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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2017 Volume 13, 003, 44 pp. (Mi sigma1203)

This article is cited in 2 papers

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

Batu Güneysua, Markus J. Pflaumb

a Institut für Mathematik, Humboldt-Universität, Rudower Chaussee 25, 12489 Berlin, Germany
b Department of Mathematics, University of Colorado, Boulder CO 80309, USA

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

MSC: 58A05; 58A20; 35A30

Received: March 30, 2016; in final form January 5, 2017; Published online January 10, 2017

Language: English

DOI: 10.3842/SIGMA.2017.003



Bibliographic databases:
ArXiv: 1308.1005


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