Higher-order breather and interaction solutions to the (3+1)-dimensional Mel’nikov equation

In this paper, we construct the high-order breather and interaction solutions of the $(3+1)$-dimensional Mel’nikov equation using the KP hierarchy reduction approach and express them in a concise determinant form. Our solutions show that the two breathers, two periodic waves, and the hybrid mode of the breather and periodic wave are all mutually parallel. Furthermore, by examining the long wave limit of the periodic wave solutions, a variety of rational solutions (lumps) and mixed solutions are obtained. Notably, the interaction between the lump and breather is found to be elastic. These novel results provide deeper insights into the interactions among different solution types.