Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg–de Vries equations

Abstract

With the aid of a generalized Darboux transformation (DT), we derive a hierarchy of rogue wave solutions to the coupled complex modified Korteweg–de Vries (mKdV) equations. Based on modulation instability (MI), two types of nth order rational rogue wave solutions in compact determinant forms are presented. Especially, the rational rogue wave solutions up to the second-order are performed explicitly and graphically. We find that there exist typical bright–dark composites and rogue wave doublets in first-order case and four or six fundamental rogue waves in second-order case. With appropriate choices of free parameters, the distribution shapes with four fundamental rogue waves admit triangular, quadrilateral, and line structures. Furthermore, triangular, ring, quadrilateral and line patterns can emerge with six fundamental rogue waves. Moreover, we exhibit the first-order semi-rational rogue wave solutions which can demonstrate the coexistence of one rational rogue wave and one breather. Our results can be applicable to the study of rogue wave manifestations in nonlinear optics.