  We use a combination of a 't Hooft limit and numerical methods to find
non-perturbative solutions of exactly solvable string theories, showing that
perturbative solutions in different asymptotic regimes are connected by smooth
interpolating functions. Our earlier perturbative work showed that a large
class of minimal string theories arise as special limits of a Painleve IV
hierarchy of string equations that can be derived by a similarity reduction of
the dispersive water wave hierarchy of differential equations. The hierarchy of
string equations contains new perturbative solutions, some of which were
conjectured to be the type IIA and IIB string theories coupled to (4,4k-2)
superconformal minimal models of type (A,D). Our present paper shows that these
new theories have smooth non-perturbative extensions. We also find evidence for
putative new string theories that were not apparent in the perturbative
analysis.
