Unraveling the complexity of solitary waves in the Klein-Fock-Gordon equation: dynamical insights into bifurcation and chaos analysis

Abstract

Soliton resonances and soliton interaction have gained significant attention in recent years within the field of nonlinear science and engineering due to their promising potential for various applications. This research provides a comprehensive analysis of the soliton interaction dynamics described by the nth-order Klein-Fock-Gordon equation. In this study, we used the improved modified Sardar subequation and modified Khater method to find the soliton solution to the nonlinear third-order Klein-Fock-Gordon equation. We construct dark, bright, kink, and periodic optical solitons using the modified Sardar subequation and Khater methods. We employ the suitable traveling wave transformation to convert the model equation into an ordinary differential equation. To analyze the physical behavior of the model, we graphically plotted some solutions, selecting appropriate parameter values in two-dimensional, three-dimensional, and contour plots. We examine the phase portrait of the equilibrium point to convert the equation into a planar dynamical system using the Galilean transformation. We conduct a sensitivity analysis to determine how sensitive our system is to the initial condition. When we apply an additional external force to the system, we also examine the chaotic analysis, observing how the dynamical system responds to various forces and comparing the patterns of periodic, quasi-periodic, and chaotic behavior. In this study, we performed the calculations using the Mathematica and Maple software programs.