The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs
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.
Keywords
Cite
@article{arxiv.1308.1005,
title = {The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs},
author = {Batu Güneysu and Markus J. Pflaum},
journal= {arXiv preprint arXiv:1308.1005},
year = {2017}
}