Sounds waves and fluctuations in one-dimensional supersolids

We consider the low energy excitations of a dilute supersolid state of matter with a one-dimensional crystal structure. We develop a hydrodynamic description based on a Lagrangian with generalized elastic parameters determined from ground state calculations. We are able to compare the predictions of this hydrodynamic theory to the direct calculation of the excitation spectrum obtained by solving the Bogoliubov-de Gennes equations. We make detailed comparisons of the speeds of sound, the various types of density fluctuations and the phase fluctuations of the gapless bands. Our results are presented for two contrasting modes: (i) a dipolar Bose-Einstein condensate in an infinite tube and (ii) a dilute Bose gas of atoms with soft-core interactions.