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Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

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 NL...

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Main Authors: Dimakis, A., Müller-Hoissen, F.
Format: Article
Language:English
Published: Інститут математики НАН України 2010
Series:Symmetry, Integrability and Geometry: Methods and Applications
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/146356
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spelling irk-123456789-1463562019-02-10T01:24:09Z Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions Dimakis, A. Müller-Hoissen, F. 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. 2010 Article Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 44 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J35; 37K10; 16E45 DOI:10.3842/SIGMA.2010.055 http://dspace.nbuv.gov.ua/handle/123456789/146356 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description 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.
format Article
author Dimakis, A.
Müller-Hoissen, F.
spellingShingle Dimakis, A.
Müller-Hoissen, F.
Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Dimakis, A.
Müller-Hoissen, F.
author_sort Dimakis, A.
title Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
title_short Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
title_full Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
title_fullStr Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
title_full_unstemmed Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
title_sort bidifferential calculus approach to akns hierarchies and their solutions
publisher Інститут математики НАН України
publishDate 2010
url http://dspace.nbuv.gov.ua/handle/123456789/146356
citation_txt Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 44 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT dimakisa bidifferentialcalculusapproachtoaknshierarchiesandtheirsolutions
AT mullerhoissenf bidifferentialcalculusapproachtoaknshierarchiesandtheirsolutions
first_indexed 2023-05-20T17:24:20Z
last_indexed 2023-05-20T17:24:20Z
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