  Spin-wave excitation plays important roles in the investigation of the
magnetic phases. In this paper, we study the spin-wave excitation spectra of
two-component Bose gases with spin-orbit coupling on a deep square optical
lattice using spin-wave theory. We find that, while the excitation spectrum of
the vortex crystal phase is gapless with a linear dispersion in the vicinity of
the minimum point, the spectra of the commensurate spiral spin phase and the
skyrmion crystal phase are gapped. Significantly, the spin fluctuations
strongly destabilize the classical ground state of the skyrmion phase. It
suggests the emergence of a new state in the phase diagram. Such features of
the spin excitation spectra provide further insights into the exploration of
the exotic spin phases.
