Localized Wave Structures Determined by Exact Solutions of the Khokhlov–Zabolotskaya Equation

Abstract

A brief description of physically meaningful exact solutions of nonlinear partial differential equations that describe concrete physical objects, processes, etc., is presented. New exact solutions of the Khokhlov–Zabolotskaya equation that have a physical meaning (including a variant of the stationary equation with respect to the direction of propagation of a wave structure) are presented that correspond to the description of localized wave structures either only with spatial localization (beams in the self-trapped propagation mode), or with spatiotemporal localization (pulse-type structures, which are sometimes referred to as wave (acoustic) bullets). The solutions and the localized structures that are described by them are not obvious for nonlinear acoustics or predictable from “physical considerations” (as, for example, for the self-trapped propagation of beams in nonlinear optics), since there are no explicit conditions for the balance of the “necessary” effects in the dispersion-free state.