Investigation of Analytical Soliton Solutions to the Non-Linear Klein–Gordon Model Using Efficient Techniques
Abstract
:1. Introduction
2. Description of Exact Approaches
2.1. Mapping Method
2.2. Generalized Riccati Equation Mapping Method
3. Implementation and Applications of the Exact Techniques
3.1. Applications of Mapping Method
- If , and , then the exact solution is
- If , and , then the exact solution is
- If , and , then the exact solution is
- If , and , then the exact solution is
- If , and , then the exact solution is
- If , and , then the exact solution isAs , then the above equation take the fromAs , then the above equation also takes the form
- If , and , then the exact solution is
- If , and , then the exact solution is
- If , and , then the exact solution is
- If , and , then the exact solution is
- If , and , then the exact solution is
- If , and , then the exact solution isAs , then the above Equation take the formAs , then the above Equation also take the form
3.2. Applications of Generalized Riccati Equation Mapping Method
4. Results and Discussion
4.1. Physical Descriptions via Mapping Method
4.2. Physical Descriptions via Generalized Riccati Equation Mapping Method
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Vivas-Cortez, M.; Nageen, M.; Abbas, M.; Alosaimi, M. Investigation of Analytical Soliton Solutions to the Non-Linear Klein–Gordon Model Using Efficient Techniques. Symmetry 2024, 16, 1085. https://doi.org/10.3390/sym16081085
Vivas-Cortez M, Nageen M, Abbas M, Alosaimi M. Investigation of Analytical Soliton Solutions to the Non-Linear Klein–Gordon Model Using Efficient Techniques. Symmetry. 2024; 16(8):1085. https://doi.org/10.3390/sym16081085
Chicago/Turabian StyleVivas-Cortez, Miguel, Maham Nageen, Muhammad Abbas, and Moataz Alosaimi. 2024. "Investigation of Analytical Soliton Solutions to the Non-Linear Klein–Gordon Model Using Efficient Techniques" Symmetry 16, no. 8: 1085. https://doi.org/10.3390/sym16081085