will consider the [*Benjamin equation*]{} $$\label{E11}
u_t+\alpha u_x+\beta u u_x-\gamma \mathcal{H}u_{xx}-\delta u_{xxx}=0,$$ where $u=u(x,t), x\in \mathbb{R}, t\geq 0, \alpha,\beta,\gamma,\delta$ are positive constants, and $\mathcal{H}$ denotes the Hilbert transform defined on the real line as $$\mathcal{H}f(x):=\frac{1}{\pi}p.v.\int_{-\infty}^{\infty}\frac{f(y)}{x-y}\,dy$$ or through its Fourier transform as $$\widehat{\mathcal{H}f}(k)=-{\rm i}{\mathop{\operator@font sign}}(k)\widehat{f}(k), k\in\mathbb{R}.$$ The Benjamin equation, cf. [@B1; @B2; @ABR], is a model for [*internal*]{} waves propagating under the effect of gravity and surface tension in the positive $x$-direction along the interface of a two-dimensional system of two homogeneous layers of incompressible, inviscid fluids consisting at rest of a thin layer of fluid 1 of depth $d_{1}$ and density $\rho_{1}$ lying above a layer of fluid 2 of very large depth $d_{2}\gg d_{1}$ and density $\rho_{2}>\rho_{1}$. The upper layer is bounded above by a horizontal rigid lid and the lower layer is bounded below by an impermeable horizontal bottom, as in Figure \[F1a\].

\[F1a\]

(0,-1.81)(10.54,1.81) (0.0,1.59)(8.96,1.59) (0.0,-1.61)(8.96,-1.61) (0.04,0.71)(8.94,0.71) (9.47,0.715)[$z=0$]{} (9.75,1.615)[$z=d_1$]{} (9.81,-1.585)[$z=-d_2$]{} (5.84,0.71)(5.84,0.21) (6.09,0.015)[$u$]{} (0.68,1.315)[$\rho_1$]{} (0.67,-1.385)[$\rho_2$]{}

It is further assumed that the following physical regime of interest is to be modelled: Let $a$ be a typical amplitude and $\lambda$ a typical wavelength of the interfacial wave. The parameters $\epsilon=a/d_{1}$ and $\mu=d_{1}^{2}/\lambda^{2}$ are assumed to be small and satisfy $\mu\sim\epsilon^{2}\ll 