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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
##### References:
 [1] U. M. Asher, S. J. Ruuth and B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823. [2] C. Canuto, M. Y. Hussaini and A. Quarteroni, Spectral Methods in Fluid Dynamics Series in Computational Physics, 1988, Springer, Berlin. [3] G. E. Karniadakis, M. Israeli and S. A. Orszag, High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97 (1991), 414-443. [4] T. Kato, Quasilinear equations of evolution with applications to partial differential equations, Proceedings of the symposium at Dundee, Lecture Notes in Mathematics, 448, Springer, (1975), 25-70. [5] T. Kato, On the Korteweg-de Vries equation, Manuscripta Mathematica, 28 (1979), 89-99. [6] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Mathematics, Advances in Mathematics, Supplementary Studies, 8, Academic Press, (1983), 92-128. [7] J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323. [8] K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body problem 2nd ed. Applied Mathematical Sciences, vol. 90,2009, Springer-Verlag. [9] P. A. Milewski and J. B. Keller, Three dimensional water waves, Studies Appl. Math., 37 (1996), 149-166. [10] L. Paumond, A rigorous link between KP and a Benney-Luke Equation, Diff. Int. Eq., 16 (2003), 1039-1064. [11] J. Quintero, Solitary water waves for a 2D Boussinesq type system, J. Part. Diff. Eqs., 23 (2010), 251-280. [12] J. Quintero, The Cauchy problem and stability of solitary waves for a 2D Boussinesq-KdV type system, Diff. Int. Eqs., 21 (2011), 325-360. [13] J. Quintero, From periodic travelling waves to solitons of a 2D water wave system, Meth. Appl. Anal., 21 (2014), 241-264. [14] J. Quintero, A water wave mixed type problem: existence of periodic travelling waves for a 2D Boussinesq system, Rev. Academia Colombiana de Ciencias Naturales, Físicas y Exactas., 38 (2015), 6-17. [15] J. R. Quintero and R. L. Pego, Two-dimensional solitary waves for a Benney-Luke equation, Physica D., 45 (1999), 476-496. [16] J. G. Verwer, J. G. Blom and W. Hundsdorfer, An implicit-explicit approach for atmospheric transport-chemistry problems, Applied Numerical Mathematics, 20 (1996), 191-209. [17] G. B. Whitham, Linear and Nonlinear Waves Wiley-Interscience, 1974.

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##### References:
 [1] U. M. Asher, S. J. Ruuth and B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823. [2] C. Canuto, M. Y. Hussaini and A. Quarteroni, Spectral Methods in Fluid Dynamics Series in Computational Physics, 1988, Springer, Berlin. [3] G. E. Karniadakis, M. Israeli and S. A. Orszag, High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97 (1991), 414-443. [4] T. Kato, Quasilinear equations of evolution with applications to partial differential equations, Proceedings of the symposium at Dundee, Lecture Notes in Mathematics, 448, Springer, (1975), 25-70. [5] T. Kato, On the Korteweg-de Vries equation, Manuscripta Mathematica, 28 (1979), 89-99. [6] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Mathematics, Advances in Mathematics, Supplementary Studies, 8, Academic Press, (1983), 92-128. [7] J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323. [8] K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body problem 2nd ed. Applied Mathematical Sciences, vol. 90,2009, Springer-Verlag. [9] P. A. Milewski and J. B. Keller, Three dimensional water waves, Studies Appl. Math., 37 (1996), 149-166. [10] L. Paumond, A rigorous link between KP and a Benney-Luke Equation, Diff. Int. Eq., 16 (2003), 1039-1064. [11] J. Quintero, Solitary water waves for a 2D Boussinesq type system, J. Part. Diff. Eqs., 23 (2010), 251-280. [12] J. Quintero, The Cauchy problem and stability of solitary waves for a 2D Boussinesq-KdV type system, Diff. Int. Eqs., 21 (2011), 325-360. [13] J. Quintero, From periodic travelling waves to solitons of a 2D water wave system, Meth. Appl. Anal., 21 (2014), 241-264. [14] J. Quintero, A water wave mixed type problem: existence of periodic travelling waves for a 2D Boussinesq system, Rev. Academia Colombiana de Ciencias Naturales, Físicas y Exactas., 38 (2015), 6-17. [15] J. R. Quintero and R. L. Pego, Two-dimensional solitary waves for a Benney-Luke equation, Physica D., 45 (1999), 476-496. [16] J. G. Verwer, J. G. Blom and W. Hundsdorfer, An implicit-explicit approach for atmospheric transport-chemistry problems, Applied Numerical Mathematics, 20 (1996), 191-209. [17] G. B. Whitham, Linear and Nonlinear Waves Wiley-Interscience, 1974.
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 $10 c$
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
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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