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

Symmetry Integrability and Geometry-Methods and Applications

article

Batu Güneysu¹  Markus J. Pflaum² 

[1] Institut für Mathematik;Department of Mathematics, University of Colorado

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

DOI  :  10.3842/SIGMA.2017.003

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown