A Lie group investigation of exact weakly nonlinear quasi-planar sound waves

The Lie point symmetries of the (1+2)D Zabolotskaya-Khokholov (ZK) equation for a gas with nonzero viscosity at rest are calculated. They form an infinite-dimensional Lie algebra which can be written as the direct sum of a 2D Lie algebra and an infinite-dimensional Lie algebra. The symmetry structure of the ZK equation is used to construct its physically relevant solutions from those of the (1 + 1)D Burgers equation. In particular, travelling wave solutions (one-solitons) as well as the multi-soliton solutions of the ZK equation are obtained. Also, we obtain the N-waves, the periodic waves and the solution to some Cauchy problems for the ZK equation. We emphasize that our method can be utilized to generate a wider class of solutions of the ZK equation.