Oscillating multi-node solitons in spin-orbit coupled spin-1 Bose-Einstein condensates

We investigate the impact of spin-orbit coupling on bright/dark solitons in spin-1 Bose-Einstein condensates subjected to a spatiotemporally varying external potential. By using the Hirota's bilinear method, the solutions of one- and two-solitons in the three-coupled Grose-Pitaevskii (GP) equation are constructed under different space-time rotated parity-time (PT) symmetric potentials. Solitons oscillate in the x-direction due to spin-orbit coupled (SOC) strength, resulting in a multi-node structure, but the SOC strength does not affect the structure of solitons in the overall system. Specifically, the background plane of dark solitons also oscillates periodically along the x-direction. The increase in SOC intensity leads to the reduction of soliton oscillation period and dark soliton background plane oscillation period, resulting in more frequent oscillations of solitons and dark soliton background planes. Through asymptotic analysis of bright/dark two-solitons, the elastic interaction is discovered in the collision of two-solitons. By calculating the total spin, it is evident that the polar bright/dark solitons are achieved owing to the SOC intensity.