An N-breather solution and hybrid solutions of rogue wave and breather for complex mKdV equation

A compact Grammian form for N-breather solution to the complex mKdV equation is derived using the bilinear Kadomtsev–Petviashvili hierarchy reduction method. The propagation trajectory, period, maximum points, and peak value of the 1-breather solution are calculated. Additionally, through the asymptotic analysis of 2-breather solution, we show that two breathers undergo an elastic collision. By applying the generalized long-wave limit method, the fundamental and second-order rogue wave solutions for the complex mKdV equation are obtained from the 1-breather and 2-breather solutions, respectively. We also construct the hybrid solution of a breather and a fundamental rogue wave for the complex mKdV equation from the 2-breather solution. Furthermore, the hybrid solution of two breathers and a fundamental rogue wave as well as the hybrid solution of a breather and a second-order rogue wave for the complex mKdV equation are derived from the 3-breather solution via the generalized long-wave limit method. By controlling the phase parameters of breathers, the diverse phenomena of interaction between the breathers and the rogue waves are demonstrated.