Traveling-wave solutions for a higher-order Boussinesq system: existence and numerical analysis

TL;DR

This paper investigates the existence and numerical computation of traveling wave solutions for a broad class of higher-order Boussinesq systems, combining variational methods and spectral numerical techniques.

Contribution

It establishes existence results using variational principles and develops spectral and collocation methods for numerical approximation of solutions.

Findings

Existence of traveling wave solutions proven for a wide class of systems.

Spectral and collocation methods effectively approximate solutions.

Numerical experiments explore solutions outside the theoretical velocity range.

Abstract

We study the existence and numerical computation of traveling wave solutions for a family of nonlinear higher-order Boussinesq evolution systems with a Hamiltonian structure. This general Boussinesq evolution system includes a broad class of homogeneous and non-homogeneous nonlinearities. We establish the existence of traveling wave solutions using the variational structure of the system and the \textit{concentration-compactness} principle by P.-L. Lions, even though the nonlinearity could be non-homogeneous. For the homogeneous case, the traveling wave equations of the Boussinesq system are approximated using a spectral approach based on a Fourier basis, along with an iterative method that includes appropriate stabilizing factors to ensure convergence. In the non-homogeneous case, we apply a collocation Fourier method supplemented by Newton's iteration. Additionally, we present numerical experiments that explore cases in which the wave velocity falls outside the theoretical range of existence.