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

Paper

Abstract

A hierarchy of

-graded integrable equations is constructed using the loop extension of the

-graded Lie superalgebra

. This hierarchy includes

-graded extensions of the Liouville, sinh-Gordon, cosh-Gordon, and, in particular, the mKdV equations. The

-graded KdV equation is also derived from the

-mKdV equation via the Miura transformation. We present explicit formulas for the conserved charges of the

-KdV and

-mKdV equations. A distinctive feature of these

-graded integrable systems is the existence of conserved charges with nontrivial grading.