Riemann--Hilbert method to the Ablowitz--Ladik equation: higher-order case

We focused on the Ablowitz–Ladik equation on a zero background, specifically considering the scenario of $N$ pairs of multiple poles. Our first goal was to establish a mapping between the initial data and the scattering data. This allowed us to introduce a direct problem by analyzing the discrete spectrum associated with $N$ pairs of higher-order zeros. Next, we constructed another mapping from the scattering data to a $2\times 2$ matrix Riemann–Hilbert problem equipped with several residue conditions set at $N$ pairs of multiple poles. By characterizing the inverse problem based on this Riemann–Hilbert problem, we were able to derive higher-order soliton solutions in the reflectionless case. Furthermore, we expressed an infinite-order soliton solution using a special Riemann–Hilbert problem formulation.