  We construct a sheaf theoretic and derived geometric machinery to study
nonlinear partial differential equations and their singular supports. We
establish a notion of derived microlocalization for solution spaces of
non-linear equations and develop a formalism to pose and solve singular
non-linear Cauchy problems globally. Using this approach we estimate the
domains of propagation for the solutions of non-linear systems. It is achieved
by exploiting the fact that one may greatly enrich and simplify the study of
derived non-linear PDEs over a space $X$ by studying its derived linearization
which is a module over the sheaf of functions on the $S^1$-equivariant derived
loop stack $\mathcal{L}X$.
