Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background

In this work we develop the inverse scattering transform (IST) for the defocusing Ablowitz–Ladik (AL) equation with arbitrarily a large nonzero background at space infinity. The IST was developed in previous works under the assumption that the amplitude of the background (Formula presented.) satisfies a “small norm” condition (Formula presented.). On the other hand, Ohta and Yang recently showed that the defocusing AL system, which is modulationally stable for (Formula presented.), becomes unstable if (Formula presented.), and exhibits discrete rogue wave solutions, some of which are regular for all times. Here, we construct the IST for the defocusing AL with (Formula presented.), analyze the spectrum, and characterize the soliton and rational solutions from a spectral point of view. We formulate the direct and inverse problems by using a suitable uniformization variable, and pose the inverse problem as an RHP across a simple contour in the complex plane of the uniform variable. As a by-product of the IST, we also obtain explicit soliton solutions, which are the discrete analog of the celebrated Kuznetsov–Ma, Akhmediev, Peregrine solutions, and which mimic the corresponding solutions for the focusing AL equation. Soliton solutions that are the analog of the dark soliton solutions of the defocusing AL equation in the case (Formula presented.) are also presented.