A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions
In this paper, the Hirota bilinear method, which is an important scheme, is used. The equation of the shallow water wave in oceanography and atmospheric science is extended to (3+1) dimensions, which is a well-known equation. A lot of classes of rational solutions by selecting the interaction betwee...
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2022-08-01
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| Series: | Mathematics |
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| author | Ruijuan Li Onur Alp İlhan Jalil Manafian Khaled H. Mahmoud Mostafa Abotaleb Ammar Kadi |
| author_facet | Ruijuan Li Onur Alp İlhan Jalil Manafian Khaled H. Mahmoud Mostafa Abotaleb Ammar Kadi |
| author_sort | Ruijuan Li |
| collection | DOAJ |
| description | In this paper, the Hirota bilinear method, which is an important scheme, is used. The equation of the shallow water wave in oceanography and atmospheric science is extended to (3+1) dimensions, which is a well-known equation. A lot of classes of rational solutions by selecting the interaction between a lump and one- or two-soliton solutions are obtained. The bilinear form is considered in terms of Hirota derivatives. Accordingly, the logarithm algorithm to obtain the exact solutions of a (3+1)-dimensional variable-coefficient (VC) generalized shallow water wave equation is utilized. The analytical treatment of extended homoclinic breather wave solutions is studied and plotted in three forms 3D, 2D, and density plots. Using suitable mathematical assumptions, the established solutions are included in view of a combination of two periodic and two solitons in terms of two trigonometric and two hyperbolic functions for the governing equation. Maple software for computing the complicated calculations of nonlinear algebra equations is used. The effect of the free parameters on the behavior of acquired figures to a few obtained solutions for two nonlinear rational exact cases was also discussed. |
| first_indexed | 2024-03-10T01:33:48Z |
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| id | doaj.art-aa4fa33b46c04d6488462f18d97fab03 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T01:33:48Z |
| publishDate | 2022-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-aa4fa33b46c04d6488462f18d97fab032023-11-23T13:37:56ZengMDPI AGMathematics2227-73902022-08-011017307410.3390/math10173074A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton SolutionsRuijuan Li0Onur Alp İlhan1Jalil Manafian2Khaled H. Mahmoud3Mostafa Abotaleb4Ammar Kadi5School of Mathematics and Statistics, Pingdingshan University, Pingdingshan 467000, ChinaDepartment of Mathematics, Faculty of Education, Erciyes University, Melikgazi, Kayseri 38039, TurkeyNatural Sciences Faculty, Lankaran State University, 50, H. Aslanov Str., Lankaran AZ4200, AzerbaijanDepartment of Physics, College of Khurma University College, Taif University, P.O. Box 11099, Taif 21944, Saudi ArabiaDepartment of System Programming, South Ural State University, 454080 Chelyabinsk, RussiaDepartment of Food and Biotechnology, South Ural State University, 454080 Chelyabinsk, RussiaIn this paper, the Hirota bilinear method, which is an important scheme, is used. The equation of the shallow water wave in oceanography and atmospheric science is extended to (3+1) dimensions, which is a well-known equation. A lot of classes of rational solutions by selecting the interaction between a lump and one- or two-soliton solutions are obtained. The bilinear form is considered in terms of Hirota derivatives. Accordingly, the logarithm algorithm to obtain the exact solutions of a (3+1)-dimensional variable-coefficient (VC) generalized shallow water wave equation is utilized. The analytical treatment of extended homoclinic breather wave solutions is studied and plotted in three forms 3D, 2D, and density plots. Using suitable mathematical assumptions, the established solutions are included in view of a combination of two periodic and two solitons in terms of two trigonometric and two hyperbolic functions for the governing equation. Maple software for computing the complicated calculations of nonlinear algebra equations is used. The effect of the free parameters on the behavior of acquired figures to a few obtained solutions for two nonlinear rational exact cases was also discussed.https://www.mdpi.com/2227-7390/10/17/3074hirota bilinear techniqueinteraction between a lump and one-, two soliton solutionsgeneralized shallow water wave equation with variable coefficients |
| spellingShingle | Ruijuan Li Onur Alp İlhan Jalil Manafian Khaled H. Mahmoud Mostafa Abotaleb Ammar Kadi A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions Mathematics hirota bilinear technique interaction between a lump and one-, two soliton solutions generalized shallow water wave equation with variable coefficients |
| title | A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions |
| title_full | A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions |
| title_fullStr | A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions |
| title_full_unstemmed | A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions |
| title_short | A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions |
| title_sort | mathematical study of the 3 1 d variable coefficients generalized shallow water wave equation with its application in the interaction between the lump and soliton solutions |
| topic | hirota bilinear technique interaction between a lump and one-, two soliton solutions generalized shallow water wave equation with variable coefficients |
| url | https://www.mdpi.com/2227-7390/10/17/3074 |
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