On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations
We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund tr...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2008 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2008
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/149004 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations / D. Levi, M. Petrera, C. Scimiterna, R. Yamilov // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 31 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862711611491352576 |
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| author | Levi, D. Petrera, M. Scimiterna, C. Yamilov, R. |
| author_facet | Levi, D. Petrera, M. Scimiterna, C. Yamilov, R. |
| citation_txt | On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations / D. Levi, M. Petrera, C. Scimiterna, R. Yamilov // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 31 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.
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| first_indexed | 2025-12-07T17:31:40Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149004 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:31:40Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Levi, D. Petrera, M. Scimiterna, C. Yamilov, R. 2019-02-19T12:54:07Z 2019-02-19T12:54:07Z 2008 On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations / D. Levi, M. Petrera, C. Scimiterna, R. Yamilov // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 31 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K10; 37L20; 39A05 https://nasplib.isofts.kiev.ua/handle/123456789/149004 We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability. DL, MP and CS have been partially supported by PRIN Project Metodi geometrici nella teoria delle onde non lineari ed applicazioni-2006 of the Italian Minister for Education and Scientific Research. RY has been partially supported by the Russian Foundation for Basic Research (Grant numbers 07-01-00081-a and 06-01-92051-KE-a) and he thanks the University of Roma Tre for hospitality. This work has been done in the framework of the Project Classification of integrable discrete and continuous models financed by a joint grant from EINSTEIN consortium and RFBR. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations Article published earlier |
| spellingShingle | On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations Levi, D. Petrera, M. Scimiterna, C. Yamilov, R. |
| title | On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations |
| title_full | On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations |
| title_fullStr | On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations |
| title_full_unstemmed | On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations |
| title_short | On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations |
| title_sort | on miura transformations and volterra-type equations associated with the adler-bobenko-suris equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149004 |
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