Is the $\mathbb{Z}_2\times \mathbb{Z}_2$-graded sine-Gordon equation integrable?

TL;DR

This paper investigates whether a newly proposed $ ext{Z}_2 imes ext{Z}_2$-graded sine-Gordon model, extending the supersymmetric version, is integrable by analyzing transformations and conservation laws.

Contribution

It introduces a method to assess integrability of the $ ext{Z}_2 imes ext{Z}_2$-graded sine-Gordon model using auto-Bäcklund transformations and conserved currents.

Findings

Identification of auto-Bäcklund transformations for the model

Construction of conserved spinor-valued currents

Establishment of infinite conservation laws

Abstract

We examine the question of the integrability of the recently defined $\mathbb{Z}_2\times \mathbb{Z}_2$-graded sine-Gordon model, which is a natural generalisation of the supersymmetric sine-Gordon equation. We do this via appropriate auto-B\"{a}cklund transformations, construction of conserved spinor-valued currents and a pair of infinite sets of conservation laws.