  The properties of a supersolid state (SS) in quasi-one-dimensional dipolar
Bose-Einstein condensate is studied, considering two possible mechanisms of
realization - due to repulsive three-body atomic interactions and quantum
fluctuations in the framework of the Lee-Huang-Yang (LHY) theory. The proposed
theoretical model, based on 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 SS
first, we numerically construct its ground state in different settings,
including periodic boundary conditions, box-like trap and parabolic potential,
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 component of the system. Two distinct
oscillation frequencies of the supersolid associated with the superfluid
fraction and crystalline components of the wave function are identified from
numerical simulations of the GPE.
