(\[E15\]), which have a handy Fourier representation in the periodic case as well, naturally suggests using spectral-type methods for approximating their solutions. The preceding discussion of the Benjamin equation applies to its associated Cauchy problem on $\mathbb{R}$. Solving it numerically requires posing it on a finite $x$-interval $[-L,L]$ with, say, periodic boundary conditions, assuming $2L$-periodic initial data. In case solitary waves, their generation and interactions, are the focus of interest, one should take into account that they decay quadratically. Consequently, the interval $[-L,L]$ should be taken sufficiently large in some experiments to ensure that the numerical solution in the temporal range of interest remains sufficiently small at the endpoints so that the simulations give valid approximations of the solutions of the Cauchy problem.

In [@ABR] the equation (\[E16\]) was discretized in space by a pseudospectral technique and the resulting nonlinear system of equations for the Fourier coefficients of $\psi=\psi_{\tilde{\gamma}}$ for a desired value of $\tilde{\gamma}\in (0,1)$ was solved by an incremental continuation method. This entailed defining a homotopic path $\tilde{\gamma}_{0}=0<\tilde{\gamma_{1}}<\ldots<\tilde{\gamma}_{M}=\tilde{\gamma}$, starting from the known profile of a solitary wave $\psi_{\tilde{\gamma}_{0}}$ of the KdV equation with a given speed $c_{s}$, and computing $\psi_{\tilde{\gamma}_{j+1}}$, given $\psi_{\tilde{\gamma}_{j}}$, by Newton’s method. With this technique 