📝 SummarXiv

Abstract

We propose Faraday waves as a probe for collective excitations in self-gravitating Bose-Einstein condensates (SGBECs), driven by periodic modulation of the $s$-wave scattering length. Linear stability analysis of the driven Gross-Pitaevskii-Newton equations reveals that parametric instability follows a Mathieu-like equation, with Faraday waves emerging resonantly when half the driving frequency matches the collective excitation frequency of the SGBEC. This framework yields a stability phase diagram that maps the competitive interplay between the unstable tongues of parametric resonance and the intrinsic Jeans instability. The diagram reveals that increasing the driving frequency compresses the Jeans-unstable region and allows well-separated parametric resonance tongues to dominate, thereby creating a clear regime for observing Faraday waves. Conversely, lowering the driving frequency expands the domain of Jeans instability, which fragments and overwhelms the parametric resonance structures. We numerically simulate Faraday wave formation and dynamics within the SGBEC, including the effects of dissipation; simulations reveal a characteristic transition from parametric-resonance-driven Faraday waves to gravity-dominated Jeans collapse as the Jeans frequency increases.