On the existence and computation of periodic travelling waves for a 2D water wave model

Abstract

In this work we establish the existence of at least one weak periodic solution in the spatial directions of a nonlinear system of two coupled differential equations associated with a 2D Boussinesq model which describes the evolution of long water waves with small amplitude under the effect of surface tension. For wave speed \begin{document} $0 , the problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space \begin{document} $H^{1}_k(\mathbb{R})$ \end{document} ( \begin{document} $k$ \end{document} -periodic functions \begin{document} $f∈ L_k^2(\mathbb{R})$ \end{document} such that \begin{document} $f' ∈ L_k^2(\mathbb{R})$ \end{document} ). For wave speed \begin{document} $|c|>1$ \end{document} , the result is a direct consequence of the Lyapunov Center Theorem since the nonlinear system can be rewritten as a \begin{document} $4× 4$ \end{document} system with a special Hamiltonian structure. In the case \begin{document} $|c|>1$ \end{document} , we also compute numerical approximations of these travelling waves by using a Fourier spectral discretization of the corresponding 1D travelling wave equations and a Newton-type iteration.