Tuesday, August 6, 2013

1308.1005 (Batu Güneysu et al.)

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

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

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