Dynamics of interaction between lumps and solitons in the Mel'nikov equation

Two families of semi-rational solutions to the Mel'nikov equation (ME), which is a known model of the interaction between long and short waves in two dimensions, are reported. These semi-rational solutions describe interaction between lumps and solitons. The first family of semi-rational solutions is derived by employing the KP-hierarchy reduction method. The fundamental (first-order) semi-rational solutions, consisting of a lump and a dark soliton, feature three different interactions, depending on speeds of the corresponding lump, V_lump, and soliton, V_soliton: fusion of the lump into the dark soliton (at V_lump < V_soliton); splitting of the lump from the soliton (at V_lump > V_soliton); and lump-soliton bound states (at V_lump=V_soliton). Three subclasses of non-fundamental semi-rational solutions, namely, higher-order, multi-, and mixed semi-rational solutions, are produced. These non-fundamental semi-rational solutions also represent different interaction: fusion of mutli-lumps into mutli-dark solitons, fission of multi-lumps from multi-dark solitons, multi-lump-soliton bound states, partial merger or partial splitting of lumps into/from lump-soliton bound states, etc. In particular, by selecting specific parameter constraints, the first family of semi-rational solutions reduces to solutions of a newly proposed partially spatial-reversed nonlocal ME. The second family of semi-rational solutions is constructed by using the Hirota method combined with a perturbative expansion and a long-wave limit, which describes a lump permanently propagating on the background of a dark soliton. The second family of the solutions indicates not all interactions between lumps and solitons in the ME give rise to fission or fusion. Besides that, a semi-rational solution to the coupled Schrödinger-Boussinesq equation, consisting of a rogue wave and a dark soliton, is obtained as a reduction of a semi-rational solution belonging to the second family.