Exploring the Lie symmetries, conservation laws, bifurcation analysis and dynamical waveform patterns of diverse exact solution to the Klein–Gordan equation

IF 3.3 3区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC
Tariq Mahmood, Ghadah Alhawael, Sonia Akram, Mati ur Rahman
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Abstract

This paper presents a comprehensive analysis of the \((1+1)\)-dimensional Klein-Gordan equation which plays a significant role in various areas of theoretical and applied physics. The main focus of this research centers on several key areas. First, infinitesimal generators of symmetries were found by Lie symmetry invariance analysis. Then, using the adjoint representation, an ideal system was created based on the found Lie vectors. Secondly, by utilizing the analytical approach, namely the modified Sardar sub equation method, we systematically derive various novel soliton solution in the form of dark, bright, periodic, singular, combo, hyperbolic as well as mixed trigonometric. Finally, bifurcation analysis is performed at the system’s fixed points, revealing chaotic behavior when an external force is introduced into the dynamic system. To identify the chaotic characteristics, a range of tools, such as 3D and 2D phase plots, time series, Lyapunov exponents, and multistability analysis, are utilized. Additionally, the sensitivity analysis of the model is examined under different initial conditions. These results enhance the understanding of nonlinear wave phenomena in mathematical physics and hold potential applications across numerous scientific disciplines.

探索克莱因-戈尔丹方程多种精确解的李对称性、守恒定律、分岔分析和动态波形模式
本文全面分析了((1+1)\)维克莱因-哥尔登方程,该方程在理论物理和应用物理的各个领域发挥着重要作用。这项研究的重点集中在几个关键领域。首先,通过李对称不变性分析找到了对称性的无穷小发生器。然后,利用邻接表示法,根据所发现的 Lie 向量创建了一个理想系统。其次,利用分析方法,即改进的萨达尔子方程法,我们系统地推导出了暗解、亮解、周期解、奇异解、组合解、双曲解以及混合三角解等各种新颖的孤子解。最后,我们在系统的固定点进行了分岔分析,揭示了在动态系统中引入外力时的混沌行为。为了识别混沌特性,我们使用了一系列工具,如三维和二维相位图、时间序列、Lyapunov 指数和多稳定性分析。此外,还研究了模型在不同初始条件下的敏感性分析。这些成果加深了人们对数学物理中非线性波现象的理解,并在众多科学学科中具有潜在的应用价值。
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来源期刊
Optical and Quantum Electronics
Optical and Quantum Electronics 工程技术-工程:电子与电气
CiteScore
4.60
自引率
20.00%
发文量
810
审稿时长
3.8 months
期刊介绍: Optical and Quantum Electronics provides an international forum for the publication of original research papers, tutorial reviews and letters in such fields as optical physics, optical engineering and optoelectronics. Special issues are published on topics of current interest. Optical and Quantum Electronics is published monthly. It is concerned with the technology and physics of optical systems, components and devices, i.e., with topics such as: optical fibres; semiconductor lasers and LEDs; light detection and imaging devices; nanophotonics; photonic integration and optoelectronic integrated circuits; silicon photonics; displays; optical communications from devices to systems; materials for photonics (e.g. semiconductors, glasses, graphene); the physics and simulation of optical devices and systems; nanotechnologies in photonics (including engineered nano-structures such as photonic crystals, sub-wavelength photonic structures, metamaterials, and plasmonics); advanced quantum and optoelectronic applications (e.g. quantum computing, memory and communications, quantum sensing and quantum dots); photonic sensors and bio-sensors; Terahertz phenomena; non-linear optics and ultrafast phenomena; green photonics.
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