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

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, if they live on a background of sufficiently large amplitude, and we construct a different continuous limit describing this regime: a NLS equation with a nonlinear and weak dispersion. 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.