Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension
We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite symmetry g...
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MDPI AG
2024-08-01
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| Series: | Fractal and Fractional |
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| author | Lihua Zhang Zitong Zheng Bo Shen Gangwei Wang Zhenli Wang |
| author_facet | Lihua Zhang Zitong Zheng Bo Shen Gangwei Wang Zhenli Wang |
| author_sort | Lihua Zhang |
| collection | DOAJ |
| description | We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite symmetry group method, and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton and elliptical function interplaying solution for the fractional KdVSKR model in (1+1)-dimension has been derived for the first time. For the fractional KdVSKR model in (2+1)-dimension, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. The effect of the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> on the dynamical behaviors of the solutions has been illustrated by figures. The three-wave interaction solution has not been studied in the current references. The novelty of this paper is that the finite symmetry group method is adopted to construct interaction solutions of fractional nonlinear systems. This research idea can be applied to other fractional differential equations. |
| format | Article |
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| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | MDPI AG |
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| series | Fractal and Fractional |
| spelling | doaj-art-2a0b36862f1b49a186e1b72a013f855d2025-08-20T01:55:31ZengMDPI AGFractal and Fractional2504-31102024-08-018951710.3390/fractalfract8090517Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-DimensionLihua Zhang0Zitong Zheng1Bo Shen2Gangwei Wang3Zhenli Wang4School of Statistics and Mathematics, Hebei University of Economics and Business, Shijiazhuang 050061, ChinaSchool of Statistics and Mathematics, Hebei University of Economics and Business, Shijiazhuang 050061, ChinaSchool of Statistics and Mathematics, Hebei University of Economics and Business, Shijiazhuang 050061, ChinaSchool of Statistics and Mathematics, Hebei University of Economics and Business, Shijiazhuang 050061, ChinaSchool of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, ChinaWe extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite symmetry group method, and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton and elliptical function interplaying solution for the fractional KdVSKR model in (1+1)-dimension has been derived for the first time. For the fractional KdVSKR model in (2+1)-dimension, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. The effect of the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> on the dynamical behaviors of the solutions has been illustrated by figures. The three-wave interaction solution has not been studied in the current references. The novelty of this paper is that the finite symmetry group method is adopted to construct interaction solutions of fractional nonlinear systems. This research idea can be applied to other fractional differential equations.https://www.mdpi.com/2504-3110/8/9/517KdVSKR equationfinite symmetry groupsinteraction solutionscaputo derivative |
| spellingShingle | Lihua Zhang Zitong Zheng Bo Shen Gangwei Wang Zhenli Wang Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension Fractal and Fractional KdVSKR equation finite symmetry groups interaction solutions caputo derivative |
| title | Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension |
| title_full | Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension |
| title_fullStr | Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension |
| title_full_unstemmed | Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension |
| title_short | Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension |
| title_sort | interaction solutions for the fractional kdvskr equations in 1 1 dimension and 2 1 dimension |
| topic | KdVSKR equation finite symmetry groups interaction solutions caputo derivative |
| url | https://www.mdpi.com/2504-3110/8/9/517 |
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