Full-time dynamics of modulational instability in spinor Bose-Einstein condensates

We describe the full-time dynamics of modulational instability in $F=1$ spinor Bose-Einstein condensates for the case of the integrable three-component model associated with the matrix nonlinear Schrödinger equation. We obtain an exact homoclinic solution of this model by employing the dressing method which we generalize to the case of the higher-rank projectors. This homoclinic solution describes the development of modulational instability beyond the linear regime, and we show that the modulational instability demonstrates the reversal property when the growth of the modulated amplitude is changed by its exponential decay.