Hierarchies of Symplectic Structures for $\mathfrak{sl}(3,\mathbb{C})$ Zakharov-Shabat Systems in Canonical and Pole Gauge with $\mathbb{Z}_2\times \mathbb{Z}_2$ Reduction of Mikhailov Type

Abstract

We consider the theory of the hierarchies of nonlinear evolution equations associated with two gauge-equivalent auxiliary systems. They are obtained from the Generalized Zakharov-Shabat system on $\mathfrak{sl}(3,\mathbb{C})$ in general position making a $\mathbb{Z}_2\times \mathbb{Z}_2$ reductions of Mikhailov type in canonical and in pole gauge respectively. Using the Recursion Operators approach we study the symplectic structures of the hierarchies of the nonlinear evolution equations associated with the above auxiliary systems and find the relation between them.