Dynamics of rational and semi-rational solutions of the general N-component nonlinear Schrödinger equations

Abstract

We investigate an integrable general N-component nonlinear Schrödinger equations. The Darboux-dressing transformation for the equations is firstly constructed according to its Lax pair. Applying the idea of separating variables, we generate the solutions of the Lax pair equations with a nontrivial background. Then we derive the rational and semi-rational vector solutions of the equations in detail. The rational solutions act as the pure rogue waves. The semi-rational solutions represent a combination of different types of rogue waves, breathers, and soliton waves. Moreover, a constraint condition is given for the semi-rational solutions being reduced to unmixed breather or nth-rogue waves. To illustrate the corresponding dynamic behaviors of these solutions, we analyze two-component and four-component cases, showing that the multi-component system exhibits richer phenomena and novel behaviors compared to lower-component systems, and also presenting some novel phenomena.