Symmetries, Bifurcations and Control for Traveling Waves of the Drinfeld‐Sokolov‐Wilson System

ABSTRACT In this paper, we employ the Lie symmetry method to derive the Lie infinitesimal generators for the Drinfeld‐Sokolov‐Wilson system. By symmetry reduction, the original partial differential equation (PDE) system is transformed into an ordinary differential equation (ODE) system. We analyze the equilibrium points and phase portraits of the resulting planar dynamical system. By using bifurcation theory, we construct various types of traveling wave solutions, including smooth periodic waves, bounded soliton waves, kink and anti‐kink waves, periodic blow‐up solutions, and blow‐up solitons. Furthermore, based on the differential geometry structure, we design a feedback continuous controller to globally stabilize the traveling waves at equilibrium points. Moreover, we perform numerical simulations to validate the analytical results.