  Boussinesq-type wave equations involve nonlinearities and dispersion. In this
paper a Boussinesq-type equation with amplitude-dependent nonlinearities is
presented. Such a model was proposed by Heimburg and Jackson (2005) for
describing longitudinal waves in biomembranes and later improved by Engelbrecht
et al. (2015) taking into account the microinertia of a biomembrane. The steady
solution in the form of a solitary wave is derived and the influence of
nonlinear and dispersive terms over a large range of possible sets of
coefficients demonstrated. The solutions emerging from arbitrary initial inputs
are found using the numerical simulation. The properties of emerging trains of
solitary waves waves are analysed. Finally, the interaction of solitary waves
which satisfy the governing equation is studied. The interaction process is not
fully elastic and after several interactions radiation effects may be
significant. This means that for the present case the solitary waves are not
solitons in the strict mathematical sense. However, like in other cases known
in solid mechanics, such solutions may be conditionally called solitons.
