Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations
We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature, and they correspond to solutions of one of...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2024 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2024
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212108 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations. Filipe Kelmer and Keti Tenenblat. SIGMA 20 (2024), 015, 27 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862550495207358464 |
|---|---|
| author | Kelmer, Filipe Tenenblat, Keti |
| author_facet | Kelmer, Filipe Tenenblat, Keti |
| citation_txt | Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations. Filipe Kelmer and Keti Tenenblat. SIGMA 20 (2024), 015, 27 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature, and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon, and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation, and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.
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| first_indexed | 2026-03-13T03:51:21Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212108 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T03:51:21Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kelmer, Filipe Tenenblat, Keti 2026-01-28T13:56:19Z 2024 Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations. Filipe Kelmer and Keti Tenenblat. SIGMA 20 (2024), 015, 27 pages 1815-0659 2020 Mathematics Subject Classification: 58J72; 35J15; 35L10; 53A35; 53A05 arXiv:2211.01247 https://nasplib.isofts.kiev.ua/handle/123456789/212108 https://doi.org/10.3842/SIGMA.2024.015 We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature, and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon, and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation, and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given. Filipe Kelmer was partially supported by CNPq grant 132908/2015-8 and CAPES/BrazilFinance Code 001. Keti Tenenblat was partially supported by CNPq, Brazil, Proc. 311916/20210 and CAPES/Brazil-Finance Code 001. The authors would like to thank the anonymous referees whose comments gave a relevant contribution to the improvement of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations Article published earlier |
| spellingShingle | Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations Kelmer, Filipe Tenenblat, Keti |
| title | Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations |
| title_full | Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations |
| title_fullStr | Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations |
| title_full_unstemmed | Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations |
| title_short | Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations |
| title_sort | superposition formulae for the geometric bäcklund transformations of the hyperbolic and elliptic sine-gordon and sinh-gordon equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212108 |
| work_keys_str_mv | AT kelmerfilipe superpositionformulaeforthegeometricbacklundtransformationsofthehyperbolicandellipticsinegordonandsinhgordonequations AT tenenblatketi superpositionformulaeforthegeometricbacklundtransformationsofthehyperbolicandellipticsinegordonandsinhgordonequations |