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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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2010-07-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2010.055 |
| _version_ | 1819293546843209728 |
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| author | Aristophanes Dimakis Folkert Müller-Hoissen |
| author_facet | Aristophanes Dimakis Folkert Müller-Hoissen |
| author_sort | Aristophanes Dimakis |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-24T04:12:09Z |
| format | Article |
| id | doaj.art-6fc6eb47fecb49838d9c6ad76232a722 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-24T04:12:09Z |
| publishDate | 2010-07-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-6fc6eb47fecb49838d9c6ad76232a7222022-12-21T17:16:02ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592010-07-016055Bidifferential Calculus Approach to AKNS Hierarchies and Their SolutionsAristophanes DimakisFolkert Müller-HoissenWe 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.http://dx.doi.org/10.3842/SIGMA.2010.055AKNS hierarchynegative flowsMiura transformationbidifferential graded algebraHeisenberg magnetmKdVNLSsine-Gordonvector short pulse equationmatrix solitons |
| spellingShingle | Aristophanes Dimakis Folkert Müller-Hoissen Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions Symmetry, Integrability and Geometry: Methods and Applications AKNS hierarchy negative flows Miura transformation bidifferential graded algebra Heisenberg magnet mKdV NLS sine-Gordon vector short pulse equation matrix solitons |
| title | 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_short | Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions |
| title_sort | bidifferential calculus approach to akns hierarchies and their solutions |
| topic | AKNS hierarchy negative flows Miura transformation bidifferential graded algebra Heisenberg magnet mKdV NLS sine-Gordon vector short pulse equation matrix solitons |
| url | http://dx.doi.org/10.3842/SIGMA.2010.055 |
| work_keys_str_mv | AT aristophanesdimakis bidifferentialcalculusapproachtoaknshierarchiesandtheirsolutions AT folkertmullerhoissen bidifferentialcalculusapproachtoaknshierarchiesandtheirsolutions |