Structure of set of symmetries for hyperbolic systems of Liouville type and generalized Laplace invariants
S. Ya. Startsev
Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevsky str. 112, 450008, Ufa, Russia
The present paper is devoted to hyperbolic systems consisting of $n$ partial differential equations and possessing symmetry drivers, i.e., differential operators mapping any function of one independent variable into a symmetry of the corresponding system. The presence of the symmetry drivers is a feature of the Liouville equation and similar systems.
The composition of a differential operator with a symmetry driver is a symmetry driver again if the coefficients of the differential operator belong to the kernel of a total derivative. We prove that the entire set of the symmetry drivers is generated via the above compositions from a basis set consisting of at most $n$ symmetry drivers whose sum of orders is the smallest possible.
We also prove that if a system admits a symmetry driver of order $k-1$ and generalized Laplace invariants are well-defined for this system, then the leading coefficient of the symmetry driver belongs to the kernel of the $k$th Laplace invariant. Basing on this statement, after calculating the Laplace invariants of a system, we can obtain the lower bound for the smallest orders of the symmetry drivers for this system. This allows us to check whether we can guarantee that a particular set of the drivers is a basis set.
higher symmetries, symmetry drivers, nonlinear hyperbolic partial differential systems, Laplace invariants, Darboux integrability.
PDF file (379 kB)
Ufa Mathematical Journal, 2018, 10:4, 103–110 (PDF, 344 kB); https://doi.org/10.13108/2018-10-4-103
MSC: 37K05, 37K10, 35L51
S. Ya. Startsev, “Structure of set of symmetries for hyperbolic systems of Liouville type and generalized Laplace invariants”, Ufimsk. Mat. Zh., 10:4 (2018), 103–110; Ufa Math. J., 10:4 (2018), 103–110
Citation in format AMSBIB
\paper Structure of set of symmetries for hyperbolic systems of Liouville type and generalized Laplace invariants
\jour Ufimsk. Mat. Zh.
\jour Ufa Math. J.
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