Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example
In this paper, the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a local time-fractional Korteweg-de vries (tfKdV) equation. More specifically, a linear spectral problem associated with the KdV equation of integer order is first equipped with...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2019/7952871 |
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| author | Sheng Zhang Yuanyuan Wei Bo Xu |
| author_facet | Sheng Zhang Yuanyuan Wei Bo Xu |
| author_sort | Sheng Zhang |
| collection | DOAJ |
| description | In this paper, the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a local time-fractional Korteweg-de vries (tfKdV) equation. More specifically, a linear spectral problem associated with the KdV equation of integer order is first equipped with local time-fractional derivative. Based on the spectral problem with the equipped local time-fractional derivative, the local tfKdV equation with Lax integrability is then derived and solved by extending the spectral transform. As a result, a formula of exact solution with Mittag-Leffler functions is obtained. Finally, in the case of reflectionless potential the obtained exact solution is reduced to fractional n-soliton solution. In order to gain more insights into the fractional n-soliton dynamics, the dynamical evolutions of the reduced fractional one-, two-, and three-soliton solutions are simulated. It is shown that the velocities of the reduced fractional one-, two-, and three-soliton solutions change with the fractional order. |
| format | Article |
| id | doaj-art-d626147f7d784c3999bf849998fe680d |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-d626147f7d784c3999bf849998fe680d2025-02-03T05:50:48ZengWileyComplexity1076-27871099-05262019-01-01201910.1155/2019/79528717952871Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete ExampleSheng Zhang0Yuanyuan Wei1Bo Xu2School of Mathematics and Physics, Bohai University, Jinzhou 121013, ChinaSchool of Mathematics and Physics, Bohai University, Jinzhou 121013, ChinaSchool of Education and Sports, Bohai University, Jinzhou 121013, ChinaIn this paper, the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a local time-fractional Korteweg-de vries (tfKdV) equation. More specifically, a linear spectral problem associated with the KdV equation of integer order is first equipped with local time-fractional derivative. Based on the spectral problem with the equipped local time-fractional derivative, the local tfKdV equation with Lax integrability is then derived and solved by extending the spectral transform. As a result, a formula of exact solution with Mittag-Leffler functions is obtained. Finally, in the case of reflectionless potential the obtained exact solution is reduced to fractional n-soliton solution. In order to gain more insights into the fractional n-soliton dynamics, the dynamical evolutions of the reduced fractional one-, two-, and three-soliton solutions are simulated. It is shown that the velocities of the reduced fractional one-, two-, and three-soliton solutions change with the fractional order.http://dx.doi.org/10.1155/2019/7952871 |
| spellingShingle | Sheng Zhang Yuanyuan Wei Bo Xu Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example Complexity |
| title | Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example |
| title_full | Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example |
| title_fullStr | Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example |
| title_full_unstemmed | Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example |
| title_short | Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example |
| title_sort | fractional soliton dynamics and spectral transform of time fractional nonlinear systems a concrete example |
| url | http://dx.doi.org/10.1155/2019/7952871 |
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