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Asymmetric periodic traveling wave patterns of two-dimensional Boussinesq systems. (English) Zbl 1143.76396

Summary: We consider a Boussinesq system which describes three-dimensional water waves in a fluid layer with the depth being small with respect to the wave length. We prove the existence of a large family of bifurcating bi-periodic patterns of traveling waves, which are non-symmetric with respect to the direction of propagation. The existence of such bifurcating asymmetric bi-periodic traveling waves is still an open problem for the Euler equation (potential flow, without surface tension).In this study, the lattice of wave vectors is spanned by two vectors \(\mathbf k_1\) and \(\mathbf k_2\) of equal or different lengths and the direction of propagation \(\mathbf c\) of the waves is close to the critical value \(\mathbf c_0\) which is a solution of the dispersion equation. The wave pattern may be understood at leading order as the superposition of two planar waves of equal or different amplitudes, respectively, with wave vectors \(\mathbf k_1\) and \(\mathbf k_2\).
Our class of non-symmetric waves bifurcates from the rest state. The four components of the two basic wave vectors are constrained by the dispersion equation, forming a 3-dimensional set of free parameters. Here we are able to avoid the small divisor problem by restricting the study to propagation directions \(\mathbf c\) such that \((\mathbf k_1 \cdot \mathbf c)/(\mathbf k_2 \cdot \mathbf c)\) is any rational number close to \((\mathbf k_1 \cdot \mathbf c_0)/(\mathbf k_2 \cdot \mathbf c_0)\). However, we need to solve a problem of weak differentiability with respect to the propagation direction for the pseudo-inverse of the linear operator. It appears that the above rationality condition influences only mildly the domain of existence of the bifurcating waves.
In the special case where the lattice is generated by wave vectors \(\mathbf k_1\) and \(\mathbf k_2\) of equal length, the bisecting direction is the critical propagation direction \(\mathbf c_0\), the parameter set is two-dimensional and the rationality condition gives bifurcating asymmetric waves which propagate in a direction \(\mathbf c\) at a small angle with the bisector of \(\mathbf k_1\) and \(\mathbf k_2\).
In the last section of the paper, we show examples of wave patterns for \(\mathbf k_1\) and \(\mathbf k_2\) of equal or different lengths, with various amplitude ratios along the two basic wave vectors and with various angles between the traveling direction \(\mathbf c \) and the critical direction \(\mathbf c_0\).

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76E17 Interfacial stability and instability in hydrodynamic stability
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References:

[1] Bona, J. L.; Colin, T.; Lannes, D., Long wave approximations for water waves, Arch. Ration. Mech. Anal., 178, 3, 373-410 (2005) · Zbl 1108.76012
[2] Bona, J. L.; Chen, M.; Saut, J.-C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory, J. Nonlinear Sci., 12, 4, 283-318 (2002) · Zbl 1022.35044
[3] Chen, M.; Iooss, G., Periodic Wave Patterns of two-dimensional Boussinesq systems, European J. Mech. B Fluids, 25, 393-405 (2006) · Zbl 1122.76018
[4] G. Iooss, P. Plotnikov, Three-dimensional doubly-periodic traveling gravity waves, Mem. AMS (in press); G. Iooss, P. Plotnikov, Three-dimensional doubly-periodic traveling gravity waves, Mem. AMS (in press) · Zbl 1231.35169
[5] Hammack, J. L.; McCallister, D.; Scheffner, N.; Segur, H., Two dimensional periodic waves in shallow water. Part 2. Asymmetric waves, J. Fluid Mech., 285, 95-122 (1995)
[6] Craig, W.; Nicholls, D. P., Traveling gravity water waves in two and three dimensions, European J. Mech. B Fluids, 21, 6, 615-641 (2002) · Zbl 1084.76509
[7] Groves, M. D.; Haragus, M., A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves, J. Nonlinear Sci., 13, 4, 397-447 (2003) · Zbl 1116.76326
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