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

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• 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.

Mathematics Subject Classification: Primary:35C07, 35C08, 65M70, 35A15;Secondary:35A24, 35B10.

 Citation: • • Figure 1.  Periodic travelling wave solution $(\eta,\varphi)$ of system (36)-(37) with $p = 1$, $\sigma = 0.52$, $\epsilon = \mu = 0.01$, $\beta = 50$, $\nu = 0.093$, $\gamma = 4.44$, $\rho = 0.02$, $\beta_1 = 0.01$, $\beta_2 = 1$, $\beta_3 = 2.59$, $c_0 = 1.2$, wave speed $c = 48.81$ and period $T = 91.7$, obtained after 6 Newton's iterations. In solid line is the numerical simulation at $t = 10$ obtained with the scheme (63)-(64) and in points is the travelling wave computed with the Newton's procedure translated a distance of $10 c$

Figure 2.  Periodic travelling wave solution $(\eta,\varphi)$ of system (36)-(37) with $p = 1$, $\sigma = 2$, $\epsilon = \mu = 0.01$, $\beta = 15$, $\nu = 0.093$, $\gamma = 4.44$, $\rho = 0.067$, $\beta_1 = 0.01$, $\beta_2 = 1$, $\beta_3 = 2.59$, $c_0 = 1.2$, wave speed $c = 13.83$ and period $T = 45.15$, obtained after 7 Newton's iterations. In solid line is the numerical simulation at $t = 10$ obtained with the scheme (63)-(64) and in points is the travelling wave computed with the Newton's procedure translated a distance of $10 c$

Figure 3.  Surface plot of the wave elevation $\tilde{\eta}(x,y,t) = \eta(x+\beta y,t)$ in the original system (35) at $t = 0$, with the parameters used in Figure 1

Figure 4.  Surface plot of the wave elevation $\tilde{\eta}(x,y,t) = \eta(x+\beta y,t)$ in the original system (35) at $t = 0$, with the parameters used in Figure 2

Figure 5.  Periodic solution $(\zeta,u)$ of system (53)-(54) with $p = 1$, $\sigma = 1$, $\epsilon = \mu = 0.1$, $\beta = 15$, wave speed $c = 30$ and period $T_0 =52.83$, obtained after 18 Newton's iterations. Observe that this solution satisfies the condition on the wave speed $c^2 > 1+ \beta^2$ as required in Theorem 3.2

Figure 6.  Periodic solution $(\zeta,u)$ of system (53)-(54) with $p = 1$, $\sigma = 1$, $\epsilon = \mu = 0.1$, $\beta = 15$, $b = -0.1482$, wave speed $c = 20$ and period $T_+(1) =31.4879$, obtained after 12 Newton's iterations. Observe that this solution satisfies the condition on the wave speed $c^2 > 1+ \beta^2$ as required in Theorem 3.3

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