  We demonstrate the controllable generation of distinct types of dispersive
shock-waves emerging in a quantum droplet bearing environment with the aid of
step-like initial conditions. Dispersive regularization of the ensuing
hydrodynamic singularities occurs due to the competition between meanfield
repulsion and attractive quantum fluctuations. This interplay delineates the
dominance of defocusing (hyperbolic) and focusing (elliptic) hydrodynamic
phenomena respectively being designated by real and imaginary speed of sound.
Specifically, the symmetries of the extended Gross-Pitaevskii model lead to a
three-parameter family, encompassing two densities and a relative velocity, of
the underlying Riemann problem utilized herein. Surprisingly, dispersive shock
waves persist across the hyperbolic-to-elliptic threshold, while a plethora of
additional wave patterns arise, such as rarefaction waves, traveling dispersive
shock waves, (anti)kinks and droplet wavetrains. The classification and
characterization of these features is achieved by deploying Whitham modulation
theory. Our results pave the way for unveiling a multitude of unexplored
coherently propagating waveforms in such attractively interacting mixtures and
should be detectable by current experiments.
