Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 003, 44 pages      arXiv:1308.1005

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

Batu Güneysu a and Markus J. Pflaum b
a) Institut für Mathematik, Humboldt-Universität, Rudower Chaussee 25, 12489 Berlin, Germany
b) Department of Mathematics, University of Colorado, Boulder CO 80309, USA

Received March 30, 2016, in final form January 05, 2017; Published online January 10, 2017

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.

Key words: profinite dimensional manifolds; jet bundles; geometric PDEs; formal integrability; scalar fields.

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