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

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 \(\mathcal{S}\).