Higher order dispersive effects in regularized Boussinesq equation

TL;DR

This paper investigates the higher order dispersive effects in the bi-directional propagation of waves modeled by the higher order Boussinesq equation, using a spectral numerical scheme to analyze solitary waves, collisions, and blow-up phenomena.

Contribution

It introduces a numerical scheme combining Fourier pseudo-spectral and Runge Kutta methods for the HBq equation and proves its convergence in Sobolev spaces.

Findings

Analysis of solitary wave propagation and collision behaviors

Observation of blow-up solutions under certain conditions

Validation of the numerical scheme's convergence

Abstract

In this paper, we consider the higher order Boussinesq (HBq) equation which models the bi-directional propagation of longitudinal waves in various continuous media. The equation contains the higher order effects of frequency dispersion. The present study is devoted to the numerical investigation of the HBq equation. For this aim a numerical scheme combining the Fourier pseudo-spectral method in space and a Runge Kutta method in time is constructed. The convergence of semi-discrete scheme is proved in an appropriate Sobolev space. To investigate the higher order dispersive effects and nonlinear effects on the solutions of HBq equation, propagation of single solitary wave, head-on collision of solitary waves and blow-up solutions are considered.