Title: Onset of a modulational instability in trapped dipolar Bose-Einstein condensates,
Abstract: We explore the phase diagram of a finite-sized dysprosium dipolar
Bose-Einstein condensate in a cylindrical harmonic trap. We monitor the final
state after the scattering length is lowered from the repulsive BEC regime to
the quantum droplet regime. Either an adiabatic transformation between a BEC
and a quantum droplet is obtained or, above a critical trap aspect ratio
$\lambda_{\rm c}=1.87(14)$, a modulational instability results in the formation
of multiple droplets. This is in full agreement with the predicted structure of
the phase diagram with a crossover region below $\lambda_{\rm c}$ and a
multistable region above. Our results provide the missing piece connecting the
previously explored regimes resulting in a single or multiple dipolar quantum
droplets.