Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV
We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal trans...
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MDPI AG
2021-04-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/9/926 |
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| author | Paz Albares Pilar Garcia Estévez |
| author_facet | Paz Albares Pilar Garcia Estévez |
| author_sort | Paz Albares |
| collection | DOAJ |
| description | We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax pairs for RKdV and RmKdV are straightforwardly obtained by means of the aforementioned reciprocal transformations. We have also identified the classical Lie symmetries for the Lax pairs of RKdV and RmKdV. Non-trivial similarity reductions are computed and they yield non-autonomous ordinary differential equations (ODEs), whose Lax pairs are obtained as a consequence of the reductions. |
| first_indexed | 2024-03-10T12:05:49Z |
| format | Article |
| id | doaj.art-824df606b8e9451aa5c13ada9e96fb0f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T12:05:49Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
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| spelling | doaj.art-824df606b8e9451aa5c13ada9e96fb0f2023-11-21T16:35:58ZengMDPI AGMathematics2227-73902021-04-019992610.3390/math9090926Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdVPaz Albares0Pilar Garcia Estévez1Departamento de Física Fundamental, Universidad de Salamanca, 37008 Salamanca, SpainDepartamento de Física Fundamental, Universidad de Salamanca, 37008 Salamanca, SpainWe present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax pairs for RKdV and RmKdV are straightforwardly obtained by means of the aforementioned reciprocal transformations. We have also identified the classical Lie symmetries for the Lax pairs of RKdV and RmKdV. Non-trivial similarity reductions are computed and they yield non-autonomous ordinary differential equations (ODEs), whose Lax pairs are obtained as a consequence of the reductions.https://www.mdpi.com/2227-7390/9/9/926symmetry groupsconservation lawspartial differential equations |
| spellingShingle | Paz Albares Pilar Garcia Estévez Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV Mathematics symmetry groups conservation laws partial differential equations |
| title | Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV |
| title_full | Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV |
| title_fullStr | Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV |
| title_full_unstemmed | Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV |
| title_short | Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV |
| title_sort | miura reciprocal transformation and symmetries for the spectral problems of kdv and mkdv |
| topic | symmetry groups conservation laws partial differential equations |
| url | https://www.mdpi.com/2227-7390/9/9/926 |
| work_keys_str_mv | AT pazalbares miurareciprocaltransformationandsymmetriesforthespectralproblemsofkdvandmkdv AT pilargarciaestevez miurareciprocaltransformationandsymmetriesforthespectralproblemsofkdvandmkdv |