On the (non)removability of spectral parameters in $Z_2$-graded zero-curvature representations and its applications

TL;DR

This paper extends a method to analyze the (non)removability of spectral parameters in zero-curvature representations within $bZ_2$-graded PDEs, linking gauge transformations and linear coverings.

Contribution

It generalizes a practical approach for inspecting spectral parameter removability to the $bZ_2$-graded setting, connecting gauge deformations with linear coverings.

Findings

Demonstrates the link between deformation of zero-curvature representations and linear coverings.

Provides criteria for the (non)removability of spectral parameters in $bZ_2$-graded PDEs.

Illustrates the method with examples of parameter generation and elimination.

Abstract

We generalise to the $\mathbb{Z}_2$-graded set-up a practical method for inspecting the (non)removability of parameters in zero-curvature representations for partial differential equations (PDEs) under the action of smooth families of gauge transformations. We illustrate the generation and elimination of parameters in the flat structures over $\mathbb{Z}_2$-graded PDEs by analysing the link between deformation of zero-curvature representations via infinitesimal gauge transformations and, on the other hand, propagation of linear coverings over PDEs using the Fr\"olicher--Nijenhuis bracket.