Functional representation of the negative AKNS hierarchy

TL;DR

This paper develops a functional representation for the extended AKNS hierarchy, including both positive and negative flows, enabling the derivation of local systems, conservation laws, and soliton solutions.

Contribution

It introduces a finite set of functional equations using Miwa's shifts that encapsulate the entire hierarchy, including negative flows, and extends to the Landau-Lifshitz hierarchy.

Findings

Functional equations for the hierarchy derived

Negative flows converted into local systems

Generated N-dark-soliton solutions

Abstract

This paper is devoted to the negative flows of the AKNS hierarchy. The main result of this work is the functional representation of the extended AKNS hierarchy, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed by means of the Miwa's shifts, which contains all equations of the hierarchy. Using the obtained functional representation we convert the nonlocal equations of the negative subhierarchy into local systems of higher order, derive the generating function of the conservation laws and the N-dark-soliton solutions for the extended AKNS hierarchy. As an additional result we obtain the functional representation of the Landau-Lifshitz hierarchy.