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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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2010 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2010
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146356 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862728610717106176 |
|---|---|
| author | Dimakis, A. Müller-Hoissen, F. |
| author_facet | Dimakis, A. Müller-Hoissen, F. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-12-07T19:09:48Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146356 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:09:48Z |
| publishDate | 2010 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dimakis, A. Müller-Hoissen, F. 2019-02-09T09:31:31Z 2019-02-09T09:31:31Z 2010 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 https://nasplib.isofts.kiev.ua/handle/123456789/146356 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. We would like to thank Sergei Sakovich and some anonymous referees for helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions Article published earlier |
| spellingShingle | Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions Dimakis, A. Müller-Hoissen, F. |
| 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 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146356 |
| work_keys_str_mv | AT dimakisa bidifferentialcalculusapproachtoaknshierarchiesandtheirsolutions AT mullerhoissenf bidifferentialcalculusapproachtoaknshierarchiesandtheirsolutions |