An integrable $\mathbb{Z}_2^2$-graded extension of Camassa-Holm equation and its bi-Hamiltonian structure

Abstract: By constructing Lax operators in the loop algebra of the $\mathbb{Z}_2<sup>2$-graded extension of the Lie superalgebra $\mathfrak{osp}(1|2)$, we derive a $\mathbb{Z}_2<sup>2$-graded extension of the Camassa-Holm equation. The resulting equation is an integrable nonlinear PDE for a system of four $\mathbb{Z}_2<sup>2$-graded commutative functions, each associated with a distinct $\mathbb{Z}_2<sup>2$-degree. We further show that the $\mathbb{Z}_2<sup>2$-Camassa-Holm equation admits a bi-Hamiltonian structure. As a consequence, it possesses infinitely many conserved quantities, including one with non-trivial $\mathbb{Z}_2<sup>2$-degree, which are mutually in involution with respect to the $\mathbb{Z}_2<sup>2$-graded Poisson brackets.