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The Ablowitz-Ladik equations, hereafter called $AL_+$ and $AL_-$, aredistinguished integrable discretizations of respectively the focusing anddefocusing nonlinear Schr\"odinger (NLS) equations. In this paper we firststudy the modulation instability of the homogeneous background solutions of$AL_{\pm}$ in the periodic setting, showing in particular that the backgroundsolution of $AL_{-}$ is unstable under a monochromatic perturbation of any wavenumber if the amplitude of the background is greater than $1$, unlike itscontinuous limit, the defocusing NLS. Then we use Darboux transformations toconstruct the exact periodic solutions of $AL_{\pm}$ describing suchinstabilities, in the case of one and two unstable modes, and we show that thesolutions of $AL_-$ are always singular on curves of spacetime. At last, usingmatched asymptotic expansion techniques, we describe in terms of elementaryfunctions 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 backgroundis greater than $1$. The quantitative agreement between the analytic formulasof this paper and numerical experiments is perfect.

Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz-Ladik lattices