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

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.