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

Abstract

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 $$t{\gg }0$$ , or a process of a lump coexisting with the dark soliton as $$t{\ll }0$$ and then fusing into the dark soliton. The higher-order semi-rational solutions exhibit $$n_i\,(n_{i}\ge 2)$$ lumps splitting from or annihilating into one dark soliton. The multi-semi-rational solutions describe $$N\,(N \ge 2)$$ lumps fissuring from or fusing into N-dark solitons.