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,
Gamma, 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 Schrodinger problem whose wavefunction is that of
an extended D-brane probe. We perform a numerical study of the spectrum of this
system for varying Gamma and establish that when Gamma 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 Gamma by an integer) is the Backlund transformation of the KdV
hierarchy specialized to (scale invariant) solitons at 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. Further, we observe an
interesting transition when Gamma=-1.Comment: 17 pages, 3 figure