Modulational instability and dynamics of multi-rogue wave solutions for the discrete Ablowitz-Ladik equation

Abstract

The higher order discrete rogue waves (RWs) of the integrable discrete Ablowitz-Ladik equation are reported using a novel discrete version of generalized perturbation Darboux transformation. The dynamical behaviors of strong and weak interactions of these RWs are analytically and numerically discussed, which exhibit the abundant wave structures. We numerically show that a small noise has the weaker effect on strong-interaction RWs than weak-interaction RWs, whose main reason may be related to main energy distributions of RWs. The interaction of two first-order RWs is shown to be non-elastic. Moreover, we find that the maximal number (Smax) of the possibly split first-order ones of higher order RWs is related to the number (Pmax) of peak points of their strongest-interaction cases, that is, Smax = (Pmax + 1)/2. The results will excite to further understand the discrete RW phenomena in nonlinear optics and relevant fields.