Quasiperiodic Solutions for Dissipative Boussinesq Systems

Abstract

In this paper we analyze the behavior of the solution of the dissipative Boussinesq systems $$\partial_t u = -\partial_x v - \alpha \partial_{xxx} v +\beta \partial_{xxt} u - \partial_x(uv)$$ $$\partial_t v = - \partial_x u +c \partial_{xxx} u + \beta \partial_{xxt} v - v \partial_x v$$ where α, β, c > 0 are parameters. Those systems model two-dimensional small amplitude long wavelength water waves. For α ≤ 1, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every β, c and almost every α ≤ 1, 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.