Multiphase flow modelling is still a major challenge in fluid dynamics and, although many different models have been derived, there is no clear evidence of their relevance to certain flow situations. That is particularly valid for bubbly flows, because most of the studies have considered the case of fluidized beds. In the present study we give a general formulation to five existing models and study their relevance to bubbly flows. The results of the linear analysis of those models clearly show that only two of them are applicable to that case. They both show a very similar qualitative linear stability behaviour. In the subsequent asymptotic analysis we derive an equation hierarchy which describes the weakly nonlinear stability of the models. Their qualitative behaviour up to first order with respect to the small parameter is again identical. A permanent-wave solution of the first two equations of the hierarchy is found. It is shown, however, that the permanent-wave (soliton) solution is very unlikely to occur for the most common case of gas bubbles in water. The reason is that the weakly nonlinear equations are unstable due to the low magnitude of the bulk modulus of elasticity. Physically relevant stabilization can eventually be achieved using some available experimental data. Finally, a necessary condition for existence of a fully nonlinear soliton is derived.