for time-stepping. This method is used to explore properties of solitary-wave solutions of (\[E11\]), such as their generation, interaction and stability.

Much of numerical work with spectral-type methods for one-dimensional, nonlocal, nonlinear dispersive wave equations has been centered around the Benjamin-Ono (BO), [@B0; @O], and the Intermediate Long Wave (ILW) equation, [@J; @ABFS]. Early computational work was reviewed in [@PD1]; here we mention only the rigorous convergence results known to us. In [@PD2] $L^{2}-$error estimates were derived for the standard Fourier-Galerkin semidiscretization of the BO and ILW equations. If the number of Fourier modes is $2N+1$ and the initial value is $2L-$periodic and belongs to the periodic Sobolev space $H_{p}^{r}$, the $L^{2}$-error bounds derived in [@PD2] are of $O(N^{1-r})$. In addition, the full discretization of the semidiscrete system of ode’s with the explicit leap-frog scheme is shown in [@PD2] to have an $L^{2}-$error bound of $O(N^{1-r}+\Delta t^{2})$ under the stability restriction that $N^{2}\Delta t\leq C$ for a sufficiently small constant $C$; here $\Delta t$ is the time step. For a class of equations with the same nonlocal terms and more general nonlinear terms it was subsequently shown in [@DM1] that the error of the Fourier-Galerkin semidiscretization is of optimal order 