<p>In this article, the Klein-Gordon equation, a model for nonlinear wave propagation in high-energy nuclear settings, is studied in terms of bifurcation, chaotic dynamics, and solitary wave solutions. The study use the planar dynamical system technique to investigate self-interactions within the wave field, with a focus on phenomena including radiation transport, plasma oscillations, and neutron wave propagation in fission and fusion. Solitons, wave steepening, and energy localization all crucial elements in nuclear explosions and plasma instabilities—are produced by these interactions. In quantum mechanics and quantum field theory, spin-0 particles are described by the Klein-Gordon equation, a basic relativistic wave equation that serves as the foundation for scalar field theory and the quantization of such fields. It is specifically significant in general relativity, where it is extended to curved spacetime to analyze scalar fields under gravitational influences. It is also commonly used in modeling free particles and studying wave propagation in many physical systems. A key mathematical model for investigating the characteristics and solutions of partial differential equations in both physics and applied mathematics, the equation also has traditional uses in characterizing vibrating systems and nonlinear wave phenomena. The analysis employs a range of advanced dynamical tools, including phase diagrams, Lyapunov exponents, Poincaré maps, time series, bifurcation diagrams, fractal dimensions, strange attractors, recurrence plots, and return maps, revealing both chaotic and quasi-periodic behaviors. A perturbation term is introduced to further enrich the system’s dynamics, yielding a variety of complex patterns. Soliton solutions such as dark, bright, kink, anti-kink, periodic, and singular solitons are derived using the novel Improved Auxiliary Problem Mapping technique. Enhanced sensitivity analysis, supported by 3D, 2D, stream, density, contour, and phase trajectory visualizations, demonstrates the model’s dependence on initial conditions and showcases its capability in modeling more intricate phenomena. A stability analysis of the solitary wave solutions using the Hamiltonian technique confirms the consistency of the results, which are systematically organized for clarity. This study not only advances our understanding of nonlinear wave dynamics and soliton behavior but also highlights the model’s applicability to high-energy nuclear physics, providing valuable insights into the role of shock waves and turbulence in energy dissipation and structural changes in complex media.</p>