Global Bifurcation In Four-Component Bose-Einstein Condensates In Space
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arXiv
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| Format: | Preprint |
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2025
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| author | García-Azpeitia, Carlos Gołȩbiewska, Anna Krawcewicz, Wieslaw Liu, Jingzhou |
| author_facet | García-Azpeitia, Carlos Gołȩbiewska, Anna Krawcewicz, Wieslaw Liu, Jingzhou |
| contents | We analyze a system of coupled Bose-Einstein condensates in the domain of a unitary ball in $\mathbb{R}^3$. The coupling is due to atom-to-atom interactions that occur between different gas components. The multi-component Bose-Einstein condensate is described by a system of Gross-Pitaevskii equations, which has an explicit trivial branch of constant solutions bifurcating from the zero-solution. Our main theorem establishes that this trivial branch undergoes multiple global bifurcations at any critical values with kernels of dimensions at least $3(2k+1)$, for $k \in \mathbb{N}^+$. Handling these high dimension kernels poses a challenge from the perspective of bifurcation theory. Our methodology, which relies on the $G$-equivariant gradient degree, effectively manages these complexities and establishes the existence of at least two global branch in the particular case of $k = 0$ and at least six branches in the case of $k = 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07235 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Bifurcation In Four-Component Bose-Einstein Condensates In Space García-Azpeitia, Carlos Gołȩbiewska, Anna Krawcewicz, Wieslaw Liu, Jingzhou Analysis of PDEs We analyze a system of coupled Bose-Einstein condensates in the domain of a unitary ball in $\mathbb{R}^3$. The coupling is due to atom-to-atom interactions that occur between different gas components. The multi-component Bose-Einstein condensate is described by a system of Gross-Pitaevskii equations, which has an explicit trivial branch of constant solutions bifurcating from the zero-solution. Our main theorem establishes that this trivial branch undergoes multiple global bifurcations at any critical values with kernels of dimensions at least $3(2k+1)$, for $k \in \mathbb{N}^+$. Handling these high dimension kernels poses a challenge from the perspective of bifurcation theory. Our methodology, which relies on the $G$-equivariant gradient degree, effectively manages these complexities and establishes the existence of at least two global branch in the particular case of $k = 0$ and at least six branches in the case of $k = 1$. |
| title | Global Bifurcation In Four-Component Bose-Einstein Condensates In Space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.07235 |