Confinement of bright matter-wave solitons on top of a pedestal-shaped potential

•We have shown that moving matter-wave solitons and breathers can be trapped on top of a pedestal-shaped potential for a sufficiently long time.•In the two-pulse configuration, we observed the phenomenon of trajectory jump in soliton collisions and estimated its magnitude.•We observed unusual splitting of the breather when the larger soliton remains motionless, while the smaller one performs shuttle motion.