LAUSR

By introducing a suitable ansatz and employing the similarity transformation technique, we construct the first- and second-order rational solutions for a quasi-one-dimensional (1D) dissipative Gross–Pitaevskii (GP) equation with a time-varying cubic nonlinearity and an external time-dependent potential. Then, by using these solutions, we engineer first- and second-order rogue waves in the Bose–Einstein condensate (BEC) contexts for the experimentally relevant systems when the gain/loss of atoms is taken into consideration. Our analysis shows that the control of the scattering length, the external harmonic, and the linear trapping potentials allows one to manage the motion and the background of dissipative rogue matter waves in BEC systems. We show that the wave amplitudes depend on the absolute value of s-wave scattering and the bias magnetic field, while its motion depends on the external trapping potentials. We show that unlike classical rogue waves, the nonzero continuous wave backgrounds of non-autonomous forced (damped) rogue matter waves in BECs with time-dependent complicated potential increases (decreases) during the wave motion. Our results also reveal that neither the gain nor the loss of the BEC atoms affects the amplitude of the rogue matter waves during their propagation. Our results may help to control and manage experimentally dissipative rogue waves in a BEC systems.