1308.1005 (Batu Güneysu et al.)

The profinite dimensional manifold structure of formal solution spaces
of formally integrable PDE's    [PDF]

Batu Güneysu, Markus J. Pflaum

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 PDE's and prove a new criterion for formal integrability of such PDE's. In particular, this result entails that the Euler-Lagrange Equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.

View original: http://arxiv.org/abs/1308.1005