Multiple higher-order poles solutions in spinor Bose-Einstein condensates

• Main Authors: Liu, Huan, Shen, Jing, Geng, Xianguo
• Format: Journal Article
• Language: English
• Published: 13.02.2024
• Subjects: Physics - Exactly Solvable and Integrable Systems; Physics - Pattern Formation and Solitons
• DOI: 10.48550/arxiv.2402.08362

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.