Investigation in this paper is the spatially modulated two-component GP system with Rabi coupling in a Bose–Einstein condensate consisting of the two hyperfine states. Based on the Kadomtsev–Petviashvili hierarchy reduction, we derive the Gramian expression of the one- and two-dark–dark soliton solutions. The nonlinearity coefficients [Formula: see text] and the external spatially varying trapping potential [Formula: see text] can be constrained as the functions of [Formula: see text], where [Formula: see text] is the spatial coordinate, [Formula: see text] is the time coordinate, [Formula: see text] is the dispersion parameter. With the Rabi coupling coefficient [Formula: see text] increasing, period along [Formula: see text] decreases. When [Formula: see text] is a constant, soliton propagates stably with the amplitude and velocity unvarying; When [Formula: see text] is a function of [Formula: see text], background is periodic and velocity of the soliton varies with [Formula: see text] increasing. Head-on and overtaking elastic interactions between the two solitons are presented analytically and graphically.
Hamilton’s equations of motion of a vortex filament in the rotating Bose–Einstein condensate and their “soliton” solutions
