  We theoretically study the collective excitations of a spin-orbit-coupled
spin-1 Bose-Einstein condensate with antiferromagnetic spin-exchange
interactions in a cigar-shaped trapping potential at zero and finite
temperatures using the Hartree-Fock-Bogoliubov theory with Popov approximation.
The collective modes at zero temperature are corroborated by the real-time
evolution of the ground state subjected to a perturbation suitable to excite a
density or a spin mode. We have also calculated a few low-lying modes
analytically and found a very good agreement with the numerical results. We
confirm the presence of excitations belonging to two broad categories, namely
density, and spin excitations, based on the calculation of dispersion. The
degeneracy between a pair of spin modes is broken by the spin-orbit coupling.
At finite temperature, spin and density excitations show qualitatively
different behavior as a function of temperature.
