The Boussinesq system: Dynamics on the center manifold

Abstract

In this paper we analyze the behavior of small amplitude solutions of the variant of the classical Boussinesq system given by ∂tu = -∂xv - α∂xxxv - ∂x(uv), ∂tv = -∂xu - v∂xv, For a ≤ 1, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every a, it admits solutions that are quasiperiodic in time. The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the Boussinesq system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem and studying the Birkhoff normal form.