Modulational instability of a harmonically trapped quantum droplet

We investigate the modulational instability (MI) of a harmonically trapped quantum droplet. The MI is the key mechanism for the formation of soliton trains in diverse physical media, as a result of the interplay between the intrinsic nonlinearity and the kinetic-energy term. Here, we analytically get the time-dependent criterion for MI of a trapped quantum droplets with Lee-Huang-Yang (LHY) term. It is shown that the external harmonic potential can dramatically change the condition of modulational instability. Moreover, we find that there exists a critical time t(c), that is the biggest time before instability of droplet will set in. We find the t(c) depends on the harmonic trapping frequency k. Meanwhile, the results show that the trapping frequency k also determine the solitons' types, a single peak soliton or a soliton train can be excited by the different k. Our theoretical predicts are confirmed by the direct numerical calculation of the generalized Gross-Pitaevskii equation with LHY correction term.