Non polynomial conservation law densities generated by the symmetry operators in some hydrodynamical models

Abstract

New extra series of conserved densities for the polytropic gas model and nonlinear elasticity equation are obtained without any references to the recursion operator or to the Lax operator formalism. Our method based on the utilization of the symmetry operators and allows us to obtain the densities of arbitrary homogenuity dimensions. The nonpolynomial densities with logarithmics behaviour are presented as an example. The special attention is paid for the singular case $(γ=1)$ for which we found new non homogenious solutions expressed in terms of the elementary functions.