  Using Bogoliubov theory we calculate the excitation spectrum of a spinor
Bose-Einstein condensed gas with equal Rashba and Dresselhaus spin-orbit
coupling in the stripe phase. The emergence of a double gapless band structure
is pointed out as a key signature of Bose-Einstein condensation and of the
spontaneous breaking of translational invariance symmetry. In the long
wavelength limit the lower and upper branches exhibit, respectively, a clear
spin and density nature. For wave vectors close to the first Brillouin zone,
the lower branch acquires an important density character responsible for the
divergent behavior of the structure factor and of the static response function,
reflecting the occurrence of crystalline order. The sound velocities are
calculated as functions of the Raman coupling for excitations propagating
orthogonal and parallel to the stripes. Our predictions provide new
perspectives for the identification of supersolid phenomena in ultracold atomic
gases.
