the Cauchy problem and also for the periodic initial-value problem for (\[E11\]) was established in [@L].

In this paper we will study (\[E11\]) numerically, paying particular attention to properties of its [*solitary-wave*]{} solutions. These are travelling-wave solutions of the form $u(x,t)=\varphi(x-c_{s}t), c_{s}>0$, such that $\varphi$ and its derivatives tend to zero as $\xi=x-c_{s}t$ approaches $\pm\infty$. Substituting this expression in (\[E11\]) and integrating once we obtain $$\begin{aligned}
(\alpha-c_{s})\varphi
+\frac{\beta}{2}\varphi^{2}-\gamma \mathrm{H}\varphi-\delta
\varphi^{\prime\prime}=0,\label{E15}\end{aligned}$$ where ${}^{\prime}=d/d\xi$, and the operator $\mathrm{H}$ is defined by $\mathrm{H}:=\mathcal{H}\partial_{x}$, i. e. by $\widehat{\mathrm{H}f}(k)=|k|\widehat{f}(k), k\in \mathbb{R}$. We will assume that $\alpha-c_{s}>0$.

If we perform the change of variables $$\begin{aligned}
\varphi(\xi)=-\frac{2(\alpha-c_{s})}{\beta}\psi(z),\quad z=\sqrt{\frac{\alpha-c_{s}}{\delta}}\xi,\end{aligned}$$ in (\[E15\]), we see that the solitary-wave profile $\psi(z)$ satisfies the ordinary differential equation (ode) $$\begin{aligned}
\psi-2\tilde{\gamma}\mathrm{H}\psi-\psi_{zz}-\psi^{2}=0,\quad z\in\mathbb{R},\label{E16}\end{aligned}$$ where $$\begin{aligned}
\tilde{\gamma}=\frac{\gamma}{2\sqrt{\delta(\alpha-c_{s})}}.\label{E17}\end{aligned}$$ This change of variables and the resulting equation (\[E16\]) was used in [@B1; @B2], and [@ABR]. (In these references $\tilde{\gamma}$ is denoted by $\gamma$.) In his papers Benjamin showed, using degree theory, that for each $\tilde{\gamma}\in [0,1)$, there exists a solution $\psi$ of (\[E16\]) which is an even function of $z$ with $\psi(0)=\max_{z\in\mathbb{R}}\psi(z)>0$. He also argued by formal asymptotics that for each $\tilde{\gamma}\in [0,1)$ there is a bounded interval centered at $z=0$, in which $\psi$ oscillates (with the 