  Consideration is given to three different full dispersion Boussinesq systems
arising as asymptotic models in the bi-directional propagation of weakly
nonlinear surface waves in shallow water. We prove that, under a non-cavitation
condition on the initial data, these three systems are well-posed on a time
scale of order $\mathcal{O}(\frac1\varepsilon)$, where $\varepsilon$ is a small
parameter measuring the weak non-linearity of the waves. This result seems new
for one of these systems, even for short time. The two other systems involve
surface tension, and for one of them, the non-cavitation condition has to be
sharpened when the surface tension is small. The proof relies on suitable
symmetrizers and the classical theory of hyperbolic systems. However, we have
to track the small parameters carefully in the commutator estimates to get the
long time well-posedness.
  Finally, combining our results with the recent ones of Emerald provide a full
justification of these systems as water wave models in a larger range of
regimes than the classical $(a,b,c,d)$-Boussinesq systems.
