We study the type 0A string theory in the (2, 4k) superconformal minimal model backgrounds, focusing on the fully non-perturbative string equations which define the partition function of the model. The equations admit a parameter, Γ, which in the spacetime interpretation controls the number of background D-branes, or R–R flux units, depending upon which weak coupling regime is taken. We study the properties of the string equations (often focusing on the (2, 4) model in particular) and their physical solutions. The solutions are the potential for an associated Schrödinger problem whose wavefunction is that of an extended D-brane probe. We perform a numerical study of the spectrum of this system for varying Γ and establish that when Γ is a positive integer the equations' solutions have special properties consistent with the spacetime interpretation. We also show that a natural solution-generating transformation (that changes Γ by an integer) is the Bäcklund transformation of the KdV hierarchy specialized to (scale invariant) solitons at the zero velocity. Our results suggest that the localized D-branes of the minimal string theories are directly related to the solitons of the KdV hierarchy.