  Stratifications and iterative differential equations are analogues in
positive characteristic of complex linear differential equations. There are few
explicit examples of stratifications. The main goal of this paper is to
construct stratifications on projective or affine curves in positive
characteristic and to determine the possibilities for their differential Galois
groups. For the related "differential Abhyankar conjecture" we present partial
answers, supplementing the literature. The tools for the construction of
regular singular stratifications and the study of their differential Galois
groups are $p$-adic methods and rigid analytic methods using Mumford curves and
Mumford groups. These constructions produce many stratifications and
differential Galois groups. In particular, some information on the tame
fundamental groups of affine curves is obtained.
