Generalized Webster equation: exact and approximate renormgroup symmetries, invariant solutions and conservation laws

The exact point symmetry group for the generalized Webster type equation, which describes nonlinear acoustic waves in lossy channels with variable cross sections, is found. It is shown that, for certain types of cross section profiles $S$, the admitted three-dimensional point symmetry group is extended and group classification problem for different types of $S$ is solved. Optimal systems of one-dimensional subalgebras of the admitted Lie algebra are revealed and the invariant solutions corresponding to these subalgebras are obtained. Approximate renormgroup symmetries and the corresponding approximate analytic solutions, as well as conservation laws to the generalized Webster equation are derived for channels with constant and smoothly varying or constant cross sections and arbitrary initial conditions.