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

The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs