  In this paper we consider the Benjamin equation, a partial differential
equation that models one-way propagation of long internal waves of small
amplitude along the interface of two fluid layers under the effects of gravity
and surface tension. We solve the periodic initial-value problem for the
Benjamin equation numerically by a new fully discrete hybrid finite-element /
spectral scheme, which we first validate by pinning down its accuracy and
stability properties. After testing the evolution properties of the scheme in a
study of propagation of single - and multi-pulse solitary waves of the Benjamin
equation, we use it in an exploratory mode to illuminate phenomena such as
overtaking collisions of solitary waves, and the stability of single-,
multi-pulse and 'depression' solitary waves.
