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

Using the finite gap method, in this paper we extend the recently developed perturbation theory for anomalous waves (AWs) of the periodic nonlinear Schrödinger (NLS) type equations to lattice equations, using as basic model the Ablowitz–Ladik (AL) lattices, integrable discretizations of the focusing and defocusing NLS equations. We study the effect of physically relevant perturbations of the AL equations, like linear loss, gain, and/or Hamiltonian corrections, on the AW recurrence, in the simplest case of one unstable mode. We show that these small perturbations induce O(1) effects on the periodic AW dynamics, generating three distinguished asymptotic patterns. Since dissipation and higher order Hamiltonian corrections can hardly be avoided in natural phenomena involving AWs, we expect that the asymptotic states described analytically in this paper will play a basic role in the theory of periodic AWs in natural phenomena described by discrete systems. The quantitative agreement between the analytic formulas of this paper and numerical experiments is excellent.