of $\gamma\in (0,1)$ that can be taken quite close to $1$.

In Section \[sec4\] we make a detailed computational study of overtaking (‘one-way’) collisions of solitary waves of the Benjamin equation and compare the inelastic character of these interactions with the analogous, clean interactions in the case of the integrable BO equation. Finally, in Section \[sec5\] we explore issues of stability and instability of single-and multi-pulse solitary waves of the Benjamin equation under small and large perturbations. Our computational study confirms the stability of the single-pulse solitary waves for small and moderate values of $\gamma$ but is inconclusive for cases of $\gamma$ very close to $1$. The multi-pulse waves appear to be unstable and our experiments suggest that after an initial orbiting or dancing phase, they produce separated solitary waves. This confirms the conjecture of [@KB] that was mentioned previously. Finally, we examine the stability of the depression solitary waves and confirm the results of [@CA] regarding their instability.

In summary, the main contributions of the paper at hand are

-   The construction of a novel, highly accurate, stable and efficient hybrid scheme that combines the accuracy of the spectral approximation of the nonlocal term and the accuracy of the 