This paper finds new wave patterns in a well‑known discrete model of nonlinear waves called the Ablowitz–Ladik (AL) system. The authors study the defocusing case, which means waves tend to spread rather than focus, and they assume a small but nonzero background amplitude 0<ρ<1. They treat both the single‑component (scalar) AL equation and a two‑component, coupled version often called the integrable discrete Manakov (IDM) system.

The main technical step is to use Hirota’s bilinear method, a direct way to rewrite the nonlinear equation so one can build exact solutions. In the scalar case they first establish a precise map between the parameters that appear in Hirota’s formulas and the spectral parameters used in the inverse scattering transform (IST). That map tells which Hirota solutions correspond to the usual discrete dark solitons (dips on a nonzero background). It also shows that if one picks Hirota parameters outside the dark‑soliton range, new classes of solutions appear.

Many of those new solutions are singular somewhere on the lattice. However the authors identify a family that is time‑periodic and homoclinic in space (a breather that returns to the background far away) and that can be made regular on every lattice site for all times by a suitable choice of parameters. They also describe interactions between a dark soliton and a regular breather, and between two regular breathers. The paper links these discrete time‑periodic breathers to discrete versions of known continuum structures called Kuznetsov–Ma breathers.

For the coupled two‑component AL (the IDM) the authors include discrete, counter‑propagating plane waves in the background. With that setup Hirota’s method produces novel Akhmediev‑type discrete breathers. “Akhmediev‑type” here means the structures are periodic in space and localized in time. These discrete Akhmediev breathers can be regular for all times when the background norm is less than one. By taking the spatial period to infinity (the wavenumber to zero) the authors obtain new coupled rogue‑wave solutions, i.e., large, transient, and spatially localized pulses on the background.