Document Type

Article

Publication Date

1-9-2017

Publication Title

Symmetry, Integrability and Geometry: Methods and Applications

ISSN

1815-0659

Volume

13

DOI

https://doi.org/10.3842/SIGMA.2017.003

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.

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