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

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.

18 pages, no figure,references updated

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