Novel Bäcklund Transformations for Integrable Equations
In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation analogous...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/19/3565 |
| _version_ | 1827654015863750656 |
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| author | Pilar Ruiz Gordoa Andrew Pickering |
| author_facet | Pilar Ruiz Gordoa Andrew Pickering |
| author_sort | Pilar Ruiz Gordoa |
| collection | DOAJ |
| description | In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation analogous to that of this matrix fourth Painlevé equation. Such auto-Bäcklund transformations, in appearance similar to those for Painlevé equations, are quite novel, having been little studied in the case of partial differential equations. Our work here shows the importance of the underlying structure of differential equations, whether ordinary or partial, in the derivation of such results. The starting point for the results in this paper is the construction of a new completely integrable equation, namely, an inverse matrix dispersive water wave equation. |
| first_indexed | 2024-03-09T21:27:36Z |
| format | Article |
| id | doaj.art-7ace174c467c43bb977fb3bc4e80bebc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:27:36Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7ace174c467c43bb977fb3bc4e80bebc2023-11-23T21:03:40ZengMDPI AGMathematics2227-73902022-09-011019356510.3390/math10193565Novel Bäcklund Transformations for Integrable EquationsPilar Ruiz Gordoa0Andrew Pickering1Área de Matemática Aplicada, ESCET Universidad Rey Juan Carlos C, Tulipán s/n, 28933 Móstoles, Madrid, SpainÁrea de Matemática Aplicada, ESCET Universidad Rey Juan Carlos C, Tulipán s/n, 28933 Móstoles, Madrid, SpainIn this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation analogous to that of this matrix fourth Painlevé equation. Such auto-Bäcklund transformations, in appearance similar to those for Painlevé equations, are quite novel, having been little studied in the case of partial differential equations. Our work here shows the importance of the underlying structure of differential equations, whether ordinary or partial, in the derivation of such results. The starting point for the results in this paper is the construction of a new completely integrable equation, namely, an inverse matrix dispersive water wave equation.https://www.mdpi.com/2227-7390/10/19/3565integrable equationsBäcklund transformationsMiura mapsinverse matrixdispersive water wave equation |
| spellingShingle | Pilar Ruiz Gordoa Andrew Pickering Novel Bäcklund Transformations for Integrable Equations Mathematics integrable equations Bäcklund transformations Miura maps inverse matrix dispersive water wave equation |
| title | Novel Bäcklund Transformations for Integrable Equations |
| title_full | Novel Bäcklund Transformations for Integrable Equations |
| title_fullStr | Novel Bäcklund Transformations for Integrable Equations |
| title_full_unstemmed | Novel Bäcklund Transformations for Integrable Equations |
| title_short | Novel Bäcklund Transformations for Integrable Equations |
| title_sort | novel backlund transformations for integrable equations |
| topic | integrable equations Bäcklund transformations Miura maps inverse matrix dispersive water wave equation |
| url | https://www.mdpi.com/2227-7390/10/19/3565 |
| work_keys_str_mv | AT pilarruizgordoa novelbacklundtransformationsforintegrableequations AT andrewpickering novelbacklundtransformationsforintegrableequations |