Bifurcation, modulation instability analysis, observation of closed solutions and effect of dispersion coefficient on soliton in relativity and quantum mechanics

The present paper aims to explore bifurcation analysis, modulation instability, soliton solutions, and the interaction of solitons for the third-order dispersion Klein-Fock-Gordon (K-F-G) equation. This equation incorporates both special relativity and quantum mechanics, providing a framework for understanding the behavior of such particles in relativistic settings. Initially, the bifurcation theory is employed to look at the dynamic behavior of nonlinear models. In figures 01–04, we provide an analytical and graphical study of the observed mechanism of static soliton through a saddle-node bifurcation for the K-F-G equation. Secondly, the Klein-Fock-Gordon (K-F-G) equation studies the devoting soliton solution, the interaction of soliton solutions, the effect of the nonlinear dispersion coefficient, and also the modulation instability. In this framework, we apply the new form of modified Kudryashov’s (NMK) technique and the unified solver technique to acquire diverse types of soliton solutions from the Klein-Fock-Gordon (K-F-G) equation. For the special value of the constraints, the shock wave, periodic wave, interaction of kink and periodic lump wave are obtained by the NMK scheme, and single soliton and periodic wave solutions are obtained by the unified solver technique. Additionally, we present the modulation instability for the K-F-G equation. The computational difficulties and outcomes highlight the clarity, effectiveness, and simplicity of the approaches, suggesting that these schemes can be applied to a variety of dynamic and static nonlinear equations governing evolutionary phenomena in computational physics as well as to other real-world situations and a wide range of academic fields.