On the discrete Kuznetsov–Ma solutions for the defocusing Ablowitz–Ladik equation with large background amplitude

The focus of this work is on a class of solutions of the defocusing Ablowitz–Ladik lattice on an arbitrarily large background which are discrete analogs of the Kuznetsov–Ma (KM) breathers of the focusing nonlinear Schrödinger equation. One such solution was obtained in 2019 as a byproduct of the Inverse Scattering Transform, and it was observed that the solution could be regular for certain choices of the soliton parameters, but its regularity was not analyzed in detail. This work provides a systematic investigation of the conditions on the background and on the spectral parameters that guarantee the KM solution to be non-singular on the lattice for all times. Furthermore, a novel KM-type breather solution is presented which is also regular on the lattice under the same conditions. We also employ Darboux transformations to obtain a multi-KM breather solution, and show that parameters choices exist for which a double KM breather solution is regular on the lattice. We analyze the features of these solutions, including their frequency which, when tending to 0, renders them proximal to rogue waveforms. Finally, numerical results on the stability and spatio-temporal dynamics of the single KM breathers are presented, showcasing the potential destabilization of the obtained states due to the modulational instability of their background.