Abstract
We introduce a subsymmetry of a differential system as an infinitesimal transformation of a subset of the system that leaves the subset invariant on the solution set of the entire system. We discuss the geometric meaning and properties of subsymmetries and also an algorithm for finding subsymmetries of a system. We show that a subsymmetry is a significantly more powerful tool than a regular symmetry with regard to deformation of conservation laws. We demonstrate that all lower conservation laws of the nonlinear telegraph system can be generated by system subsymmetries.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. J. Olver, Applications of Lie Groups to Differential Equations (Grad. Texts Math., Vol. 107), Springer, New York (2000).
G. W. Bluman, A. Cheviakov, and S. Anco, Applications of Symmetry Methods to Partial Differential Equations (Appl. Math. Sci., Vol. 168), Springer, New York (2010).
V. Rosenhaus and R. Shankar, “Sub-symmetries and the decoupling of differential systems,” to be submitted for publication.
G. W. Bluman and J. D. Cole, “The general similarity solution of the heat equation,” J. Math. Mech., 18, 1025–1042 (1969).
V. I. Fushchych, “Conditional symmetry of the equations of nonlinear mathematical physics,” Ukrainian Math. J., 43, 1350–1364 (1991).
P. J. Olver and P. Rosenau, “Group-invariant solutions of differential equations,” SIAM J. Appl. Math., 47, 263–278 (1987).
D. Levi and P. Winternitz, “Nonclassical symmetry reduction: Example of the Boussinesq equation,” J. Phys. A, 22, 2915–2924 (1989).
P. J. Olver and E. M. Vorob’ev, “Nonclassical and conditional symmetries,” in: Lie Group Analysis of Differential Equations (N. H. Ibragimov, ed.), Vol. 3, New Trends in Theoretical Developments and Computational Methods, CRC Press, Boca Raton, Fla. (1994), pp. 291–328.
G. Cicogna and G. Gaeta, “Partial Lie-point symmetries of differential equations,” J. Phys. A: Math. Gen., 34, 491–512 (2001).
V. Rosenhaus, “An infinite set of conservation laws for infinite symmetries,” Theor. Math. Phys., 151, 869–878 (2007).
V. Rosenhaus and R. Shankar, “Second Noether theorem for quasi-Noether systems,” J. Phys. A: Math. Gen., 49, 175205 (2016).
S. P. Tsarev, “The geometry of Hamiltonian systems of hydrodynamic type: The generalized hodograph method,” Math. USSR-Izv., 37, 397–419 (1991).
G. W. Bluman and Temuerchaolu, “Comparing symmetries and conservation laws of nonlinear telegraph equations,” J. Math. Phys., 46, 073513 (2005).
G. W. Bluman, “Conservation laws for nonlinear telegraph equations,” J. Math. Anal. Appl., 310, 459–476 (2005).
A. Jeffrey, “Acceleration wave propagation in hyperelastic rods of variable cross-section,” Wave Motion, 4, 173–180 (1982).
Author information
Authors and Affiliations
Corresponding author
Additional information
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 197, No. 1, pp. 138–152, October, 2018.
Rights and permissions
About this article
Cite this article
Rosenhaus, V., Shankar, R. Subsymmetries and Their Properties. Theor Math Phys 197, 1514–1526 (2018). https://doi.org/10.1134/S0040577918100082
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577918100082