Bifurcations and Dynamics in a Generalized KdV-mKdV-like System

This paper presents a comprehensive bifurcation analysis of the generalized KdV–mKdV-like system, employing dynamical systems theory to investigate its nonlinear wave dynamics and asymptotic behaviors. Through meticulous analysis across diverse parameter regimes, we uncover intricate wave phenomena closely associated with system parameters. Our work reveals a critical phase transition under specific parametric conditions. These findings provide fundamental insights into nonlinear wave dynamics and system stability, with potential applications in wave control and system optimization across various physical contexts.