Rational Sine-Gordon expansion method to analyze the dynamical behavior of the time-fractional phi-four and (2 + 1) dimensional CBS equations
Abstract This study uses the rational Sine-Gordon expansion (RSGE) method to investigate the dynamical behavior of traveling wave solutions of the water wave phenomena for the time-fractional phi-four equation and the (2 + 1) dimensional Calogero-Bogoyavlanskil schilf (CBS) equation based on the con...
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Nature Portfolio
2024-04-01
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| Online Access: | https://doi.org/10.1038/s41598-024-60156-w |
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| author | Abdulla-Al- Mamun Chunhui Lu Samsun Nahar Ananna Md Mohi Uddin |
| author_facet | Abdulla-Al- Mamun Chunhui Lu Samsun Nahar Ananna Md Mohi Uddin |
| author_sort | Abdulla-Al- Mamun |
| collection | DOAJ |
| description | Abstract This study uses the rational Sine-Gordon expansion (RSGE) method to investigate the dynamical behavior of traveling wave solutions of the water wave phenomena for the time-fractional phi-four equation and the (2 + 1) dimensional Calogero-Bogoyavlanskil schilf (CBS) equation based on the conformable derivative. The technique uses the sine-Gordon equation as an auxiliary equation to generalize the well-known sine-Gordon expansion. It adopts a more broad strategy, a rational function rather than a polynomial one, of the solutions of the auxiliary equation, in contrast to the traditional sine-Gordon expansion technique. Several explanations for hyperbolic functions may be produced using the previously stated approach. The approach mentioned above is employed to provide diverse solutions of the time-fractional phi-four equation and the (2 + 1) dimensional CBS equations involving hyperbolic functions, such as soliton, single soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, and others. The RSGE approach enhances our comprehension of nonlinear processes, offers precise solutions to nonlinear equations, facilitates the investigation of solitons, propels the development of mathematical tools, and is applicable in many scientific and technical fields. The solutions are graphically shown in three-dimensional (3D) surface and contour plots using MATLAB software. All screens display the absolute wave configurations in the resolutions of the equation with the proper parameters. Furthermore, it can be deduced that the physical properties of the found solutions and their characteristics may help us comprehend how shallow water waves move in nonlinear dynamics. |
| format | Article |
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| institution | OA Journals |
| issn | 2045-2322 |
| language | English |
| publishDate | 2024-04-01 |
| publisher | Nature Portfolio |
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| spelling | doaj-art-e207b5ff90c347a6ab1f229933f23e7b2025-08-20T02:32:50ZengNature PortfolioScientific Reports2045-23222024-04-0114111810.1038/s41598-024-60156-wRational Sine-Gordon expansion method to analyze the dynamical behavior of the time-fractional phi-four and (2 + 1) dimensional CBS equationsAbdulla-Al- Mamun0Chunhui Lu1Samsun Nahar Ananna2Md Mohi Uddin3College of Hydrology and Water Resources, Hohai UniversityCollege of Hydrology and Water Resources, Hohai UniversitySchool of Mathematics, Hohai UniversityCollege of Water Conservancy and Hydropower Engineering, Hohai UniversityAbstract This study uses the rational Sine-Gordon expansion (RSGE) method to investigate the dynamical behavior of traveling wave solutions of the water wave phenomena for the time-fractional phi-four equation and the (2 + 1) dimensional Calogero-Bogoyavlanskil schilf (CBS) equation based on the conformable derivative. The technique uses the sine-Gordon equation as an auxiliary equation to generalize the well-known sine-Gordon expansion. It adopts a more broad strategy, a rational function rather than a polynomial one, of the solutions of the auxiliary equation, in contrast to the traditional sine-Gordon expansion technique. Several explanations for hyperbolic functions may be produced using the previously stated approach. The approach mentioned above is employed to provide diverse solutions of the time-fractional phi-four equation and the (2 + 1) dimensional CBS equations involving hyperbolic functions, such as soliton, single soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, and others. The RSGE approach enhances our comprehension of nonlinear processes, offers precise solutions to nonlinear equations, facilitates the investigation of solitons, propels the development of mathematical tools, and is applicable in many scientific and technical fields. The solutions are graphically shown in three-dimensional (3D) surface and contour plots using MATLAB software. All screens display the absolute wave configurations in the resolutions of the equation with the proper parameters. Furthermore, it can be deduced that the physical properties of the found solutions and their characteristics may help us comprehend how shallow water waves move in nonlinear dynamics.https://doi.org/10.1038/s41598-024-60156-wThe rational sine-Gordon expansion (RSGE) methodPhi-four equationSoliton waveTravelling wave solutionCalogero-Bogoyavlanskil Schilf equationSine-Gordon |
| spellingShingle | Abdulla-Al- Mamun Chunhui Lu Samsun Nahar Ananna Md Mohi Uddin Rational Sine-Gordon expansion method to analyze the dynamical behavior of the time-fractional phi-four and (2 + 1) dimensional CBS equations Scientific Reports The rational sine-Gordon expansion (RSGE) method Phi-four equation Soliton wave Travelling wave solution Calogero-Bogoyavlanskil Schilf equation Sine-Gordon |
| title | Rational Sine-Gordon expansion method to analyze the dynamical behavior of the time-fractional phi-four and (2 + 1) dimensional CBS equations |
| title_full | Rational Sine-Gordon expansion method to analyze the dynamical behavior of the time-fractional phi-four and (2 + 1) dimensional CBS equations |
| title_fullStr | Rational Sine-Gordon expansion method to analyze the dynamical behavior of the time-fractional phi-four and (2 + 1) dimensional CBS equations |
| title_full_unstemmed | Rational Sine-Gordon expansion method to analyze the dynamical behavior of the time-fractional phi-four and (2 + 1) dimensional CBS equations |
| title_short | Rational Sine-Gordon expansion method to analyze the dynamical behavior of the time-fractional phi-four and (2 + 1) dimensional CBS equations |
| title_sort | rational sine gordon expansion method to analyze the dynamical behavior of the time fractional phi four and 2 1 dimensional cbs equations |
| topic | The rational sine-Gordon expansion (RSGE) method Phi-four equation Soliton wave Travelling wave solution Calogero-Bogoyavlanskil Schilf equation Sine-Gordon |
| url | https://doi.org/10.1038/s41598-024-60156-w |
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