Abstract
We study invariants of linear partial differential operators in two variables under gauge transformations. Using the Beals-Kartashova factorization, we construct a hierarchy of generalized invariants for operators of an arbitrary order. We study the properties of these invariants and give some examples. We also show that the classic Laplace invariants correspond to some particular cases of generalized invariants.
Similar content being viewed by others
References
A. B. Shabat, Theor. Math. Phys., 103, 482–485 (1995).
L. Bianchi, Lezioni di Geometria Differenziale, Zanichelli, Bologna (1924); A. N. Leznov and M. V. Saveliev, Group Methods for Integration of Nonlinear Dynamical Systems [in Russian], Nauka, Moscow (1985); English transl.: Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems, Birkhäuser, Basel (1992); S. P. Tsarev, “An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator,” in: Proc. Intl. Symp. on Symbolic and Algebraic Computation (ISSAC’96) (Zurich, Switzerland, 24–26 July 1996, Y. N. Lakshman, ed.), ACM Press, New York (1996), pp. 226–231; S. P. Tsarev, “Factorization of linear partial differential operators and Darboux integrability of nonlinear PDEs,” Poster at ISSAC’98, Rostock, Germany, 13–15 August 1998 (1998); cs.SC/9811002 (1998); G. Tzitzeica, Compt. Rend. Acad. Sci., 150, 955–956, 971–974 (1910).
D. Grigoriev and F. Schwarz, J. Comput., 73, 179–197 (2004).
E. Kaltofen, “Factorization of polynomials,” in: Computing (B. Buchberger, G. E. Collins, and R. Loos, eds.), Springer, Wien (1982), pp. 95–113.
R. Beals and E. A. Kartashova, Theor. Math. Phys., 145, 1511–1524 (2005).
E. Kartashova, “BK-factorization and Darboux-Laplace transformations,” in: Proc. 2005 Intl. Conf. on Scientific Computing (CSC’05, Las Vegas, Nevada, USA, 20–23 June 2005, H. R. Arabnia and G. A. Gravvanis, eds.), C.S.R.E.A. Press, Athens, Georgia (2005), pp. 144–150.
Author information
Authors and Affiliations
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 470–478, June, 2006.
Rights and permissions
About this article
Cite this article
Kartashova, E.A. A hierarchy of generalized invariants for linear partial differential operators. Theor Math Phys 147, 839–846 (2006). https://doi.org/10.1007/s11232-006-0079-4
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11232-006-0079-4