Riemann–Hilbert method to the Ablowitz–Ladik equation: Higher-order cases

Abstract

We focus on the Ablowitz–Ladik equation on the 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, which 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 (RH) problem equipped with several residue conditions set at $N$ pairs of multiple poles. By characterizing the inverse problem on the basis of this RH problem, we are able to derive higher-order soliton solutions in the reflectionless case.