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 $0 < |c| < 1$ , the problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space $H^{1}_k(\mathbb{R})$ ( $k$ -periodic functions $f∈ L_k^2(\mathbb{R})$ such that $f' ∈ L_k^2(\mathbb{R})$ ). For wave speed $|c|>1$ , the result is a direct consequence of the Lyapunov Center Theorem since the nonlinear system can be rewritten as a $4× 4$ system with a special Hamiltonian structure. In the case $|c|>1$ , 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.
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Figure 1.
Periodic travelling wave solution
Figure 2.
Periodic travelling wave solution
Figure 3.
Surface plot of the wave elevation
Figure 4.
Surface plot of the wave elevation
Figure 5.
Periodic solution
Figure 6.
Periodic solution
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