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

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.

References in corpus (4)

• Secondary Calculus and the Covariant Phase Space
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• Involutivity of field equations
• Bosonic Loop Diagrams as Perturbative Solutions of the Classical Field Equations in ${\varphi }^4$-Theory

Cited by in corpus (5)

• Synthetic geometry of differential equations: I. Jets and comonad structure
• ${\mathsf{L}}^1$-elliptic regularity and $H=W$ on the whole ${\mathsf{L}}^p$-scale on arbitrary manifolds
• Field Theory via Higher Geometry I: Smooth Sets of Fields
• Lagrangian field theories: ind/pro-approach and L-infinity algebra of local observables
• Jets and differential linear logic