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

TL;DR

This paper develops a Riemann--Hilbert approach to analyze higher-order soliton solutions of the Ablowitz--Ladik equation with multiple poles, extending the inverse scattering method for integrable discrete systems.

Contribution

It introduces a novel Riemann--Hilbert framework for the higher-order Ablowitz--Ladik equation, including the construction of solutions with multiple poles and infinite-order solitons.

Findings

Derived higher-order soliton solutions in the reflectionless case

Established a mapping between initial data and scattering data for multiple poles

Formulated an infinite-order soliton solution using Riemann--Hilbert problem

Abstract

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 $2\times2$ 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.