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

TL;DR

This work proves the existence of finite-energy traveling-wave solutions for a generalized, higher-order Boussinesq system with Hamiltonian structure by a variational approach and Lions' concentration-compactness, under a velocity constraint and assumptions on the nonlinear function $F$. It then develops and validates spectral numerical methods for both homogeneous and non-homogeneous nonlinearities: a Fourier-based solver for homogeneous cases and a Fourier collocation method with Newton iterations for non-homogeneous cases. Numerical experiments compute solitary waves across diverse parameter regimes, including velocities outside the theoretically predicted existence range, and verify the time-evolution of these waves remains shape-preserving, supporting the theoretical results. The study highlights how nonlinear power and other parameters influence wave shape, revealing potential non-monotone or oscillatory profiles, and provides a robust framework for analyzing solitary waves in higher-order Boussinesq systems with practical implications for wave dynamics in fluid contexts.

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.

Figures (10)

• Figure 1: Solitary wave of the Boussinesq system \ref{['1bbl']} computed with $b=d =2$, $b_2 = d_2=5$, $a=-2$, $c=-2$, $a_2=20$, $c_2=20$, $p=8$ and wave velocity $\omega = 0.8$.
• Figure 2: Solitary wave of the Boussinesq system \ref{['1bbl']} computed with $b=d =4$, $b_2 = d_2=2$, $a=-4$, $c=-4$, $a_2=4$, $c_2=4$, $p=5$ and wave velocity $\omega = 0.8$.
• Figure 3: Solitary wave of the Boussinesq system \ref{['1bbl']} computed with $b=d =4$, $b_2 = d_2=2$, $a=-4$, $c=-4$, $a_2=0.5$, $c_2=0.5$, $p=1$ and wave velocity $\omega = 0.4$.
• Figure 4: Solitary wave of the Boussinesq system \ref{['1bbl']} computed with $b=d =4$, $b_2 = d_2=3$, $a=-4$, $c=-4$, $a_2=1$, $c_2=1$, $p=2$ and wave velocity $\omega = 0.4$.
• Figure 5: Solitary wave solution $(u, \eta)$ of the Boussinesq system \ref{['1bbl']} computed at $t=10$, with parameters $b=d =2$, $b_2 = d_2=5$, $a=-2$, $c=-2$, $a_2=20$, $c_2=20$, $p=8$ and wave velocity $\omega = 0.8$. Dotted line: numerical solution computed using the scheme in \ref{['evol1']}-\ref{['evol2']}, with the initial condition given in Figure \ref{['Solit_u_eta1']}. Solid line: Approximate solitary wave shown in Figure \ref{['Solit_u_eta1']}, translated to the right by $\omega t = 8$.

Theorems & Definitions (19)

• Theorem 1.1
• Lemma 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 2.5
• proof
• Lemma 2.6
• proof