### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 055, 27 pages      arXiv:1004.1627      https://doi.org/10.3842/SIGMA.2010.055

### Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

Aristophanes Dimakis a and Folkert Müller-Hoissen b
a) Department of Financial and Management Engineering, University of the Aegean, 41, Kountourioti Str., GR-82100 Chios, Greece
b) Max-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10, D-37073 Göttingen, Germany

Received April 12, 2010, in final form June 21, 2010; Published online July 16, 2010

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.

Key words: AKNS hierarchy; negative flows; Miura transformation; bidifferential graded algebra; Heisenberg magnet; mKdV; NLS; sine-Gordon; vector short pulse equation; matrix solitons.

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