  We numerically demonstrate the unidirectional flow of flat-top solitons when
interacting with two reflectionless potential wells with slightly different
depths. The system is described by a nonlinear Schr\"{o}dinger equation with
dual nonlinearity. The results show that for shallow potential wells, the
velocity window for unidirectional flow is larger than for deeper potential
wells. A wider flat-top solitons also have a narrow velocity window for
unidirectional flow than those for thinner flat-top solitons.
