Riemann--Hilbert method to the Ablowitz--Ladik equation: higher-order case
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929658728022016 |
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| author | Liu, Huan Shen, Jing Geng, Xianguo |
| author_facet | Liu, Huan Shen, Jing Geng, Xianguo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Riemann--Hilbert method to the Ablowitz--Ladik equation: higher-order case Liu, Huan Shen, Jing Geng, Xianguo Exactly Solvable and Integrable Systems Pattern Formation and Solitons 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. |
| title | Riemann--Hilbert method to the Ablowitz--Ladik equation: higher-order case |
| topic | Exactly Solvable and Integrable Systems Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2402.08352 |