Unveiling novel insights: Nonlinear dispersion dynamics of optical solitons through new φ6-Model expansion in the Klein–Gordon equation

This study employs a nonlinear differential equation to model diverse phenomena, encompassing dislocation movement in crystals, characteristics of elementary particles, and the propagation of fluxions in Josephson junctions, with the Klein–Gordon equation serving as an illustrative example. The ${\phi }^6$-model expansion method, a multitude of solution types are explicitly obtained, which encompass kink-type solitons, recognized as topological solitons within the realm of water waves. Notably, these solitons exhibit velocities independent of wave amplitude, alongside other variations like dark, singular, periodic, and combined singular soliton solutions. The research outcomes hold the potential to enhance the nonlinear dynamical characteristics of the Klein–Gordon equation. The suggested ${\phi }^6$-model expansion technique provides a pragmatic and efficient strategy for addressing a wide range of nonlinear partial differential equations. The findings are visually presented through insightful graphs that elucidate the dynamic aspects of the results, demonstrating the accuracy of the obtained solutions when applied to the Klein–Gordon equation. The physical properties of surface waves are comprehensively analyzed, with a particular focus on Rayleigh waves. By modeling the regular oscillations, energy transfer and complex behavior of Rayleigh waves under nonlinear effects, this study provides theoretical support that these waves can exhibit solitary wave properties under certain boundary conditions.