Multi-dimensional quasi-simple waves in weakly dissipative flows

A multi-dimensional simple wave formalism is employed to formulate a multi-dimensional quasi-simple wave for a weakly dissipative fluid. This is a natural but nontrivial generalization of the so-called unidirectional quasi-simple wave. The method is more close to multi-orders analysis which is different from the standard perturbation method. Detailed solving up to the second order is presented for a 2D sound simple wave. A new 2D Burgers equation is derived for the wave phase. It essentially differs from the known 2D generalizations of the Burgers equation, e.g., from the Zabolotskaya–Khokhlov equation. The derived equation is investigated in the framework of group analysis of differential equations. Multi-parameter families of its exact solutions are constructed. Simplest solutions are chosen for analysis of their physical relevance in the initial variables.