1$; it is also assumed that capillarity effects along the interface are not negligible. Under these assumptions (\[E11\]) was derived in [@B1] from the two-dimensional, two-layer Euler equations in the presence of interface surface tension by dispersion relation arguments. The variables in (\[E11\]) are nondimensional and scaled, and the coefficients are given by $$\alpha=\sqrt{\frac{\rho_{2}-\rho_{1}}{\rho_{1}}}, \quad \beta=\frac{3}{2}\alpha \epsilon, \quad\gamma=\frac{1}{2}\alpha\sqrt{\mu}\frac{\rho_{2}}{\rho_{1}}, \quad\delta=\frac{\alpha T}{2g\lambda^{2}(\rho_{2}-\rho_{1})},$$ where $T$ is the interfacial surface tension and $g$ the acceleration of gravity. The variables $x$ and $t$ are proportional to distance along the channel and time, respectively, and $u(x,t)$ denotes the downward vertical displacement of the interface from its level of rest at $(x,t)$. The interfacial surface tension $T$ is assumed to be much larger than $g(\rho_{2}-\rho_{1})d_{1}^{2}$. (For a further discussion of the physical regime of validity of (\[E11\]) cf. [@ABR].) Note that if the parameter $\delta$ is taken equal to zero, (\[E11\]) reduces to the Benjamin-Ono (BO) equation, [@B0; @O], while, if we put $\gamma=0$ we obtain the KdV equation with negative dispersion coefficient.

It is well known, cf. [@B1], that sufficiently smooth solutions of (\[E11\]) that vanish suitably at infinity preserve the functionals $$\begin{aligned}
m(u)&=&\int_{-\infty}^{\infty} udx,\label{E12}\\
I(u)&=&\frac{1}{2}\int_{-\infty}^{\infty} u^{2}dx,\label{E13}\\
E(u)&=&\int_{-\infty}^{\infty}
\left(\frac{\beta}{6}u^{3}-\frac{1}{2}\gamma
u\mathcal{H}u_{x}+\frac{1}{2}\delta u_{x}^{2}\right)dx.\label{E14}\end{aligned}$$ Global well-posedness in $L^{2}$ for 