Strict versions of various matrix hierarchies related to SL(n)-loops and their combinations

Abstract

Let $\mathfrak{t}$ be a commutative Lie subalgebra of ${\rm sl}_{n}(\mathbb{C})$ of maximal dimension. We consider in this paper three spaces of $\mathfrak{t}$-loops that each get deformed in a different way. We require that the deformed generators of each of them evolve w.r.t. the commuting flows they generate according to a certain, different set of Lax equations. This leads to three integrable hierarchies: the $({\rm sl}_{n}(\mathbb{C}), \mathfrak{t})$-hierarchy, its strict version and the combined $({\rm sl}_{n}(\mathbb{C}), \mathfrak{t})$-hierarchy. For $n=2$ and $\mathfrak{t}$ the diagonal matrices, the $({\rm sl}_{2}(\mathbb{C}), \mathfrak{t})$-hierarchy is the AKNS-hierarchy. We treat their interrelations and show that all three have a zero curvature form. Furthermore, we discuss their linearization and we conclude by giving the construction of a large class of solutions.