Green’s function solution of nonlinear wave equation depending on the absolute value of the unknown function

Abstract

This paper is devoted to possibilities of a semi-analytical approximation of nonlinear wave equations with nonlinearities depending on the absolute value of the unknown function, which arise in different areas of physics and mechanics. The main difficulty of analysis of such equations is that the derivation of their rigorous solution is highly sophisticated, while their numerical solution requires burdensome computational costs. Using the traveling wave ansatz, we first reduce the wave equation to a nonlinear ordinary differential equation. Then, applying Frasca’s method, we construct its general solution in terms of the nonlinear Green’s function. For particular nonlinearities, it is shown that the first-order approximation of the Green’s function solution is numerically comparable with the solution obtained by the well-known numerical method of lines. The contribution of the higher order terms is studied for a particular nonlinearity.