Sup. 3, 1–63 (1902), supplément Google Scholar
  13. 13.
    Clelland, J.N.: Geometry of conservation laws for a class of parabolic partial differential equations. Sel. Math., New Ser. 3, 1–77 (1997) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fels, M.: The equivalence problem for systems of second order ordinary differential equations. Proc. Lond. Math. Soc. 71, 221–240 (1995) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Fels, M., Olver, P.J.: Moving coframes. I. A practical algorithm. Acta. Appl. Math. 51, 161–213 (1998) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Gardner, R.B.: The Method of Equivalence and Its Applications. CBMS–NSF Regional Conference Series in Applied Math. SIAM, Philadelphia (1989) MATHGoogle Scholar
  17. 17.
    Harrison, B.K.: On methods of finding Bäcklund transformations in systems with more than two independent variables. J. Nonlinear Math. Phys. 2, 201–215 (1995) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Harrison, B.K.: Matrix methods of searching for Lax pairs and a paper by Estévez. Proc. Inst. Math. NAS Ukr. 30 (2000), Part 1, 17–24 Google Scholar
  19. 19.
    Hsu, L., Kamran, N.: Classification of second order ordinary differential equations admitting Lie groups of fiber-preserving symmetries. Proc. Lond. Math. Soc., Ser. 3 58, 387–416 (1989) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Dordrecht (1985) MATHGoogle Scholar
  21. 21.
    Kamran, N.: Contributions to the study of the equivalence problem of Élie Cartan and its applications to partial and ordinary differential equations. Mem. Cl. Sci. Acad. R. Belg. 45 (1989), Fac. 7 Google Scholar
  22. 22.
    Kamran, N., Shadwick, W.F.: Équivalence locale des équations aux dérivées partielles quasi lineaires du deixième ordre et pseudo-groupes infinis. C. R. Acad. Sci. (Paris) Ser. I 303, 555–558 (1986) MATHMathSciNetGoogle Scholar
  23. 23.
    Kamran, N., Shadwick, W.F.: A differential geometric characterization of the first Painlevé transcendents. Math. Ann. 279, 117–123 (1987) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kamran, N., Lamb, K.G., Shadwick, W.F.: The local equivalence problem for y ′′=f(x,y,y ) and the Painlevé transcendents. J. Differ. Geom. 22, 139–150 (1985) MATHMathSciNetGoogle Scholar
  25. 25.
    Krasil’shchik, I.S., Vinogradov, A.M.: Nonlocal symmetries and the theory of coverings. Acta Appl. Math. 2, 79–86 (1984) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Krasil’shchik, I.S., Vinogradov, A.M.: Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math. 15, 161–209 (1989) CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Krasil’shchik, I.S., Vinogradov, A.M. (eds.): Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Transl. Math. Monographs, vol. 182. Am. Math. Soc., Providence (1999) MATHGoogle Scholar
  28. 28.
    Kuz’mina, G.M.: On a possibility to reduce a system of two first-order partial differential equations to a single equation of the second order. Proc. Moscow State Pedagog. Inst. 271, 67–76 (1967) (in Russian) MathSciNetGoogle Scholar
  29. 29.
    Lie, S.: Gesammelte Abhandlungen, vol. 1–6. Leipzig, Teubner (1922–1937) Google Scholar
  30. 30.
    Lisle, I.G., Reid, G.J.: Geometry and structure of Lie pseudogroups from infinitesimal defining equations. J. Symb. Comput. 26, 355–379 (1998) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Lisle, I.G., Reid, G.J., Boulton, A.: Algorithmic determination of structure of infinite Lie pseudogroups of symmetries of PDEs. In: Proc. ISSAC’95. ACM Press, New York (1995) Google Scholar
  32. 32.
    Morozov, O.I.: Moving coframes and symmetries of differential equations. J. Phys. A, Math. Gen. 35, 2965–2977 (2002) MATHCrossRefGoogle Scholar
  33. 33.
    Morozov, O.I.: Symmetries of differential equations and Cartan’s equivalence method. Proc. Inst. Math. NAS Ukr. 50 (2004), Part 1, 196–203 Google Scholar
  34. 34.
    Morozov, O.I.: Structure of symmetry groups via Cartan’s method: survey of four approaches. Symmetry Integr. Geom.: Methods Appl. 1 (2005), Paper 006, 14 p Google Scholar
  35. 35.
    Morozov, O.I.: Contact-equivalence problem for linear hyperbolic equations. J. Math. Sci. 135, 2680–2694 (2006) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Morozov, O.I.: Contact equivalence problem for linear parabolic equations. arXiv:math-ph/0304045 Google Scholar
  37. 37.
    Morris, H.C.: Prolongation structures and nonlinear evolution equations in two spatial dimensions. J. Math. Phys. 17, 1870–1872 (1976) CrossRefGoogle Scholar
  38. 38.
    Morris, H.C.: Prolongation structures and nonlinear evolution equations in two spatial dimensions: a general class of equations. J. Phys. A, Math. Gen. 12, 261–267 (1979) CrossRefGoogle Scholar
  39. 39.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986) MATHGoogle Scholar
  40. 40.
    Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995) MATHGoogle Scholar
  41. 41.
    Olver, P.J., Pohjanpelto, J.: Moving frames for Lie pseudo-groups. Can. J. Math. (to appear) Google Scholar
  42. 42.
    Olver, P.J., Pohjanpelto, J.: Maurer–Cartan forms and the structure of Lie pseudo-groups. Sel. Math., New Ser. 11, 99–126 (2005) MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic, New York (1982) MATHGoogle Scholar
  44. 44.
    Wahlquist, H.D., Estabrook, F.B.: Prolongation structures of nonlinear evolution equations. J. Math. Phys. 16, 1–7 (1975) MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Zabolotskaya, E.A., Khokhlov, R.V.: Quasi-plane waves in the nonlinear acoustics of confined beams. Sov. Phys. Acoust. 15, 35–40 (1969) Google Scholar
  46. 46.
    Zakharov, V.E.: Integrable systems in multidimensional spaces. Lect. Notes Phys. 153, 190–216 (1982) CrossRefGoogle Scholar
  47. 47.
    Zakharov, V.E., Shabat, A.B.: Integration of nonlinear equations of mathematical physics by the method of inverse scattering, II. Funct. Anal. Appl. 13, 166–174 (1980) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Moscow State Technical University of Civil AviationMoscowRussia

Personalised recommendations