Continuation of Time Bounds for a Regularized Boussinesq System

We study the periodic solution of a perturbed regularized Boussinesq system (Bona et al., J. Nonlinear Sci. 12:283–318, 2002, Bona et al., Nonlinearity 17:925–952, 2004), namely the system η _t +u _x +β(−η _xxt +u _xxx )+α((ηu) _x +ηη _x +uu _x )=0,u _t +η _x +β(η _xxx −u _xxt )+α((ηu) _x +ηη _x +uu _x )=0, with 0<α,β≤1. We prove that the solution, starting from an initial datum of size ε, remains smaller than ε for a time scale of order (ε ⁻¹ α ⁻¹ β)², whereas the natural time is of order ε ⁻¹ α ⁻¹ β.