Interaction of lumps and dark solitons in the Mel'nikov equation

Herein, we employ the bilinear method and the KP hierarchy reduction technique for obtaining a hierarchy of semi-rational solutions to the Mel'nikov equation. These semi-rational solutions are given in terms of determinants whose matrix elements have plain algebraic expressions. Under suitable parametric conditions, these semi-rational solutions reduce to rational solutions of the Mel'nikov equation. These semi-rational solutions reveal two opposite types of excitation phenomena: fusion and fission, which are determined by a input parameter gamma. The fundamental (first-order) semi-rational solutions describe a process of a lump originating from a dark soliton and then coexisting with the dark soliton as t >> 0, or a process of a lump coexisting with the dark soliton as t << 0 and then fusing into the dark soliton. The higher-order semi-rational solutions exhibi n(i) (n(i) >= 2) lumps splitting from or annihilating into one dark soliton. The multi-semi-rational solutions describe N (N >= 2) lumps fissuring from or fusing into N-dark solitons.