Modulation instability of a spin-1 Bose-Einstein condensate with spin-orbit coupling

We investigate the modulation instability (MI) of plane-wave states in a spin-orbit-coupled spin-1 Bose-Einstein condensate (BEC). The MI of the system can be adjusted by the intensities of the spin-orbit coupling (SOC), Raman coupling and atom-atom interactions. Our analysis shows that the MI of a system without the SOC and Raman coupling mainly occurs for repulsive density-density and spin-exchange interactions due to the combined effects coming from both of them. But when considering the SOC, a multi-peak structure appears in the momentum distribution for the instability, which may be thought of as a precursor to the generation of the stripe-type nonlinear excitations in such systems. The MI becomes more remarkable with increasing intensity of Raman coupling. In addition, the SOC inevitably contributes to the instability for the ternary superfluid regardless of the nature of the density-density and spin-exchange interactions.