spline discretization of the rest of the terms of the Benjamin equation with an accurate, unconditionally stable time stepping procedure which is effective in approximating highly stiff problems such as semidiscretizations of the Benjamin equation in the presence of the KdV term.

-   The validation of the accuracy of the numerically generated single- and multi- pulse solitary wave solutions by showing that when used as initial values of the hybrid scheme they produce highly accurate approximations to travelling wave solutions of the evolution problem. These approximate solitary waves were computed by a Fourier spatial discretization of the solitary wave ode (\[E15\]) coupled with a continuation conjugate gradient-Newton nonlinear system solver that was proposed by the authors in [@DDM1] and can produce accurate solitary waves for any desired values of the speed $c_{s}>\alpha$ and $\gamma\in [0,1)$, avoiding the drawbacks of the iterative cleaning .

-   The illumination, by computational means, of important phenomena associated with solitary waves of nonlinear dispersive wave equations, such as their one-way interaction (overtaking collision) and stability properties in the case of the Benjamin equation.

In the paper, we denote , for integer $r\geq 0$, by $C_{p}^{r}$ the periodic functions, on $[-L,L]$ or $[0,2\pi]$ as the case 