Existence of quasiperiodic solutions for the second-order approximation Boussinesq system

In this paper we study analytically a class of waves in the variant of the classical second-order approximation Boussinesq system given by ∂tu−b∂xxtu+c∂xxxxtu=−∂xv−∂x(uv)−(13−2b)∂xxxv+b∂xxx(uv)+b∂x(u∂xxv)−a∂xxxxxv,∂tv−b∂xxtv+c∂xxxxtv=−∂xu+12∂xxxu−v∂xv−∂x(u∂xxu)+b∂x(v∂xxv)−d∂xxxxxu, where aa, bb, cc, dd are some real constants. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every aa, bb, cc and dd it admits solutions that are quasiperiodic in time.