  In this paper, we study the formal solution space of a nonlinear PDE in a
fiber bundle. To this end, we start with foundational material and introduce
the notion of a pfd structure to build up a new concept of profinite
dimensional manifolds. We show that the infinite jet space of the fiber bundle
is a profinite dimensional manifold in a natural way. The formal solution space
of the nonlinear PDE then is a subspace of this jet space, and inherits from it
the structure of a profinite dimensional manifold, if the PDE is formally
integrable. We apply our concept to scalar PDEs and prove a new criterion for
formal integrability of such PDEs. In particular, this result entails that the
Euler-Lagrange equation of a relativistic scalar field with a polynomial
self-interaction is formally integrable.
