  In this work, we propose a management method for controlling the speed and
direction of self-bound quantum droplets (QDs) in a binary Bose-Einstein
condensate mixture under time-modulated external harmonic confinement.
Utilizing the 1D extended Gross-Pit"{a}evskii equation, QDs are constructed
within both regular and expulsive parabolic traps, considering temporally
varying attractive quadratic beyond mean field and repulsive cubic mean-field
atom-atom interactions. Through the derived wavefunction solution, we
illustrate the dynamics of slowing, stopping, reversing, fragmentation,
collapse, and revival of droplets. Additionally, the solutions reveal a
crystalline order with a superfluid background, indicative of supersolid
behavior in various parameter domains. Notably, one-third of the constant
background matches the lowest residual condensate. These findings hold
potential applications in matter-wave interferometry and quantum information
processing.
