Numerical solution for a general class of nonlocal nonlinear wave equations

TL;DR

This paper develops and analyzes a numerical method for solving a broad class of nonlocal nonlinear wave equations, demonstrating convergence and exploring solution behaviors influenced by kernel functions.

Contribution

It introduces a Fourier spectral discretization scheme for nonlocal nonlinear wave equations and proves its convergence, also applying it to study solitary wave solutions.

Findings

Convergence of the semidiscrete Fourier spectral scheme is established.

The fully-discrete scheme effectively captures solution dynamics.

Kernel functions significantly affect solitary wave solutions.

Abstract

A class of nonlocal nonlinear wave equation arises from the modeling of a one dimensional motion in a nonlinearly, nonlocally elastic medium. The equation involves a kernel function with nonnegative Fourier transform. We discretize the equation by using Fourier spectral method in space and we prove the convergence of the semidiscrete scheme. We then use a fully-discrete scheme, that couples Fourier pseudo-spectral method in space and 4th order Runge-Kutta in time, to observe the effect of the kernel function on solutions. To generate solitary wave solutions numerically, we use the Petviashvili's iteration method.