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