Dynamical analysis and soliton solutions to (3+1)-dimensional extended quantum nonlinear Zakharov–Kuznetsov equation

This study employs an improved modified extended tanh-function method to address the $$(3+1)$$-dimensional extended quantum nonlinear Zakharov-Kuznetsov equation, which models nonlinear wave propagation in quantum plasmas and magnetized fluid systems. The proposed method uses a generalized quartic-root auxiliary equation to derive kink, anti-kink, bright/dark solitons, Jacobi elliptic, Weierstrass doubly periodic, and complex hybrid solutions within a unified framework. To visually analyze the characteristics of obtained solutions, 3D, 2D and contour plots are plotted. Beyond exact analytical solutions, a dynamical system is obtained by using Galilean transformation, along with detailed bifurcation diagrams. To analyze the chaotic behavior and sensitivity of the system on initial conditions, comparative time-series evolution and 2D, 3D phase-space trajectories are illustrated. This work not only adds to the collection of analytical tools developed for high-dimensional nonlinear evolution equations but also forms a bridge between symbolic solution construction and theory of nonlinear dynamical systems.