Symmetry properties of a nonlinear acoustics model

We give a classification into conjugacy classes of subalgebras of the symmetry algebra generated by the Zabolotskaya-Khokhlov equation, and obtain all similarity reductions of this equation into (1 + 1)-dimensional equations. We thus show that the Lie classical reduction approach may also give rise to more general reduced equations as those expected from the direct method of Clarkson and Kruskal. By transforming the determining system for the similarity variables into the equivalent adjoint system of total differential equations, similarity reductions to ODEs which are independent of the three arbitrary functions defining the symmetries are also obtained. These results are again compared with those obtained by the direct method of Clarkson and Kruskal, by finding in particular equivalence transformations mapping some of the reduced equations to each other. Various families of new exact solutions are also derived.