  In this paper the permanent profile waves governed by a Boussinesq-type wave
equation are analysed. The model involves displacement-type nonlinearities and
dispersion terms. Physically such a model equation describes longitudinal waves
(density change) in biomembranes which have an internal structure composed by
lipid molecules. The possible solutions are constructed and analysed. The phase
plane analysis and numerical simulation reveal a novel phenomenon: the possible
existence of a soliton doublet.
