Breathers, rational solutions, and their exact physical spectra in F  = 1 spinor Bose–Einstein condensates

Three-component spinor Bose–Einstein condensates (BECs) provide an ideal platform to systematically investigate nonlinear wave dynamics under non-zero background. Using the Darboux transformation, we derive exact breather and rational solutions in which only one component maintains a zero background, while the other two reside on finite backgrounds. Based on the residue theorem associated with fourth-order singularities, we further derive physical spectra of these exact solutions. Our analysis shows that the energy-transfer feature identified in the spectra should be understood as the spectral manifestation of spin-exchange-induced population redistribution from the two non-zero-background components to the zero-background component. This redistribution causes the zero-background component to exhibit breather-like density characteristics. On the other hand, for rational solutions, the components with non-zero background exhibit classical rogue wave behavior, while the component with zero-background forms a novel valley-free double-peak structure. The numerical simulations show that these rational solutions are quite stable in their initial stages. In addition, we also derived high-order rational solutions and their spatial distribution patterns. Our results provide deeper insights into the dynamics of localized waves and enrich nonlinear excitations in coupled multi-component BEC systems.