Dynamical Control and Topological Bifurcations of Propagating Wave Patterns via Parametric Sensitivity Analysis in Nonlinear Evolution Equations

In this research, we present a unified analytical framework for the dynamical control and topological classification of propagating wave patterns in two nonlinear evolution equations, namely the nano-ionic current (NIC) equation and the classical Sawada-Kotera (CSK) equation. We reduce two archetypal models-governing nanoscale ionic transport and dispersive wave phenomena-to planar dynamical systems by employing -expansion method. We conduct the rigorous bifurcation analysis via the discriminant of the auxiliary equation with revealing three two nondegenerate NIC configurations in which the saddle and center exchange locations between D > 0 and D < 0, together with a degenerate transition at D = 0. But the case the yields a degenerate rational solution, which is omitted as it does not contribute to the physically relevant wave structures. Moreover, we give a comprehensive parametric sensitivity analysis to quantify how small variations in the system parameters dynamically control the amplitude, width, and phase velocity of the resulting wave packets. We plot our obtained solutions in 3D, 2D and contour, which encompass a diverse spectrum of waveforms within their regular domains, including localized bell-shaped, kink-type, and non-compact wave profiles. The obtaining results establish that phase-plane bifurcations serve as intrinsic control knobs for tuning wave topology, offering deeper insights into wave propagation stability. For sensitivity-driven control in complex nonlinear media, this framework is not limited to specific physical models but provides a robust mathematical toolkit.