In this chapter, we construct a class of examples of differential modules on open annuli which carry Frobenius structures, and hence are solvable at a boundary. These modules are derived from continuous linear representations of the absolute Galois group of a positive-characteristic local field. We first construct a correspondence between Galois representations and differential modules over E carrying a unit-root Frobenius structure. We then refine the construction to compare Galois representations with finite image of inertia and differential modules over the bounded Robba ring; the main result here is an equivalence of categories due to Tsuzuki. We finally describe (without proof) a numerical relationship between wild ramification of a Galois representation and convergence of solutions of p-adic differential equations. Besides making explicit the analogy between wild ramification of Galois representations and irregularity of meromorphic differential systems, it also suggests an approach to higher-dimensional ramification theory.