New interaction solutions in Mel’nikov equation obtained by modulating the phase shift

The degradation and nonlinear interactions of a two-breather solution of the Mel’nikov equation are analyzed. By modulating the phase shift and limit method, we prove that in different regions near the non-singular boundaries, there are four kinds of solutions with repulsive interaction or attractive interaction in addition to the two-breather solution. They are the interaction solution between soliton and breather, the two-soliton solution, and the two-breather solution with small amplitude, which all exhibit repulsive interactions; and the two-breather solution with small amplitude, which exhibits attractive interaction. Interestingly, a new breather acts as a messenger to transfer energy during the interaction between two breather solutions with small amplitude.