Multiple higher-order poles solutions in spinor Bose-Einstein condensates
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917588860141568 |
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| author | Liu, Huan Shen, Jing Geng, Xianguo |
| author_facet | Liu, Huan Shen, Jing Geng, Xianguo |
| contents | In this study, we explore multiple higher-order pole solutions in spinor Bose--Einstein condensates. These solutions are associated with different pairs of higher-order poles of the transmission coefficient in the inverse scattering transform, and they represent solutions of the spin-1 Gross--Pitaevskii equation. We introduce a direct scattering map that maps initial data to scattering data, which includes discrete spectrums, reflection coefficients, and a polynomial that replaces normalization constants. To analyze symmetries and discrete spectrums in the direct problem, we introduce a generalized cross product in 4-dimensional vector space. Additionally, we characterize the inverse problem in terms of a $4\times 4$ matrix Riemann--Hilbert problem that is subject to residue conditions at these higher-order poles. In the reflectionless scenario, the Riemann--Hilbert problem can be converted into a linear algebraic system. The resulting algebraic system has a unique solution, which allows us to display multiple higher-order poles solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_08362 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiple higher-order poles solutions in spinor Bose-Einstein condensates Liu, Huan Shen, Jing Geng, Xianguo Exactly Solvable and Integrable Systems Pattern Formation and Solitons In this study, we explore multiple higher-order pole solutions in spinor Bose--Einstein condensates. These solutions are associated with different pairs of higher-order poles of the transmission coefficient in the inverse scattering transform, and they represent solutions of the spin-1 Gross--Pitaevskii equation. We introduce a direct scattering map that maps initial data to scattering data, which includes discrete spectrums, reflection coefficients, and a polynomial that replaces normalization constants. To analyze symmetries and discrete spectrums in the direct problem, we introduce a generalized cross product in 4-dimensional vector space. Additionally, we characterize the inverse problem in terms of a $4\times 4$ matrix Riemann--Hilbert problem that is subject to residue conditions at these higher-order poles. In the reflectionless scenario, the Riemann--Hilbert problem can be converted into a linear algebraic system. The resulting algebraic system has a unique solution, which allows us to display multiple higher-order poles solutions. |
| title | Multiple higher-order poles solutions in spinor Bose-Einstein condensates |
| topic | Exactly Solvable and Integrable Systems Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2402.08362 |