  We analyse the nonlinear dynamics of Fabry-Perot cavities of arbitrary
finesse filled by a dispersive Kerr medium, pumped by a continuous wave laser
or a synchronous train of flat-top pulses. The combined action of feedback,
group velocity dispersion and Kerr nonlinearity leads to temporal instability
with respect to perturbations at specified frequencies. We characterise the
generation of new spectral bands by deriving the exact dispersion relation and
we find approximate analytical expressions for the instabilities threshold and
gain spectrum of modulation instability (MI). We show that, in contrast to
ring-resonators, both the stationary solutions and the gain spectrum are
dramatically affected by the duration of the pump pulse. We derive the extended
Lugiato-Lefever equation for the Fabry-Perot resonator (FP-LLE) starting from
coupled nonlinear Schr\"odinger equations (rather than Maxwell-Bloch equations)
and we compare the outcome of the stability analysis of the two models. While
FP-LLE gives overall good results, we show regimes that are not captured by the
mean-field limit, namely the period-two modulation instability, which may
appear in highly detuned or nonlinear regimes. We report numerical simulations
of the generation of MI-induced Kerr combs by solving FP-LLE and the coupled
Schr\"odinger equations.
