  We perform a detailed study of the collective mode across the whole BEC-BCS
crossover in fermionic gases at zero temperature, covering the whole range of
energy beyond the linear regime. This is done on the basis of the dynamical BCS
model. We recover first the results of the linear regime in a simple form. Then
specific attention is payed to the non linear part of the dispersion relation
and its interplay with the continuum of single fermionic excitations. In
particular we consider in detail the merging of collective mode into the
continuum of single fermionic excitations. This occurs not only on the BCS side
of the crossover, but also slightly beyond unitarity on the BEC side. Another
remarkable feature is the very linear behaviour of the dispersion relation in
the vicinity of unitarity almost up to merging with the continuum. Finally,
while on the BEC side the mode is quite analogous to the Bogoliubov mode, a
difference appear at high wavevectors. On the basis of our results we determine
the Landau critical velocity in the BEC-BCS crossover which is found to be
largest close to unitarity. Our investigation has revealed interesting
qualitative features which would deserve experimental exploration as well as
further theoretical studies by more sophisticated means.
