Abstract
We find a full system of invariants with respect to gauge transformations L →g − 1 L g for third-order hyperbolic linear partial differential operators on the plane. The operators are considered in a normalized form, in which they have the symbol Sym L = (pX + qY)XY for some non-zero bivariate functions p and q. For this normalized form, explicit formulae are given. The paper generalizes a previous result for the special, but important, case p = q = 1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Darboux, G.: Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, vol. 2. Gauthier-Villars (1889)
Ibragimov, N.: Invariants of hyperbolic equations: Solution of the Laplace problem. Prikladnaya Mekhanika i Tekhnicheskaya Fizika 45(2), 11–21 (2004), English Translation in Journal of Applied Mechanics and Technical Physics 45(2), 158166 (2004)
Kartashova, E.: Hierarchy of general invariants for bivariate LPDOs. J. Theoretical and Mathematical Physics, 1–8 (2006)
Morozov, O.: Contact Equivalence Problem for Linear Hyperbolic Equations. In: arxiv.org/preprintmath-ph/0406004
Shemyakova, E.: A Full System of Invariants for Third-Order Linear Partial Differential Operators. In: Calmet, J., Ida, T., Wang, D. (eds.) AISC 2006. LNCS (LNAI), vol. 4120, pp. 973–978. Springer, Heidelberg (2006)
Shemyakova, E., Winkler, F.: Obstacles to the Factorization of Linear Partial Differential Operators into Several Factors. Programming and Computer Software 2 (accepted, 2007)
Tsarev, S.P.: Generalized Laplace Transformations and Integration of Hyperbolic Systems of Linear Partial Differential Equations. In: Proc. ISSAC 2005 (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shemyakova, E., Winkler, F. (2007). 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) Computer Algebra in Scientific Computing. CASC 2007. Lecture Notes in Computer Science, vol 4770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75187-8_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-75187-8_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75186-1
Online ISBN: 978-3-540-75187-8
eBook Packages: Computer ScienceComputer Science (R0)