Relative Dynamics of Vortices in Confined Bose--Einstein Condensates
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914947843227648 |
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| author | Ohsawa, Tomoki |
| author_facet | Ohsawa, Tomoki |
| contents | We consider the relative dynamics -- the dynamics modulo rotational symmetry in this particular context -- of $N$ vortices in confined Bose--Einstein Condensates (BEC) using a finite-dimensional vortex approximation to the two-dimensional Gross--Pitaevskii equation. We give a Hamiltonian formulation of the relative dynamics by showing that it is an instance of the Lie--Poisson equation on the dual of a certain Lie algebra. Just as in our accompanying work on vortex dynamics with the Euclidean symmetry, the relative dynamics possesses a Casimir invariant and evolves in an invariant set, yielding an Energy--Casimir-type stability condition. We consider three examples of relative equilibria -- those solutions that are undergoing rigid rotations about the origin -- with $N=2, 3, 4$, and investigate their stability using the stability condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07657 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative Dynamics of Vortices in Confined Bose--Einstein Condensates Ohsawa, Tomoki Mathematical Physics Quantum Gases Pattern Formation and Solitons We consider the relative dynamics -- the dynamics modulo rotational symmetry in this particular context -- of $N$ vortices in confined Bose--Einstein Condensates (BEC) using a finite-dimensional vortex approximation to the two-dimensional Gross--Pitaevskii equation. We give a Hamiltonian formulation of the relative dynamics by showing that it is an instance of the Lie--Poisson equation on the dual of a certain Lie algebra. Just as in our accompanying work on vortex dynamics with the Euclidean symmetry, the relative dynamics possesses a Casimir invariant and evolves in an invariant set, yielding an Energy--Casimir-type stability condition. We consider three examples of relative equilibria -- those solutions that are undergoing rigid rotations about the origin -- with $N=2, 3, 4$, and investigate their stability using the stability condition. |
| title | Relative Dynamics of Vortices in Confined Bose--Einstein Condensates |
| topic | Mathematical Physics Quantum Gases Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2409.07657 |