  We develop the theory of Riemann-Hilbert problems necessary for the results
in part one of this series of papers. In particular, we obtain solutions for a
family of non-linear Riemann-Hilbert problems through classical contraction
principles and saddle-point estimates. We use compactness arguments to obtain
the required smoothness property on solutions. We also consider limit cases of
these Riemann-Hilbert problems where the jump function develops discontinuities
of the first kind together with zeroes of a specific order at isolated points
in the contour. Solutions through Cauchy integrals are still possible and they
have at worst a branch singularity at points where the jump function is
discontinuous and a zero for points where the jump vanishes.
