Semi-rational solutions for a \[(2+1)\] -dimensional Davey–Stewartson system on the surface water waves of finite depth

Abstract

Under investigation in this work is a \[(2+1)\]-dimensional Davey–Stewartson system, which describes the surface water wave packets of finite depth. With respect to the velocity potential of the mean flow interacting with the surface wave and the amplitude of the surface wave packet, we derive two types of the solutions in terms of the Gramian, including the semi-rational solutions, and the solutions containing certain solitons and breathers based on the Kadomtsev–Petviashvili hierarchy reduction. Amplitude of the surface wave packet is graphically presented: (i) We find the interactions between the rogue waves/lump solitons and dark solitons, and the dark solitons keep their shapes unchanged after the interactions: The focusing/defocusing parameter does not affect the rogue wave and dark soliton, while the surface tension affects the locations of the rogue wave and dark soliton; (ii) We observe the interactions between the two dark–dark solitons, and three cases of the interactions between the dark solitons and breathers: The focusing/defocusing parameter only affects the propagation direction of the dark soliton, while the surface tension does not affect the two dark–dark solitons.