  The phase diagram of a Bose-Einstein condensate with Raman-induced spin-orbit
coupling includes a stripe phase with supersolid features. In this work we
develop a perturbation approach to study the ground state and the Bogoliubov
modes of this phase, holding for small values of the Raman coupling. We obtain
analytical predictions for the most relevant observables (including the
periodicity of stripes, sound velocities, compressibility, and magnetic
susceptibility) which are in excellent agreement with the exact (non
perturbative) numerical results, obtained for significantly large values of the
coupling. We further unveil the nature of the two gapless Bogoliubov modes in
the long-wavelength limit. We find that the spin branch of the spectrum,
corresponding in this limit to the dynamics of the relative phase between the
two spin components, describes a translation of the fringes of the equilibrium
density profile, thereby providing the crystal Goldstone mode typical of a
supersolid configuration. Finally, using sum-rule arguments, we show that the
superfluid density can be experimentally accessed by measuring the ratio of the
sound velocities parallel and perpendicular to the direction of the spin-orbit
coupling.
