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

Mathematical modeling and numerical analysis for the higher order Boussinesq system