Simple waves of conic motions

Continuous media models of a gas dynamical type admit $11$-dimensional Lie algebra of Galileo group extended by an uniform dilatation of all independent variables. The object of the study is the constructing of submodels of the chain of embedded subalgebras with dimensions from $1$ till $4$ describing conical motions of the gas. For the chosen chain we find consistent invariant in the cylindrical coordinate system. On their base we obtain the representations for an invariant solution for each submodel in the chain. By substituting them into the system of gas dynamics equations we obtain embedded invariant submodels of ranks from $0$ to $3$. We prove that the solutions of submodels constructed by a subalgebra of a higher dimension are solutions to submodels constructed by subalgebras of smaller dimensions.
In the chosen chain, we consider a $4$-dimensional subalgebra generating irregular partially invariant solutions of rank $1$ defect $1$ in the cylindrical coordinates. In the gas dynamics, such solutions are called simple waves. We study the compatibility of the corresponding submodel by means of the system of alternative assumptions obtained from the submodel equations. We obtain solutions depending on arbitrary functions as well as partial solutions which can be invariant with respect to the subalgebras embedded into the considered subalgebra but are not necessarily from the considered chain.