The exact singular solutions of the Khokhlov–Zabolotskaya equations and first-order quasilinear equations

Abstract

Background. The Khokhlov-Zabolotskaya equations are one of the important tools for analyzing the propagation of sound waves in a gaseous medium and liquids, as well as in problems of compressible fluid flow around profiles. The nonlinearity of these equations requires special methods for constructing solutions and analyzing them. The purpose of the work is to construct exact solutions of the Khokhlov-Zabolotskaya equations by connecting them with first-order quasilinear equations, as well as to analyze the geometry of these solutions in three-dimensional space. Materials and methods. In this work, solutions to the Khokhlov-Zabolotskaya equations are constructed using the riverton method (solutions to systems of first-order quasi-linear equations of a special type). The general procedure for deriving the Khokhlov-Zabolotskaya equations from a system of first-order quasilinear equations is described. Results. The main result is the implicit construction of a set of exact solutions to the Khokhlov-Zabolotskaya equation, depending on three functional parameters. This allows you to construct solutions under given conditions along the coordinate axes. A general method for analyzing such solutions is presented, indicating the base curves along which the plane wave fronts of the solutions move, as well as the regions in which the number of sheets of multivalued solutions is fixed. Conclusions. The proposed method for constructing solutions allows one to construct exact solutions of the Khokhlov- Zabolotskaya equation that correspond to given conditions along the coordinate axes and analyze their geometric properties.

Keywords

• generalized khokhlov-zabolotskaya equations
• rivertons
• first-order quasilinear equations