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

Abstract

AbstractWe focus on the Ablowitz–Ladik equation on the zero background, specifically considering the scenario of 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 pairs of higher‐order zeros. Next, we constructed another mapping from the scattering data to a matrix Riemann–Hilbert (RH) problem equipped with several residue conditions set at 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.