Asymmetrical Periodic Wave Patterns of two-dimensional Boussinesq systems

Abstract

We consider a Boussinesq system which describes three-dimensional water waves in fluid layers with a depth small with respect to the wave length. We prove the existence of a large family of bifurcating bi-periodic patterns of travelling waves, which are \emph{non symmetric with respect to the direction of propagation}.

Up to now, the existence of bifurcating asymmetrical bi-periodic travelling wave is still an open problem for the Euler equation (potential flow, without surface tension). Here the lattice of wave vectors is spanned by two vectors $\mathbf{k}_{1},% \mathbf{k}_{2}$ of non equal lengths, and the direction of propagation of the waves is close to the critical one (solution of the dispersion equation). The wave pattern may be understood at main order as the superposition of two plane waves of different amplitudes, respectively propagating along directions $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$. Our class of non symmetric waves bifurcates from a 3-dimensional set of parameters which come from the components of the two basic wave vectors, constrained by the dispersion equation. Here we are able to escape from the \emph{small divisor problem} in restricting the study with \emph{one rationality condition} relating the bifurcation set and the direction of propagation close to the critical direction. However, we need to solve a problem of lack of smoothness with respect to the propagation direction, of the pseudo-inverse of the linearized operator. The rationality condition influences mildly the domain of existence of the bifurcating waves. This theory also applies when the lattice is built with wave vectors $\mathbf{k}% _{1}$, $\mathbf{k}_{2}$ of equal lengths with the bisector direction as the critical propagation direction. In such a case, the parameter set is two-dimensional and there is still one rationality condition for the bifurcating asymmetrical waves which propagate in a direction making a small angle with the bisector of $\mathbf{k}_{1}$, $\mathbf{k}_{2}$. Examples of wave patterns for $\mathbf{k}_{1}$, $\mathbf{k}_{2}$ of equal or different length, with various amplitude ratios along the two basic wave vectors, and with various angles between the traveling direction and the critical direction, are shown in the last section of the paper.