Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański
We present first heavenly equation of Plebański in a two-component evolutionary form and obtain Lagrangian and Hamiltonian representations of this system. We study all point symmetries of the two-component system and, using the inverse Noether theorem in the Hamiltonian form, obtain all the integral...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2016 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2016
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147857 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański / M.B. Sheftel, D. Yazıcı // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 34 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862550296916393984 |
|---|---|
| author | Sheftel, M.B. Yazıcı, D. |
| author_facet | Sheftel, M.B. Yazıcı, D. |
| citation_txt | Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański / M.B. Sheftel, D. Yazıcı // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 34 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present first heavenly equation of Plebański in a two-component evolutionary form and obtain Lagrangian and Hamiltonian representations of this system. We study all point symmetries of the two-component system and, using the inverse Noether theorem in the Hamiltonian form, obtain all the integrals of motion corresponding to each variational (Noether) symmetry. We derive two linearly independent recursion operators for symmetries of this system related by a discrete symmetry of both the two-component system and its symmetry condition. Acting by these operators on the first Hamiltonian operator J0 we obtain second and third Hamiltonian operators. However, we were not able to find Hamiltonian densities corresponding to the latter two operators. Therefore, we construct two recursion operators, which are either even or odd, respectively, under the above-mentioned discrete symmetry. Acting with them on J0, we generate another two Hamiltonian operators J+ and J− and find the corresponding Hamiltonian densities, thus obtaining second and third Hamiltonian representations for the first heavenly equation in a two-component form. Using P. Olver's theory of the functional multi-vectors, we check that the linear combination of J0, J+ and J− with arbitrary constant coefficients satisfies Jacobi identities. Since their skew symmetry is obvious, these three operators are compatible Hamiltonian operators and hence we obtain a tri-Hamiltonian representation of the first heavenly equation. Our well-founded conjecture applied here is that P. Olver's method works fine for nonlocal operators and our proof of the Jacobi identities and bi-Hamiltonian structures crucially depends on the validity of this conjecture.
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| first_indexed | 2025-11-25T20:40:32Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147857 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-25T20:40:32Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sheftel, M.B. Yazıcı, D. 2019-02-16T09:21:32Z 2019-02-16T09:21:32Z 2016 Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański / M.B. Sheftel, D. Yazıcı // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q75; 83C15; 37K05; 37K10 DOI:10.3842/SIGMA.2016.091 https://nasplib.isofts.kiev.ua/handle/123456789/147857 We present first heavenly equation of Plebański in a two-component evolutionary form and obtain Lagrangian and Hamiltonian representations of this system. We study all point symmetries of the two-component system and, using the inverse Noether theorem in the Hamiltonian form, obtain all the integrals of motion corresponding to each variational (Noether) symmetry. We derive two linearly independent recursion operators for symmetries of this system related by a discrete symmetry of both the two-component system and its symmetry condition. Acting by these operators on the first Hamiltonian operator J0 we obtain second and third Hamiltonian operators. However, we were not able to find Hamiltonian densities corresponding to the latter two operators. Therefore, we construct two recursion operators, which are either even or odd, respectively, under the above-mentioned discrete symmetry. Acting with them on J0, we generate another two Hamiltonian operators J+ and J− and find the corresponding Hamiltonian densities, thus obtaining second and third Hamiltonian representations for the first heavenly equation in a two-component form. Using P. Olver's theory of the functional multi-vectors, we check that the linear combination of J0, J+ and J− with arbitrary constant coefficients satisfies Jacobi identities. Since their skew symmetry is obvious, these three operators are compatible Hamiltonian operators and hence we obtain a tri-Hamiltonian representation of the first heavenly equation. Our well-founded conjecture applied here is that P. Olver's method works fine for nonlocal operators and our proof of the Jacobi identities and bi-Hamiltonian structures crucially depends on the validity of this conjecture. The research of M.B. Sheftel is partly supported by the research grant from Bo˘gazi¸ci University
 Scientific Research Fund (BAP), research project No. 11643. The authors are thankful to the
 referees for their important remarks. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański Article published earlier |
| spellingShingle | Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański Sheftel, M.B. Yazıcı, D. |
| title | Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański |
| title_full | Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański |
| title_fullStr | Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański |
| title_full_unstemmed | Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański |
| title_short | Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański |
| title_sort | recursion operators and tri-hamiltonian structure of the first heavenly equation of plebański |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147857 |
| work_keys_str_mv | AT sheftelmb recursionoperatorsandtrihamiltonianstructureofthefirstheavenlyequationofplebanski AT yazıcıd recursionoperatorsandtrihamiltonianstructureofthefirstheavenlyequationofplebanski |