We report exotic hybrid rogue wave and breather solutions for a complex mKdV equation describing soliton dynamics in shallow water wave and optics. Starting from the Lax pair, the higher-order Darboux transformations for the equation are constructed. Further, we obtain a series of theorems to compute the hybrid rogue wave and breather solutions. We also demonstrate the interaction features between rogue waves and breathers by controlling parameters. These results are of importance to understand nonlinear wave dynamics in water wave systems.

In this paper, a new (3+1)-dimensional integrable Kadomtsev–Petviashvili equation is developed. Its integrability is verified by the Painlev\’e analysis. The bilinear form, multiple-soliton, breather and lump solutions are obtained via using the Hirota bilinear method. Furthermore, the abundant dynamical behaviors for these solutions are discovered. It is interesting that there are splitting and fusing phenomena when the lumps interact.