Acta Applicandae Mathematica

, Volume 41, Issue 1–3, pp 135–144 | Cite as

Conservation laws and the variational bicomplex for second-order scalar hyperbolic equations in the plane

  • Ian M. Anderson
  • Niky Kamran
Article

Abstract

In this paper, we announce several new results concerning the cohomology of the variational bicomplex for a second-order scalar hyperbolic equation in the plane. These cohomology groups are represented by the conservation laws, and certain form-valued generalizations, for the equation. Our methods are based upon the introduction of an adapted coframe for the the variational bicomplex which is constructed by generalizing the classical Laplace transformation used to integrate certain linear hyperbolic equations in the plane.

Mathematics subject classifications (1991)

58G16 35A30 35L65 

Key words

variational bicomplex hyperbolic second-order equations conservation laws 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andersen, M.: Introduction to the variational bicomplex, in M. Gotay, J. Marsden and V. Moncrief (eds),Mathematical Aspects of Classical Field Theory, Contemporary Mathematics 132, Amer. Math Soc., Providence, 1992, pp. 51–73.Google Scholar
  2. 2.
    Anderson, Ian M. and Kamran, N.: The variational bicomplex for second order scalar partial differential equations in the plane, Centre de recherches mathématiques, Technical report, September 1994.Google Scholar
  3. 3.
    Anderson, Ian M. and Kamran, N.: The variational bicomplex for hyperbolic second order scalar partial differential equations in the plane (submitted May 1995).Google Scholar
  4. 4.
    Bryant, R. L. and Griffiths, P. A.: Characteristic cohomology of differential systems, I: General theory, Duke University, Mathematics Preprint Series, January, 1993.Google Scholar
  5. 5.
    Bryant, R. L. and Griffiths, P. A.: Characteristic cohomology of differential systems, II: Conservation laws for a class of parabolic equations, Duke University, Mathematics Preprint Series, January, 1993.Google Scholar
  6. 6.
    Darboux, G.:Leçons sur la théorie générale des suraces et les applications géométriques du calcul infinitésimal, Gauthier-Villars, Paris, 1896.Google Scholar
  7. 7.
    Forsyth, A.:Theory of Differential Equations, Vol. 6, Dover, New York, 1959.Google Scholar
  8. 8.
    Goursat, E.:Leçon sur l'intégration des équations aux dérivées partielles du second ordre á deux variables indépendantes, Tome 2, Hermann, Paris, 1896.Google Scholar
  9. 9.
    Olver, P. J.:Applications of Lie Groups to Differential Equations, Springer, New York, 1986.Google Scholar
  10. 10.
    Tsujishita, T.: On variation bicomplexes associated to differential equations,Osaka J. Math. 19(1982), 311–363.Google Scholar
  11. 11.
    Tsujishita, T.: Formal geometry of systems of differential equations,Sugaku Exposition 2 (1989), 1–40.Google Scholar
  12. 12.
    Vinogradov, A. M.: The C-spectral sequence, Lagrangian formalism and conservation laws, I, II,J. Math. Anal. Appl. 100 (1984), 1–129.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Ian M. Anderson
    • 1
  • Niky Kamran
    • 2
  1. 1.Department of MathematicsUtah State UniversityLoganUSA
  2. 2.Department of MathematicsMcGill UniversityMontrealCanada

Personalised recommendations