Bright and dark solitons under spatiotemporal modulation in (2+1)-dimensional Spin-1 Bose-Einstein condensates

The study delves into the impact of spatiotemporally modulated external potentials on both bright and dark solitons within spin-1 Bose-Einstein condensates. With the Hirota bilinear method, the one-soliton and two-soliton solutions of the (2+1)-dimensional three-coupled Gross-Pitaevskii equation under a spatiotemporal modulated capture potential are obtained. The analysis uncovers the formation of ferromagnetic bright and dark solitons, as evidenced by the computation of the total spin. The findings indicate that a multitude of soliton configurations arise contingent upon the selected external potentials, with their velocities and amplitudes being influenced by spatial and temporal factors. Asymptotic analysis of the bright and dark two-soliton systems reveals the nature of their elastic interactions. Additionally, leveraging a trained Fourier Neural Operator as an alternative predictive model, the temporal evolution of solitons as governed by the Gross-Pitaevskii equation are projected. This approach yields forecasted solutions that are notably in close agreement with the exact solutions.