Modulation Instability and Dynamical Analysis of New Abundant Closed-Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M-Fractional Derivative
In this work, we study the modulation instability (MI) and closed-form soliton solution of the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV-ZK) equation with a truncated M-fractional derivative. The mKdV-ZK equation can be used to describe the behavior of ion-acoustic waves in plasma and the...
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| Format: | Article |
| Language: | English |
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Wiley
2024-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2024/1754782 |
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| author | Md. Shafiqul Islam Md. Mamunur Roshid Mahtab Uddin Ashek Ahmed |
| author_facet | Md. Shafiqul Islam Md. Mamunur Roshid Mahtab Uddin Ashek Ahmed |
| author_sort | Md. Shafiqul Islam |
| collection | DOAJ |
| description | In this work, we study the modulation instability (MI) and closed-form soliton solution of the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV-ZK) equation with a truncated M-fractional derivative. The mKdV-ZK equation can be used to describe the behavior of ion-acoustic waves in plasma and the propagation of surface waves in deep water with nonlinear and dispersive effects in fluid dynamics. To execute a closed soliton solution, we implement two dominant techniques, namely, the improved F-expansion scheme and unified solver techniques for the mKdV-ZK equation. Under the condition of parameters, the obtained solutions exhibit hyperbolic, trigonometric, and rational functions with free parameters. Using the Maple software, we present three-dimensional (3D) plots with density plots and two-dimensional (2D) graphical representations for appropriate values of the free parameters. Under the conditions of the numerical values of the free parameters, the obtained closed-form solutions provided some novel phenomena such as antikink shape wave, dark bell shape, collision of kink and periodic lump wave, periodic wave, collision of antikink and periodic lump wave, collision of linked lump wave with kink shape, periodic lump wave by using improved F-expansion method and kink shape, diverse type of periodic wave, singular soliton, and bright bell and dark bell-shape wave phenomena by using unified solver method. The comparative effects of the fractional derivative are illustrated in 2D plots. We also provided a comparison between the results obtained through the suggested scheme and those obtained by other approaches, showing some similar solutions and some that are different. Besides, to check of stability and instability of the solution, the MI analysis of the given system is investigated based on the standard linear stability analysis and the MI gain spectrum analysis. With the use of symbolic calculations, the applied approach is clear, simple, and elementary, as demonstrated by the more broad and novel results that are obtained. It may also be applied to more complex phenomena. |
| format | Article |
| id | doaj-art-b9aee72165cd4df78f7eee55b1a15e76 |
| institution | Kabale University |
| issn | 1687-0042 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-b9aee72165cd4df78f7eee55b1a15e762025-02-03T11:37:17ZengWileyJournal of Applied Mathematics1687-00422024-01-01202410.1155/2024/1754782Modulation Instability and Dynamical Analysis of New Abundant Closed-Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M-Fractional DerivativeMd. Shafiqul Islam0Md. Mamunur Roshid1Mahtab Uddin2Ashek Ahmed3Department of MathematicsDepartment of MathematicsInstitute of Natural SciencesInstitute of Natural SciencesIn this work, we study the modulation instability (MI) and closed-form soliton solution of the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV-ZK) equation with a truncated M-fractional derivative. The mKdV-ZK equation can be used to describe the behavior of ion-acoustic waves in plasma and the propagation of surface waves in deep water with nonlinear and dispersive effects in fluid dynamics. To execute a closed soliton solution, we implement two dominant techniques, namely, the improved F-expansion scheme and unified solver techniques for the mKdV-ZK equation. Under the condition of parameters, the obtained solutions exhibit hyperbolic, trigonometric, and rational functions with free parameters. Using the Maple software, we present three-dimensional (3D) plots with density plots and two-dimensional (2D) graphical representations for appropriate values of the free parameters. Under the conditions of the numerical values of the free parameters, the obtained closed-form solutions provided some novel phenomena such as antikink shape wave, dark bell shape, collision of kink and periodic lump wave, periodic wave, collision of antikink and periodic lump wave, collision of linked lump wave with kink shape, periodic lump wave by using improved F-expansion method and kink shape, diverse type of periodic wave, singular soliton, and bright bell and dark bell-shape wave phenomena by using unified solver method. The comparative effects of the fractional derivative are illustrated in 2D plots. We also provided a comparison between the results obtained through the suggested scheme and those obtained by other approaches, showing some similar solutions and some that are different. Besides, to check of stability and instability of the solution, the MI analysis of the given system is investigated based on the standard linear stability analysis and the MI gain spectrum analysis. With the use of symbolic calculations, the applied approach is clear, simple, and elementary, as demonstrated by the more broad and novel results that are obtained. It may also be applied to more complex phenomena.http://dx.doi.org/10.1155/2024/1754782 |
| spellingShingle | Md. Shafiqul Islam Md. Mamunur Roshid Mahtab Uddin Ashek Ahmed Modulation Instability and Dynamical Analysis of New Abundant Closed-Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M-Fractional Derivative Journal of Applied Mathematics |
| title | Modulation Instability and Dynamical Analysis of New Abundant Closed-Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M-Fractional Derivative |
| title_full | Modulation Instability and Dynamical Analysis of New Abundant Closed-Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M-Fractional Derivative |
| title_fullStr | Modulation Instability and Dynamical Analysis of New Abundant Closed-Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M-Fractional Derivative |
| title_full_unstemmed | Modulation Instability and Dynamical Analysis of New Abundant Closed-Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M-Fractional Derivative |
| title_short | Modulation Instability and Dynamical Analysis of New Abundant Closed-Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M-Fractional Derivative |
| title_sort | modulation instability and dynamical analysis of new abundant closed form solutions of the modified korteweg de vries zakharov kuznetsov model with truncated m fractional derivative |
| url | http://dx.doi.org/10.1155/2024/1754782 |
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