An integrable hierarchy associated with loop extension of $\mathbb{Z}_2^2$-graded $\mathfrak{osp}(1|2)$

TL;DR

This paper constructs a hierarchy of integrable equations based on a $ obreak ext{Z}_2^2$-graded Lie superalgebra, extending classical models like KdV and Liouville, and explores their conserved charges and algebraic structures.

Contribution

It introduces a novel $ obreak ext{Z}_2^2$-graded integrable hierarchy derived from loop extensions of $ obreak ext{Z}_2^2$-graded Lie superalgebras, including explicit conserved charges.

Findings

Includes $ obreak ext{Z}_2^2$-graded extensions of classical integrable equations.

Derives $ obreak ext{Z}_2^2$-KdV from $ obreak ext{Z}_2^2$-mKdV via Miura transformation.

Identifies conserved charges with nontrivial grading in the $ obreak ext{Z}_2^2$-graded systems.

Abstract

A hierarchy of $\mathbb{Z}_2^2$-graded integrable equations is constructed using the loop extension of the $\mathbb{Z}_2^2$-graded Lie superalgebra $\mathfrak{osp}(1|2)$. This hierarchy includes $\mathbb{Z}_2^2$-graded extensions of the Liouville, sinh-Gordon, cosh-Gordon, and, in particular, the mKdV equations. The $\mathbb{Z}_2^2$-graded KdV equation is also derived from the $\mathbb{Z}_2^2$-mKdV equation via the Miura transformation. We present explicit formulas for the conserved charges of the $\mathbb{Z}_2^2$-KdV and $\mathbb{Z}_2^2$-mKdV equations. A distinctive feature of these $\mathbb{Z}_2^2$-graded integrable systems is the existence of conserved charges with nontrivial grading.