The strict AKNS hierarchy: its structure and solutions

TL;DR

This paper introduces a new integrable hierarchy extending the AKNS system, analyzing its structure, compatibility, and solutions through a broader algebraic deformation and zero curvature formulation.

Contribution

It develops a wider deformation of the algebra in the loop space of sl(2), leading to a new integrable hierarchy with a novel solution construction.

Findings

The hierarchy's Lax equations are compatible and equivalent to zero curvature conditions.

A linearization method for the hierarchy is established.

A broad class of solutions for the hierarchy is constructed.

Abstract

In this paper we discuss an integrable hierarchy of compatible Lax equations that is obtained by a wider deformation of a commutative algebra in the loop space of ${\rm sl}_{2}$ than that in the AKNS-case and whose Lax equations are based on a different decomposition of this loop space. We show the compatibility of these Lax equations and that they are equivalent to a set of zero curvature relations. We present a linearization of the system and conclude by giving a wide construction of solutions of this hierarchy.