Solitons and rogue waves in spinor Bose-Einstein condensates

We present a general classification of one-soliton solutions as well as novel families of rogue-wave solutions for $F=1$ spinor Bose-Einstein condensates (BECs). These solutions are obtained from the inverse scattering transform for a focusing matrix nonlinear Schr\"odinger equation which models condensates in the case of attractive mean field interactions and ferromagnetic spin-exchange interactions. In particular, we show that, when no background is present, all one-soliton solutions are reducible via unitary transformations to a combination of oppositely-polarized solitonic solutions of single-component BECs. On the other hand, we show that, when a non-zero background is present, not all matrix one-soliton solutions are reducible to a simple combination of scalar solutions. We show that some solitons are topological ones and others are dark-bright solitons. Finally, by taking suitable limits of all the solutions on a non-zero background we also obtain three families of rogue-wave (i.e., rational) solutions, two of which are novel to the best of our knowledge.