On the characteristic form of $\mathfrak{g}$-valued zero-curvature representations

TL;DR

This paper introduces a characteristic form for $rak{g}$-valued zero-curvature representations (ZCRs) of PDEs, generalizing conservation law structures and providing a normal form that aids in classification and analysis.

Contribution

It proves that every $rak{g}$-valued ZCR admits a characteristic representative in a normalized form, extending known conservation law forms to more general ZCRs, and derives a new necessary condition for these representatives.

Findings

Every ZCR has a characteristic form preserved under gauge transformations.

A new necessary condition for characteristic representatives is established.

The results confirm structural assumptions and aid in classifying ZCRs.

Abstract

We study $\mathfrak{g}$-valued zero-curvature representations (ZCRs) for partial differential equations in two independent variables from the perspective of their extension to the entire infinite jet space, focusing on their characteristic elements. Since conservation laws -- more precisely, conserved currents -- and their generating functions for a given equation are precisely the $\mathbb{R}$-valued ZCRs and their characteristic elements, a natural question arises: to what extent can results known for conservation laws be extended to general $\mathfrak{g}$-valued ZCRs. For a fixed matrix Lie algebra $\mathfrak{g} \subset \mathfrak{gl}(n)$, we formulate ZCRs as equivalence classes of $\mathfrak{g}$-valued function pairs on the infinite jet space that satisfy the Maurer--Cartan condition. Our main result establishes that every such ZCR admits a characteristic representative -- i.e., a representative in which the Maurer--Cartan condition takes a characteristic form -- generalizing the characteristic form known for conservation laws. This form is preserved under gauge transformations and can thus be regarded as a kind of normal form for ZCRs. We derive a new necessary condition, independent of the Maurer--Cartan equation, that must be satisfied by any characteristic representative. This condition is trivial in the abelian case but nontrivial whenever $\mathfrak{g}$ is nonabelian. These findings not only confirm structural assumptions used in previous works but also suggest potential applications in the classification and computation of ZCRs.