Long-time behavior of solutions to the general class of coupled nonlocal nonlinear wave equations

Coupled nonlocal nonlinear wave equations describe the dynamics of wave systems with multiple interacting components in a wide range of physical applications. Investigating the long-time behavior of the solutions is crucial for understanding the stability, dynamics, and qualitative properties of these systems. In this study, solitary wave solutions to the coupled nonlocal nonlinear wave equations are generated numerically using the Petviashvili iteration method and the long-time behavior of the obtained solutions is investigated. In order to investigate the long-time behavior of the solutions obtained for the coupled nonlocal nonlinear wave equations, a Fourier pseudospectral method for space discretization and a fourth-order Runge-Kutta scheme for time discretization are proposed. Various numerical experiments are conducted to test the performance of the Petviashvili iteration method and the Fourier pseudo-spectral method. The proposed scheme can be used to solve the coupled nonlocal nonlinear system for arbitrary kernel functions representing the details of the atomic scale effects.