Advertisement

On the Invariant Properties of Hyperbolic Bivariate Third-Order Linear Partial Differential Operators

  • Ekaterina Shemyakova
  • Franz Winkler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5081)

Abstract

Bivariate, hyperbolic third-order linear partial differential operators under the gauge transformations Lg(x,y)− 1 ∘ L ∘ g(x,y) are considered. The existence of a factorization, the existence of a factorization that extends a given factorization of the symbol of the operator are expressed in terms of the invariants of some known generating set of invariants. The operation of taking the formal adjoint can be also defined for equivalent classes of LPDOs, and explicit formulae defining this operation in the space invariants were obtained.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, I., Juras, M.: Generalized Laplace invariants and the method of Darboux. Duke J. Math. 89, 351–375 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Anderson, I., Kamran, N.: The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane. Duke J. Math. 87, 265–319 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Athorne, C.: A z ×r toda system. Phys. Lett. A. 206, 162–166 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Grigoriev, D., Schwarz, F.: Generalized loewy-decomposition of d-modules. In: ISSAC 2005: Proceedings of the 2005 international symposium on Symbolic and algebraic computation, pp. 163–170. ACM, New York (2005)CrossRefGoogle Scholar
  5. 5.
    Tsarev, S.: Generalized laplace transformations and integration of hyperbolic systems of linear partial differential equations. In: ISSAC 2005: Proceedings of the 2005 international symposium on Symbolic and algebraic computation, pp. 325–331. ACM Press, New York (2005)CrossRefGoogle Scholar
  6. 6.
    Tsarev, S.: Factorization of linear partial differential operators and darboux’ method for integrating nonlinear partial differential equations. Theo. Math. Phys. 122, 121–133 (2000)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Darboux, G.: Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, vol. 2. Gauthier-Villars (1889)Google Scholar
  8. 8.
    Goursat, E.: Leçons sur l’intégration des équations aux dérivées partielles du seconde ordre a deux variables indépendants, Paris, vol. 2 (1898)Google Scholar
  9. 9.
    Shemyakova, E., Winkler, F.: A full system of invariants for third-order linear partial differential operators in general form. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 360–369. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ekaterina Shemyakova
    • 1
  • Franz Winkler
    • 1
  1. 1.Research Institute for Symbolic Computation (RISC)J. Kepler UniversityLinzAustria

Personalised recommendations