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

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.