Higher-order rogue waves and modulation instability of the two-component derivative nonlinear Schrödinger equation

Abstract

The determinant representation of the n-th order different kinds of solutions to the two-component derivative nonlinear schrödinger (DNLS) equation is constructed by performing the n-fold Darboux transformation (DT). Based on this representation, the Peregrine-type, double, triple, quadruple, and sextuple vector rogue waves are obtained. Interestingly, the Peregrine-type rogue wave is not usual Peregrine soliton, but anomalous Peregrine soliton whose peak amplitude can go beyond threefold limit. Besides, coexisting rogue waves display more abundant dynamical behaviors which have not been found in scalar DNLS equation. Finally, we analyze modulation instability (MI) and find that there exist two types of MI: one MI has an usual M profile, but the other has a deformed double M-type profile. The parameter space for the two types of baseband MI regimes is agreement with the one which results in the generation of rogue waves. It hence further confirms that the baseband MI is a possible mechanism to generate the rogue wave in a nonlinear system from a plane wave background.