In this paper, we obtain a uniform Darboux transformation for multi-component coupled nonlinear Schrödinger (NLS) equations, which can be reduced to all previously presented Darboux transformations. As a direct application, we derive the single dark soliton and multi-dark soliton solutions for multi-component NLS equations with a defocusing case and a mixed focusing and defocusing case. Some exact single and two-dark solitons of three-component NLS equations are investigated explicitly. The results are meaningful for vector dark soliton studies in many physical systems, such as Bose–Einstein condensate, nonlinear optics, etc.

We study the relations between modulational instability and several well-known nonlinear excitations in a nonlinear fiber such as bright soliton, nonlinear continuous wave, Akhmediev breather, Peregrine rogue wave, and Kuznetsov–Ma breather. We present a quantitative correspondence between them based on the dominant frequency and propagation constant of each perturbation on a continuous-wave background. We especially demonstrate that rogue wave comes from modulational instability with the resonance perturbation on continuous-wave background. The numerical simulations are performed to test these theoretical results. These results will deepen our understanding of rogue wave excitation and be helpful for controllable nonlinear wave excitations in nonlinear fiber and other nonlinear systems.