The Stokes multipliers in the matrix models are invariants in the
string-theory moduli space and related to the D-instanton chemical potentials.
They not only represent non-perturbative information but also play an important
role in connecting various perturbative string theories in the moduli space.
They are a key concept to the non-perturbative completion of string theory and
also expected to imply some remnant of strong coupling dynamics in M theory. In
this paper, we investigate the non-perturbative completion problem consisting
of two constraints on the Stokes multipliers. As the first constraint, Stokes
phenomena which realize the multi-cut geometry are studied in the Z_k symmetric
critical points of the multi-cut two-matrix models. Sequence of solutions to
the constraints are obtained in general k-cut critical points. A discrete set
of solutions and a continuum set of solutions are explicitly shown, and they
can be classified by several constrained configurations of the Young diagram.
As the second constraint, we discuss non-perturbative stability of backgrounds
in terms of the Riemann-Hilbert problem. In particular, our procedure in the
2-cut (1,2) case (pure-supergravity case) completely fixes the D-instanton
chemical potentials and results in the Hastings-McLeod solution to the
Painlev\'e II equation. It is also stressed that the Riemann-Hilbert approach
realizes an off-shell background independent formulation of non-critical string
theory.Comment: 71 pages, v3: organization of Sec. 3, Sec. 4, App. C and App. D
improved, final version to be published in Nucl. Phys.
