Exponentiable Lie-Backlund vector fields in $J^\infty(\mathbb{R}^n, \mathbb{R}^m)$

TL;DR

This paper characterizes Lie-Backlund vector fields in infinite jet bundles that can be exponentiated to flows with finite-variable dependence, revealing differences between cases where the target dimension is one or greater.

Contribution

It provides a complete characterization of exponentiable Lie-Backlund vector fields in infinite jet spaces, including necessary and sufficient conditions and explicit examples.

Findings

For m=1, such fields are extensions of finite-dimensional jet space fields.

For m>1, not all such fields are extensions; conditions for exponentiation are established.

Explicit non-trivial examples are constructed for (n,m)=(1,2).

Abstract

We characterize Lie -Backlund vector fields in infinite dimensional jet bundles $J^\infty(\mathbb{R}^n, \mathbb{R}^m)$ that can be exponentiated to flows with each component depending on a finite set of variables. We show that for $m=1$ each such field is an extension of one in a finite dimensional jet space. For $m>1$ this is no longer the case and we give necessary and sufficient conditions for exponentiation. Non-trivial examples are provided for $(n,m)=(1,2)$.