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The Ablowitz-Ladik (AL) equations are distinguished integrablediscretizations of the focusing and defocusing nonlinear Schr\"odinger (NLS)equations. In a previous paper (arXiv:2305.04857) we have studied the effect ofthe modulation instability of the homogeneous background solution of the ALequations in the periodic setting, showing in particular that both modelsexhibit instability properties, and studying, in terms of elementary functions,how a generic periodic perturbation of the unstable background evolves into arecurrence of anomalous waves (AWs). Using the finite gap method, in this paperwe extend the recently developed perturbation theory for periodic NLS AWs tolattice equations, studying the effect of physically relevant perturbations ofthe $AL$ equations on the AW recurrence, like: linear loss, gain, and/orHamiltonian corrections, in the simplest case of one unstable mode. We showthat these small perturbations induce $O(1)$ effects on the periodic AWdynamics, generating three distinguished asymptotic patterns. Since dissipationand higher order Hamiltonian corrections can hardly be avoided in naturalphenomena involving AWs, and since these perturbations induce $O(1)$ effects onthe periodic AW dynamics, we expect that the asymptotic states describedanalytically in this paper will play a basic role in the theory of periodic AWsin natural phenomena described by discrete systems. The quantitative agreementbetween the analytic formulas of this paper and numerical experiments isexcellent.

The effect of loss/gain and hamiltonian perturbations of the Ablowitz-Ladik lattice on the recurrence of periodic anomalous waves