Madani Yasir A, Mohamed Khidir Shaib, Yasin Sadia, Ramzan Sehrish, Aldwoah Khaled, Hassan Mohammed
Department of Mathematics, College of Science, University of Ha'il, Ha'il, 55473, Saudi Arabia.
Department of Mathematics, College of Science, Qassim University, Buraydah, Saudi Arabia.
Sci Rep. 2025 Jan 13;15(1):1834. doi: 10.1038/s41598-025-85461-w.
In this study, the -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 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 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.
在本研究中，-模型展开法被证明可用于寻找克莱因-戈登（KG）方程的孤立波解。我们得到了多种解，包括雅可比椭圆函数、双曲形式和三角形式，从而极大地扩展了可得到的精确解的范围。二维、三维和等高线图清晰地展示了不同类型的孤立波，如亮孤子、暗孤子、奇异孤子和周期孤子。这为我们提供了许多关于KG方程非线性行为的信息。我们的研究结果突出了该模型作为研究非线性波动方程的有力工具，有助于增进我们对其复杂动力学的理解，并拓展理论探索的范围。该模型展开技术具有极强的适应性，可用于众多非线性偏微分方程。尽管具有通用性，但该技术可能并不适用于所有非线性偏微分方程，特别是那些不符合特定要求或无法用该技术处理其结构的方程。在理论物理学中，尤其是在场论和量子力学中，克莱因-戈登方程是一个经典模型。通过研究这个模型，我们可以阐明相对论速度下的波和粒子运动。在其他领域中，其在宇宙学、量子场论和非线性光学研究中的重要性也得到广泛认可。此外，它提供的精确解和非线性动力学在应用数学和物理学中有各种应用。这项研究具有新颖性，因为它通过详细评估为控制模型的复杂行为和各种波形提供了新的理解。未来的研究可以集中在进一步探索这些解在不同条件下的稳定性和物理意义，从而推动我们对非线性波动现象及其在物理学及其他领域应用的认识。
