Stability of some waves in the dissipative Boussinesq system

In this paper we study analytically a class of waves in the variant of the classical dissipative Boussinesq system given by ∂tu=−∂xv−α∂xxxv+β∂xxtu−ϵ∂x(uv),∂tu=−∂xv−α∂xxxv+β∂xxtu−ϵ∂x(uv),∂tv=−∂xu+c∂xxxu+β∂xxtv−ϵv∂xv,∂tv=−∂xu+c∂xxxu+β∂xxtv−ϵv∂xv, where β,c>0β,c>0, ϵϵ is a small parameter and α∈(0,1)α∈(0,1). This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every ββ, cc and almost every α≤1α≤1, it contains solutions that are defined for large values of time and they are very close (of order O(ϵ)O(ϵ)) to a linear torus for long times (of order O(ϵ−1)O(ϵ−1)). The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.