  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 \psi_{\pm1} are all stable but the moving breathing solitons
\psi_0 is unstable. We also present the experimental parameters to realize
these phenomena in the future experiments.
