  The properties of a supersolid state (SS) in quasi-one-dimensional dipolar
Bose-Einstein condensate is studied, considering two possible mechanisms of
realization: (i) due to repulsive three-body atomic interactions, (ii) and
quantum fluctuations in the framework of the Lee-Huang-Yang (LHY) theory. The
proposed theoretical model, based on the minimization of the energy functional,
allows evaluating the amplitude of the SS for an arbitrary set of parameters in
the governing Gross-Pitaevskii equation (GPE). To explore the dynamics of the
supersolid first we numerically construct its ground state in a box-like or
parabolic trap then impose a perturbation. In oscillations of the perturbed
supersolid we observe the key manifestation of SS, namely the free flow of the
superfluid fraction through the crystalline fraction of the system. Two
distinct oscillation frequencies of the supersolid associated with the
amplitude (Higgs mode) and phase (Goldstone mode) of the wave function are
identified from numerical simulations of the GPE.
