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

TL;DR

This paper introduces three related integrable hierarchies associated with matrix loops and their deformations, generalizing known hierarchies like AKNS, and explores their properties, interrelations, and solutions.

Contribution

It defines and analyzes three new matrix hierarchies related to SL(n)-loops, including their zero curvature form, linearization, and solution construction, extending existing integrable systems.

Findings

All three hierarchies possess a zero curvature form.

The hierarchies can be linearized.

A large class of solutions is constructed.

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.