Newton iteration, was the generally most efficient technique of implementing the incremental step and produced very accurate approximations of the solitary waves for $0\leq \gamma <1$. With this method we also computed other branches of solutions of (\[E15\]), namely [*multi-pulse*]{} solitary waves, by starting the homotopy path from linear combinations of solitary waves of the KdV equation. We verified the accuracy of these profiles as travelling waves of the Benjamin equation by using them as initial values in a full discretization of the periodic initial-value problem for (\[E11\]) and integrating forward in time. The solver combined the pseudospectral spatial discretization with the third-order accurate two-stage DIRK time-stepping technique, modified to preserve discrete analogs of the invariants (\[E12\]) and (\[E13\]). It was found that several quantities of interest, such as the speed, the amplitude and the third invariant (\[E14\]) of the discrete travelling waves, were preserved to very high accuracy, lending confidence in the validity of this technique for computing solitary waves.

In the paper at hand we continue our numerical study of the Benjamin equation. We construct and test numerically a new, efficient time-stepping method based on a spectral-finite element [hybrid]{} spatial discretization combined with a fourth-order implicit Runge-Kutta scheme 