  While a spin-orbit-coupled spin-1 Bose-Einstein condensate has been
experimentally observed, its elementary excitations remain unclear in the
stripe phase. Here, we systematically study the elementary excitations in three
distinct phases of a spin-orbit-coupled spin-1 Bose-Einstein condensate. We
find that the excitation spectrum as well as the corresponding static response
function and structure factor depend strongly on spin-orbit coupling parameters
such as the quadratic Zeeman field and the Rabi frequency. In the stripe phase,
besides two gapless Goldstone modes, we show the existence of roton
excitations. Finally, we demonstrate that quantum phase transitions between
these different phases including the zero-momentum, plane wave and stripe
phases are characterized by the sound velocities and the quantum depletion.
