Stable-Range Approach to Short Wave and Khokhlov-Zabolotskaya Equations

TL;DR

This paper introduces a new algebraic method using stable ranges to find explicit solutions for short wave and Khokhlov-Zabolotskaya equations, aiding in modeling nonlinear wave phenomena.

Contribution

It presents a novel algebraic approach leveraging stable ranges to derive explicit solutions for complex nonlinear wave equations.

Findings

Derived large families of explicit solutions

Enabled solutions for related practical models

Provided a new algebraic framework for these equations

Abstract

Short wave equations were introduced in connection with the nonlinear reflection of weak shock waves. They also relate to the modulation of a gas-fluid mixture. Khokhlov-Zabolotskaya equation are used to describe the propagation of a diffraction sound beam in a nonlinear medium. We give a new algebraic method of solving these equations by using certain finite-dimensional stable range of the nonlinear terms and obtain large families of new explicit exact solutions parameterized by several functions for them. These parameter functions enable one to find the solutions of some related practical models and boundary value problems.