Small-Mass Asymptotics of Massive Point Vortex Dynamics in Bose--Einstein Condensates

TL;DR

This work analyzes the dynamics of massive point vortices in immiscible two-component Bose–Einstein condensates in the small-mass limit $\varepsilon\to0$. Using a geometric framework, it identifies a kinematic subspace $\mathcal{K}$ on which the massive system projects to the massless Kirchhoff equations, proving that the massive trajectory remains $O(\varepsilon)$-close to $\mathcal{K}$ for short times and deriving a rigorous Gronwall-type bound for the deviation. It additionally derives a first-order correction to the massless dynamics and develops an inner (initial-layer) analysis with a time rescaling $T=t/\varepsilon$ to capture fast inertial oscillations when initial data lie off $\mathcal{K}$, showing the 0th-order inner problem is solvable and the 1st-order correction is integrable by quadrature. Collectively, these results quantify when inertial effects matter for vortex motion, provide practical approximations for realistic BEC systems, and offer a framework for comparing massless vortex models with inertial corrections across short and early-time dynamics.

Abstract

We perform an asymptotic analysis of the massive point vortex dynamics in Bose--Einstein condensates in the small-mass limit $\varepsilon \to 0$. We show that the orthogonal projection of the massive dynamics to a certain subspace $\mathcal{K}$ (called kinematic subspace) in the phase space yields the standard massless vortex dynamics or the Kirchhoff equations. We show that the massless dynamics is the zeroth-order approximation to the massive equation as $\varepsilon \to 0$, and also derive a first-order correction to the zeroth-order massless dynamics. The massive dynamics near $\mathcal{K}$ is well approximated by the corresponding massless dynamics. In fact, our main result proves that the massive dynamics starting in $\mathcal{K}$ stays $O(\varepsilon)$-close to $\mathcal{K}$ for short time. On the other hand, the massive dynamics starting outside $\mathcal{K}$ exhibits an oscillatory initial layer due to its inertia and deviates significantly from the massless dynamics. We also perform an asymptotic analysis with a rescaled time to derive the inner approximation to capture the initial layer. The 0th-order inner equations are exactly solvable, and the 1st-order inner equations can be solved by quadrature. The combined inner approximation captures the initial oscillatory layer of the massive dynamics.

Figures (21)

• Figure 1: Subspace $\mathcal{K}$ and tangent vectors $v_{j}, w_{j} \in T_{\varphi(\mathbf{r})}\mathcal{K}$ forming a basis $\{ v_{j}, w_{j} \}_{j=1}^{N}$ for $T_{\varphi(\mathbf{r})}\mathcal{K}$. Note that, strictly speaking, the tangent vectors in the figure live in tangent spaces of $\mathbb{R}^{2N}$ and $\mathcal{K}$ but are drawn in the base spaces for simplicity.
• Figure 2: Orthogonally projecting $X_{H}$---the vector field defining the massive point vortex equation \ref{['eq:Hamilton']}---from the tangent space of $T^{*}\mathbb{R}^{2N}$ to the tangent space of $\mathcal{K}$ defines vector field $\mathcal{P}(X_{H})$ on $\mathcal{K}$; this in turn corresponds via $\varphi$ to vector field $X_{E}$ on $\mathbb{R}^{2N}$ defining the Kirchhoff equation \ref{['eq:Kirchhoff']}.
• Figure 3: $x$-coordinate of vortex 1
• Figure 4: $y$-coordinate of vortex 1
• Figure 6: $x$-coordinate of vortex 2

Theorems & Definitions (5)

• Remark 1
• Proposition
• proof
• Theorem
• proof