Extreme waves statistics for Ablowitz-Ladik system

We examine statistics of waves for the problem of modulation instability development in the framework of discrete integrable Ablowitz–Ladik (AL) system. Modulation instability depends on one free parameter $h$ that has the meaning of the coupling between the nodes on the lattice. For strong coupling $h\ll 1$ the probability density functions (PDFs) for waves amplitudes coincide with that for the continuous classical Nonlinear Schrodinger (NLS) equation; the PDFs for both systems are very close to Rayleigh ones. When the coupling is weak $h\sim 1$, there appear highly localized waves with very large amplitudes, that drastically change the PDFs to significantly non-Rayleigh ones, with so-called “fat tails” when the probability of a large wave occurrence is by several orders of magnitude higher than that predicted by the linear theory. Evolution of amplitudes for such rogue waves with time is similar to that of the Peregrine solution for the classical NLS equation.