The Cauchy problem and stability of solitary waves for a 2D Boussinesq-KdV type system

Abstract

We address the well posedness of the Cauchy problem and the stability of solitary waves for a Boussinesq system in $\mathbb{R}^{1+2}$. We exploit the fact that this 2D system has a ``KdV'' structure in the sense that it takes the form $U_t =\mathcal A_0U+ \mathcal{\mathcal A}(U) U$, where $\mathcal A_0$ is a third-order linear operator and the entries of the operator $\mathcal{A}(U)(U)$ are linear combinations of products of powers of components of $U$ with its order one spatial derivatives, as in the well-known 1D-KdV model. Using this ``2D-KdV'' structure, we establish existence and uniqueness for the Cauchy problem associated with the Boussinesq type system by following Kato's approach for the generalized KdV equation. By a variational argument, we obtain global well posedness in time for small initial data. We prove orbital stability of solitary waves directly, by using a variational approach involving the characterization of the ground state solutions, as is done for some 2-D models.