Analytical solutions to the spin-1 Bose-Einstein condensates

We analytically solve the one-dimensional coupled Gross-Pitaevskii equations which govern the motion of the spinor- 1 Bose-Einstein condensates. The nonlinear density-density interactions are decoupled by making use of the unique properties of the Jacobian elliptical functions. Several types of time-invariant solutions are deduced. Furthermore, exact time-evolving solutions to the time-dependent Gross-Pitaevskii equations are constructed by making use of the spin-rotational symmetry of the Hamiltonian. The spin-polarizations exhibit kinked configurations.