  The Ablowitz-Ladik equations, hereafter called $AL_+$ and $AL_-$, are
distinguished integrable discretizations of respectively the focusing and
defocusing nonlinear Schr\"odinger (NLS) equations. In this paper we first
study the modulation instability of the homogeneous background solutions of
$AL_{\pm}$ in the periodic setting, showing in particular that the background
solution of $AL_{-}$ is unstable under a monochromatic perturbation of any wave
number if the amplitude of the background is greater than $1$, unlike its
continuous limit, the defocusing NLS. Then we use Darboux transformations to
construct the exact periodic solutions of $AL_{\pm}$ describing such
instabilities, in the case of one and two unstable modes, and we show that the
solutions of $AL_-$ are always singular on curves of spacetime. At last, using
matched asymptotic expansion techniques, we describe in terms of elementary
functions how a generic periodic perturbation of the background solution i)
evolves according to $AL_{+}$ into a recurrence of the above exact solutions,
in the case of one and two unstable modes, and ii) evolves according to
$AL_{-}$ into a singularity in finite time if the amplitude of the background
is greater than $1$. The quantitative agreement between the analytic formulas
of this paper and numerical experiments is perfect.
