Complexitons of the modified KdV equation by Darboux transformation

Darboux transformation (DT), a comprehensive approach to construct the explicit solutions of the nonlinear evolutionary equation, is applied to construct the new complexiton solution of the negative mKdV equation. We find that the complexiton solution is related to the single complex spectral parameter and that it is completely different from the breather solution for the positive mKdV equation. Consequently, we generalize the concept of complexiton of the KdV equation to the mKdV equation. In addition, the relationship of the spectral parameter and the known solutions of the mKdV equation is clarified and the multi-complexiton solution, multi-complexiton–positon, multi-complexiton–negaton and multi-complexiton–soliton solutions are obtained in the uniform manner by DT. The interaction of complexiton and soliton is also discussed in detail. It is shown that the new complexiton and soliton remain unchanged except for phase shifts after their interaction. At the same time, the superreflectionless property of one-positon potential for the mKdV equation is shown in detail.