The self-consistent spatiotemporal evolution of a drift-wave (DW) radial envelope and a zonal-flow (ZF) amplitude is investigated in a slab model. The stationary solution of the coupled partial differential equations in a simple limit yields the formation of DW-ZF soliton structures, which propagate radially with speed depending on the envelope peak amplitude. Additional interesting physics, e.g., the generation, destruction, collision, and reflection of solitons, as well as turbulence bursting can also be observed due to the effects of linear growth or damping, dissipation, equilibrium nonuniformities and soliton dynamics. The propagation of soliton causes significant radial spreading of DW turbulence and therefore can affect transport scaling with the system size by broadening of the turbulent region. The correspondence of the present analysis with the description of DW-ZF interactions in toroidal geometry is also discussed.