Small-Mass Asymptotics of Massive Point Vortex Dynamics in Bose--Einstein Condensates I: Averaging and Normal Forms

Abstract

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 $\mathcal{S}$.