Three-component nonlinear Schrödinger equations: Modulational instability, Nth-order vector rational and semi-rational rogue waves, and dynamics

The integrable three-component nonlinear Schrödinger equations are systemically explored in this paper. We firstly find the conditions for the modulational instability of plane-wave solutions of the system. Secondly, we present the general formulae for the Nth-order vector rational and semi-rational rogue wave solutions by the generalized Darboux transformation and formal series method. Particularly, we find that the second-order vector rational RWs contain five, seven, and nine fundamental vector RWs, which can arrange with many novel excitation dynamical patterns such as pentagon, triangle, 'clawlike', line, hexagon, arrow, and trapezoid structures. Moreover, we also find two different kinds of Nth-order vector semi-rational RWs: one of which can demonstrate the coexistence of Nth-order vector rational RW and N parallel vector breathers and the other can demonstrate the coexistence of Nth-order vector rational RWs and Nth-order Y-shaped vector breathers. We also exhibit distribution patterns of superposition of RWs, which can be constituted of different fundamental RW patterns. Finally, we numerically explore the dynamical behaviors of some chosen RWs. The results could excite the interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids.