Exploring soliton solutions of coupled dispersionless equations with new insights into bifurcation, chaos, and sensitivity through advanced analytical techniques

In this study, we explore the dynamics of coupled dispersionless equations using the Galilean transformation and planar dynamical systems theory. These nonlinear equations are pivotal in various physics and engineering domains, such as optical fibers and ferromagnetic materials. We analyze bifurcation and chaotic behavior, finding that slight variations in initial conditions minimally affect solution sensitivity, as confirmed by the Runge–Kutta method. Using the improved modified Sardar sub-equation and (\(\frac{\mathrm {G'}}{\mathrm{G}}, \frac{1}{\mathrm{G}}\))-expansion methods, we derive exact solutions, including bright, kink, anti-kink, and dark solitons. These results demonstrate the effectiveness of the proposed methods for solving nonlinear partial differential equations.