Breather and its interaction with rogue wave of the coupled modified nonlinear Schrödinger equation

We investigate the coupled modified nonlinear Schrödinger equation. Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions. To obtain the higher-order localized wave solution, the N-fold generalized Darboux transformation is given. Under the condition that the characteristic equation admits a double-root, we present the expression of the first-order interactional solution. Then we graphically analyze the dynamics of the breather and rogue wave. Due to the simultaneous existence of nonlinear and self-steepening terms in the equation, different profiles in two components for the breathers are presented.