Structure of set of symmetries for hyperbolic systems of Liouville type and generalized Laplace invariants

Abstract

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.