Exploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion method
Abstract In this study, the $$\phi ^{6}$$ ϕ 6 -model expansion method is showed to be useful for finding solitary wave solutions to the Klein–Gordon (KG) equation. We develop a variety of solutions, including Jacobi elliptic functions, hyperbolic forms, and trigonometric forms, so greatly enhancing...
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Nature Portfolio
2025-01-01
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| author | Yasir A. Madani Khidir Shaib Mohamed Sadia Yasin Sehrish Ramzan Khaled Aldwoah Mohammed Hassan |
| author_facet | Yasir A. Madani Khidir Shaib Mohamed Sadia Yasin Sehrish Ramzan Khaled Aldwoah Mohammed Hassan |
| author_sort | Yasir A. Madani |
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| description | Abstract In this study, the $$\phi ^{6}$$ ϕ 6 -model expansion method is showed to be useful for finding solitary wave solutions to the Klein–Gordon (KG) equation. We develop a variety of solutions, including Jacobi elliptic functions, hyperbolic forms, and trigonometric forms, so greatly enhancing the range of exact solutions attainable. The 2D, 3D, and contour plots clearly show different types of solitary waves, like bright, dark, singular, and periodic solitons. This gives us a lot of information about how the KG equation doesn’t work in a straight line. Our findings highlight the $$\phi ^{6}$$ ϕ 6 model as a powerful tool to study nonlinear wave equations, improve our understanding of their complex dynamics, and increase the scope for theoretical exploration. The $$\phi ^{6}$$ ϕ 6 model expansion technique is exceptionally adaptable and may be utilised for a wide array of nonlinear partial differential equations. Despite its versatility, the technique may not be applicable to all nonlinear PDEs, especially those that do not meet the specified requirements or structures manageable by this technique. In theoretical physics, particularly in field theory and quantum mechanics, the Klein–Gordon equation is a classical model. By studying this model, we can illustrate the waves and particles movements at relativistic speeds. Among other areas, its significance in cosmology, quantum field theory, and the study of nonlinear optics are widely considered. Additionally, it provides exact solutions and nonlinear dynamics have various applications in applied mathematics and physics. The study is novel because it provides a new understanding of the complex behaviours and various waveforms of the controlling model by means of detailed evaluation. Future research could focus on further exploring the stability and physical implications of these solutions under different conditions, thereby advancing our knowledge of nonlinear wave phenomena and their applications in physics and beyond. |
| format | Article |
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| institution | Kabale University |
| issn | 2045-2322 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Nature Portfolio |
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| series | Scientific Reports |
| spelling | doaj-art-7537e7ea036e499e878219c07dac65a82025-01-19T12:17:29ZengNature PortfolioScientific Reports2045-23222025-01-0115112110.1038/s41598-025-85461-wExploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion methodYasir A. Madani0Khidir Shaib Mohamed1Sadia Yasin2Sehrish Ramzan3Khaled Aldwoah4Mohammed Hassan5Department of Mathematics, College of Science, University of Ha’ilDepartment of Mathematics, College of Science, Qassim UniversityDepartment of Mathematics, Government College UniversityDepartment of Mathematics, Government College UniversityDepartment of Mathematics, Faculty of Science, Islamic University of MadinahDepartment of Mathematics, Faculty of Science, University of TabukAbstract In this study, the $$\phi ^{6}$$ ϕ 6 -model expansion method is showed to be useful for finding solitary wave solutions to the Klein–Gordon (KG) equation. We develop a variety of solutions, including Jacobi elliptic functions, hyperbolic forms, and trigonometric forms, so greatly enhancing the range of exact solutions attainable. The 2D, 3D, and contour plots clearly show different types of solitary waves, like bright, dark, singular, and periodic solitons. This gives us a lot of information about how the KG equation doesn’t work in a straight line. Our findings highlight the $$\phi ^{6}$$ ϕ 6 model as a powerful tool to study nonlinear wave equations, improve our understanding of their complex dynamics, and increase the scope for theoretical exploration. The $$\phi ^{6}$$ ϕ 6 model expansion technique is exceptionally adaptable and may be utilised for a wide array of nonlinear partial differential equations. Despite its versatility, the technique may not be applicable to all nonlinear PDEs, especially those that do not meet the specified requirements or structures manageable by this technique. In theoretical physics, particularly in field theory and quantum mechanics, the Klein–Gordon equation is a classical model. By studying this model, we can illustrate the waves and particles movements at relativistic speeds. Among other areas, its significance in cosmology, quantum field theory, and the study of nonlinear optics are widely considered. Additionally, it provides exact solutions and nonlinear dynamics have various applications in applied mathematics and physics. The study is novel because it provides a new understanding of the complex behaviours and various waveforms of the controlling model by means of detailed evaluation. Future research could focus on further exploring the stability and physical implications of these solutions under different conditions, thereby advancing our knowledge of nonlinear wave phenomena and their applications in physics and beyond.https://doi.org/10.1038/s41598-025-85461-wJacobi elliptic function (JEF)Nonlinear equationsSingular solutionsPeriodic solutions |
| spellingShingle | Yasir A. Madani Khidir Shaib Mohamed Sadia Yasin Sehrish Ramzan Khaled Aldwoah Mohammed Hassan Exploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion method Scientific Reports Jacobi elliptic function (JEF) Nonlinear equations Singular solutions Periodic solutions |
| title | Exploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion method |
| title_full | Exploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion method |
| title_fullStr | Exploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion method |
| title_full_unstemmed | Exploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion method |
| title_short | Exploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion method |
| title_sort | exploring novel solitary wave phenomena in klein gordon equation using phi 6 ϕ 6 model expansion method |
| topic | Jacobi elliptic function (JEF) Nonlinear equations Singular solutions Periodic solutions |
| url | https://doi.org/10.1038/s41598-025-85461-w |
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