Hirota Difference Equation and Darboux System: Mutual Symmetry
We considered the relation between two famous integrable equations: The Hirota difference equation (HDE) and the Darboux system that describes conjugate curvilinear systems of coordinates in <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvari...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/3/436 |
| _version_ | 1818037776717709312 |
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| author | Andrei Pogrebkov |
| author_facet | Andrei Pogrebkov |
| author_sort | Andrei Pogrebkov |
| collection | DOAJ |
| description | We considered the relation between two famous integrable equations: The Hirota difference equation (HDE) and the Darboux system that describes conjugate curvilinear systems of coordinates in <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> </semantics> </math> </inline-formula>. We demonstrated that specific properties of solutions of the HDE with respect to independent variables enabled introduction of an infinite set of discrete symmetries. We showed that degeneracy of the HDE with respect to parameters of these discrete symmetries led to the introduction of continuous symmetries by means of a specific limiting procedure. This enabled consideration of these symmetries on equal terms with the original HDE independent variables. In particular, the Darboux system appeared as an integrable equation where continuous symmetries of the HDE served as independent variables. We considered some cases of intermediate choice of independent variables, as well as the relation of these results with direct and inverse problems. |
| first_indexed | 2024-12-10T07:32:13Z |
| format | Article |
| id | doaj.art-d51b4a02a0734b92b61d8105e1fd0331 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-12-10T07:32:13Z |
| publishDate | 2019-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-d51b4a02a0734b92b61d8105e1fd03312022-12-22T01:57:32ZengMDPI AGSymmetry2073-89942019-03-0111343610.3390/sym11030436sym11030436Hirota Difference Equation and Darboux System: Mutual SymmetryAndrei Pogrebkov0Steklov Mathematical Institute, Moscow 119991, RussiaWe considered the relation between two famous integrable equations: The Hirota difference equation (HDE) and the Darboux system that describes conjugate curvilinear systems of coordinates in <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> </semantics> </math> </inline-formula>. We demonstrated that specific properties of solutions of the HDE with respect to independent variables enabled introduction of an infinite set of discrete symmetries. We showed that degeneracy of the HDE with respect to parameters of these discrete symmetries led to the introduction of continuous symmetries by means of a specific limiting procedure. This enabled consideration of these symmetries on equal terms with the original HDE independent variables. In particular, the Darboux system appeared as an integrable equation where continuous symmetries of the HDE served as independent variables. We considered some cases of intermediate choice of independent variables, as well as the relation of these results with direct and inverse problems.https://www.mdpi.com/2073-8994/11/3/436Hirota difference equationDarboux systemintegrable equationssymmetriescompatibilityinverse problem |
| spellingShingle | Andrei Pogrebkov Hirota Difference Equation and Darboux System: Mutual Symmetry Symmetry Hirota difference equation Darboux system integrable equations symmetries compatibility inverse problem |
| title | Hirota Difference Equation and Darboux System: Mutual Symmetry |
| title_full | Hirota Difference Equation and Darboux System: Mutual Symmetry |
| title_fullStr | Hirota Difference Equation and Darboux System: Mutual Symmetry |
| title_full_unstemmed | Hirota Difference Equation and Darboux System: Mutual Symmetry |
| title_short | Hirota Difference Equation and Darboux System: Mutual Symmetry |
| title_sort | hirota difference equation and darboux system mutual symmetry |
| topic | Hirota difference equation Darboux system integrable equations symmetries compatibility inverse problem |
| url | https://www.mdpi.com/2073-8994/11/3/436 |
| work_keys_str_mv | AT andreipogrebkov hirotadifferenceequationanddarbouxsystemmutualsymmetry |