Analytical and numerical solutions to the nonlinear Klein-Gordon equation using improved Kudryashov and modified rational methods

This study investigates the nonlinear Klein–Gordon (𝕂𝔾) equation, a fundamental relativistic wave equation that governs the propagation of scalar fields across various physical contexts, including quantum field theory, solid-state physics, and nonlinear optics. The 𝕂𝔾 equation is closely related to other nonlinear evolution equations, such as the sine-Gordon and ϕ 4 equations. Using analytical techniques, specifically the improved Kudryashov (𝕀𝕂ud) and modified rational (𝕄ℝat) methods, we construct exact traveling wave solutions to the nonlinear 𝕂𝔾 equation. To validate the accuracy of the obtained solutions, we employ He’s variational iteration (ℍ𝕍𝕀) method as a numerical scheme, demonstrating excellent agreement between analytical and numerical results, thus ensuring the applicability of the derived solutions. The research reveals a rich variety of localized and periodic wave solutions, elucidating the intricate dynamics governed by the nonlinear 𝕂𝔾 equation. These solutions have significant implications for understanding various physical phenomena, including soliton propagation, nonlinear wave interactions, and scalar field behavior in diverse systems. The novelty of this work lies on the systematic application of the 𝕀𝕂ud and 𝕄ℝat methods to the nonlinear 𝕂𝔾 equation, providing new insights into its analytical treatment and corroborating analytical solutions with numerical results. This research contributes to the field of nonlinear mathematical physics, particularly, the study of nonlinear evolution equations and their applications in quantum field theory, condensed matter physics, and nonlinear optics.