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arXiv · 2402.08352

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

Huan Liu·Jing Shen·Xianguo Geng
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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.

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Huan Liu, Jing Shen, Xianguo Geng. 2024-02-13. Riemann--Hilbert method to the Ablowitz--Ladik equation: higher-order case. https://arxiv.org/abs/2402.08352

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