Asymmetric periodic traveling wave patterns of two-dimensional Boussinesq systems

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 k1 and k2 of equal or different lengths and the direction of propagation c of the waves is close to the critical value c0 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 k1 and k2.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 c such that (k1⋅c)/(k2⋅c) is any rational number close to (k1⋅c0)/(k2⋅c0). 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 k1 and k2 of equal length, the bisecting direction is the critical propagation direction c0, the parameter set is two-dimensional and the rationality condition gives bifurcating asymmetric waves which propagate in a direction c at a small angle with the bisector of k1 and k2.In the last section of the paper, we show examples of wave patterns for k1 and k2 of equal or different lengths, with various amplitude ratios along the two basic wave vectors and with various angles between the traveling direction c and the critical direction c0.