On the Benjamin and related equations

We consider in this paper various theoretical and numerical issues on classical one dimensional models of internal waves with surface tension.They concern the Cauchy problem, including the long time dynamic, localized solitons or multisolitons, the soliton resolution property. We survey known results, present a few new ones together with open questions and conjectures motivated by numerical simulations. A major issue is to emphasize the differences of the qualitative behavior of solutions with those of the same equations without the capillary term.