# American Institute of Mathematical Sciences

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March  2018, 17(2): 557-578. doi: 10.3934/cpaa.2018030

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

 Departamento de Matemáticas, Universidad del Valle, Calle 13 No 100-00, Cali, Colombia

* Corresponding author

Received  February 2017 Revised  July 2017 Published  March 2018

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.

Citation: José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030
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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 $10 c$
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
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
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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