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

TL;DR

This paper introduces the concept of profinite dimensional manifolds to formal solution spaces of nonlinear PDEs, demonstrating their structure and applying this to establish a new criterion for formal integrability, including for scalar field equations.

Contribution

It develops the notion of profinite dimensional manifolds for formal solution spaces and provides a new criterion for the formal integrability of nonlinear PDEs.

Findings

The infinite jet space of a fiber bundle is a profinite dimensional manifold.

Formal solution spaces inherit profinite manifold structures if PDEs are formally integrable.

The criterion confirms the formal integrability of the Euler-Lagrange equation for certain scalar fields.

Abstract

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.