Small-Mass Asymptotics of Massive Point Vortex Dynamics in Bose--Einstein Condensates I: Averaging and Normal Forms
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2025
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| author | Ohsawa, Tomoki Richaud, Andrea Goodman, Roy |
| author_facet | Ohsawa, Tomoki Richaud, Andrea Goodman, Roy |
| contents | We perform an asymptotic analysis of massive point-vortex dynamics in Bose--Einstein condensates in the small-mass limit $\varepsilon \to 0$. We define two distinguished manifolds in the phase space of the dynamics. We call the first the kinematic subspace $\mathcal{K}$, whereas the second is an almost-invariant set $\mathcal{S}$ called a ``slow manifold.'' The orthogonal projection of the massive dynamics to $\mathcal{K}$ yields the standard massless vortex dynamics or the Kirchhoff equations -- also the 0th-order approximation to the massive equation as $\varepsilon \to 0$. Our first main result proves that the massive dynamics starting $O(\varepsilon)$-close to $\mathcal{K}$ remains $O(\varepsilon)$-close to the massless dynamics for short times. The second main result is the derivation of a normal form for the system's Hamiltonian for the two-vortex case; it describes the coupling between motion within $\mathcal{S}$ and that transverse to it. Specifically, we use the Lie transformation perturbation method to derive the first few terms in a formal expansion for $\mathcal{S}$ and demonstrate numerically that fast oscillations due to the vortices' mass are suppressed, given initial conditions sufficiently close to $\mathcal{S}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_19222 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small-Mass Asymptotics of Massive Point Vortex Dynamics in Bose--Einstein Condensates I: Averaging and Normal Forms Ohsawa, Tomoki Richaud, Andrea Goodman, Roy Quantum Gases Mathematical Physics 34C29, 34D05, 34C20, 35Q55, 37G05, 37J40, 70K45, 76M23 We perform an asymptotic analysis of massive point-vortex dynamics in Bose--Einstein condensates in the small-mass limit $\varepsilon \to 0$. We define two distinguished manifolds in the phase space of the dynamics. We call the first the kinematic subspace $\mathcal{K}$, whereas the second is an almost-invariant set $\mathcal{S}$ called a ``slow manifold.'' The orthogonal projection of the massive dynamics to $\mathcal{K}$ yields the standard massless vortex dynamics or the Kirchhoff equations -- also the 0th-order approximation to the massive equation as $\varepsilon \to 0$. Our first main result proves that the massive dynamics starting $O(\varepsilon)$-close to $\mathcal{K}$ remains $O(\varepsilon)$-close to the massless dynamics for short times. The second main result is the derivation of a normal form for the system's Hamiltonian for the two-vortex case; it describes the coupling between motion within $\mathcal{S}$ and that transverse to it. Specifically, we use the Lie transformation perturbation method to derive the first few terms in a formal expansion for $\mathcal{S}$ and demonstrate numerically that fast oscillations due to the vortices' mass are suppressed, given initial conditions sufficiently close to $\mathcal{S}$. |
| title | Small-Mass Asymptotics of Massive Point Vortex Dynamics in Bose--Einstein Condensates I: Averaging and Normal Forms |
| topic | Quantum Gases Mathematical Physics 34C29, 34D05, 34C20, 35Q55, 37G05, 37J40, 70K45, 76M23 |
| url | https://arxiv.org/abs/2503.19222 |