标题： 黎曼-希尔伯特方法到阿博特-拉迪克方程：高阶情况 显示英文标题

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

摘要： 我们关注的是零背景下的Ablowitz--Ladik方程，具体考虑了$N$对多重极点的情况。 我们的第一个目标是建立初始数据与散射数据之间的映射。 这使我们通过分析与$N$对高阶零点相关的离散谱来引入一个直接问题。 接下来，我们构建了从散射数据到一个带有多个在$N$对多重极点处设置的残差条件的$2\times2$矩阵黎曼-希尔伯特问题的另一个映射。 通过基于此黎曼-希尔伯特问题表征逆问题，我们得出了无反射情况下的高阶孤子解。 此外，我们使用一种特殊的黎曼-希尔伯特问题公式表达了无限阶孤子解。 显示英文摘要

摘要： 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.