number of oscillations increasing as $\tilde{\gamma}$ approaches $1$), while outside this interval he concluded in [@B2] that $|\psi|$ decays like $1/z^{2}$. In addition, in the same paper he outlined an orbital stability theory for these solitary waves for small $\tilde{\gamma}$. In [@ABR] a complete theory of existence and orbital stability of the solitary waves for small $\tilde{\gamma}$ was presented, based on the implicit function theorem, perturbation theory of operators, and the fact that $\tilde{\gamma}=0$ corresponds to solitary waves of the KdV equation. Further issues of existence and rigorous asymptotics of the solitary waves of (\[E11\]) and related equations were explored in [@CB]. In [@A] concentration compactness arguments were used to establish existence and a weaker version of stability of the solitary waves of (\[E11\]) for $0<\tilde{\gamma}<1$.

In this paper we will employ the solitary-wave equation in the form (\[E15\]). As a result, normally the solitary waves will have negative maximum excursions from their level of rest.

Since explicit formulas for the solitary waves of the Benjamin equation are not known (except when one of $\gamma$ or $\delta$ is set equal to zero), one must resort to approximate techniques for their construction. The presence of the nonlocal terms in (\[E11\]) and 