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.
B.G. has been financially supported by the SFB 647: Raum–Zeit–Materie, and would like to thank the University of Colorado at Boulder for its hospitality. The second named author (M.P.) has been partially supported by NSF grant DMS 1105670 and by a Simons Foundation collaboration grant, award nr. 359389.
Received:March 30, 2016; in final form January 5, 2017; Published online January 10, 2017
Citation:
Batu Güneysu, Markus J. Pflaum, “The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs”, SIGMA, 13 (2017), 003, 44 pp.