Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

TL;DR

This paper introduces a bidifferential calculus framework to generate solutions for AKNS hierarchies and related integrable systems, unifying various equations and their solutions through algebraic transformations.

Contribution

It develops a universal algebraic approach to derive solutions for multiple integrable hierarchies, including AKNS, NLS, mKdV, and sine-Gordon, using a bidifferential graded algebra.

Findings

Generated infinite families of exact solutions, including matrix solitons.

Revealed connections between hierarchies via Miura transformations.

Extended hierarchies to include negative flows and related equations.

Abstract

We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the two derivations of the bidifferential graded algebra, we recover "negative flows", leading to an extension of the respective hierarchy. In this way we also meet a matrix and vector version of the short pulse equation and also the sine-Gordon equation. For these equations corresponding solution formulas are also derived. In all these cases the solutions are parametrized in terms of matrix data that have to satisfy a certain Sylvester equation.