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

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2017
• Hauptverfasser: Güneysu, B., Pflaum, M.J.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Інститут математики НАН України 2017
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/148568
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs / B. Güneysu, M.J. Pflaum // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 50 назв. — англ.

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.