  We developed functional integral formulation for the stripe phase of a spinor
Bose-Einstein condensates with Rashba spin-orbit coupling. The excitation
spectrum is found to exhibit double gapless band structures, identified to be
two Goldstone modes resulting from spontaneously broken internal gauge symmetry
and translational invariance symmetry. The sound velocities display anisotropic
behaviors with the lower branch vanishes in the direction perpendicular to the
stripe in the x-y plane. At the transition point between the plane wave phase
and the stripe phase, physical quantities such as fluctuation correction to the
ground state energy and quantum depletion of the condensates exhibit
discontinuity, characteristic of the first order phase transition. Despite
strong quantum fluctuations induced by Rashba spin-orbit coupling, we show that
the supersolid phase is stable against quantum depletion. Finally we extend our
formulation to finite temperatures to account for interactions between
excitations.
