Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations
In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/7/1268 |
| _version_ | 1797526020719378432 |
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| author | Yarong Xia Ruoxia Yao Xiangpeng Xin Yan Li |
| author_facet | Yarong Xia Ruoxia Yao Xiangpeng Xin Yan Li |
| author_sort | Yarong Xia |
| collection | DOAJ |
| description | In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions <i>u</i> and <i>v</i>. |
| first_indexed | 2024-03-10T09:22:17Z |
| format | Article |
| id | doaj.art-f6d95cd8a910489e9d4a95ca5c200bae |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T09:22:17Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-f6d95cd8a910489e9d4a95ca5c200bae2023-11-22T05:09:47ZengMDPI AGSymmetry2073-89942021-07-01137126810.3390/sym13071268Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV EquationsYarong Xia0Ruoxia Yao1Xiangpeng Xin2Yan Li3School of Computer Science, Shaanxi Normal University, Xi’an 710062, ChinaSchool of Computer Science, Shaanxi Normal University, Xi’an 710062, ChinaSchool of Mathematical Sciences, Liaocheng University, Liaocheng 252029, ChinaSchool of Computer Science, Shaanxi Normal University, Xi’an 710062, ChinaIn this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions <i>u</i> and <i>v</i>.https://www.mdpi.com/2073-8994/13/7/1268nonlocal symmetryPainlevé analysisinteraction solutionLie point symmetry |
| spellingShingle | Yarong Xia Ruoxia Yao Xiangpeng Xin Yan Li Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations Symmetry nonlocal symmetry Painlevé analysis interaction solution Lie point symmetry |
| title | Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations |
| title_full | Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations |
| title_fullStr | Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations |
| title_full_unstemmed | Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations |
| title_short | Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations |
| title_sort | nonlocal symmetry painleve integrable and interaction solutions for ckdv equations |
| topic | nonlocal symmetry Painlevé analysis interaction solution Lie point symmetry |
| url | https://www.mdpi.com/2073-8994/13/7/1268 |
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