Dynamics of the localized nonlinear waves in spin-1 Bose-Einstein condensates with time-space modulation

We investigate the dynamics of the localized nonlinear matter wave in spin-1 Bose-Einstein condensates with trapping potentials and nonlinearities dependent on time and space. We solve the three coupled Gross-Pitaevskii equation by similarity transformation and obtain two families of exact matter wave solutions in terms of Jacobi elliptic functions and Mathieu equation. The localized states of the spinor matter wave describe the dynamics of vector breathing solitons, moving breathing solitons, quasibreathing solitons and resonant solitons. The results of stability show that one order vector breathing solitons, quasibreathing solitons, resonant solitons, and the moving breathing solitons ψ_±1 are all stable but the moving breathing solitons ψ_0 is unstable. We also present the experimental parameters to realize these phenomena in the future experiments.