  This study deals with higher-ordered asymptotic equations for the water-waves
problem. We considered the higher-order/extended Boussinesq equations over a
flat bottom topography in the well-known long wave regime. Providing an
existence and uniqueness of solution on a relevant time scale of order
$1/\sqrt{\eps}$ and showing that the solution's behavior is close to the
solution of the water waves equations with a better precision corresponding to
initial data, the asymptotic model is well-posed in the sense of Hadamard. Then
we compared several water waves solitary solutions with respect to the
numerical solution of our model. At last, we solve explicitly this model and
validate the results numerically.
