Novel discrete Kutznetsov–Ma breather solutions of the focusing Ablowitz–Ladik equation

In this work, we present a novel solution of the Ablowitz–Ladik lattice on a nonzero background, which is a discrete analog of the Kuznetsov–Ma (KM) breather of the focusing nonlinear Schrödinger equation, i.e., a solution that is periodic in time and homoclinic in the (discrete) space variable. In earlier works, the Inverse Scattering Transform (IST) was developed to solve the initial-value problem for the focusing AL equation on a non-zero background, and several explicit solutions were constructed as a byproduct of the IST. In particular, discrete KM breathers were obtained for specific choices of the discrete eigenvalue. We show here that an additional KM-type breather solution exists, corresponding to a different choice for the discrete eigenvalue, which we refer to as KM2 breather, in analogy to the solutions recently obtained for the defocusing AL equation on a large background. Like the discrete KM solution already known in the literature (here referred to as KM1), the novel KM2 breather is also regular on the lattice for all times. On the other hand, unlike KM1, the KM2 breather is a purely discrete solution; namely, it is only defined for values of the spatial variable n∈Z. Furthermore, we derive Darboux transformations, which allow us to obtain multi-KM breather solutions of both types, and we present some illustrative examples.