Bifurcation, Chaos, Sensitivity Analysis, and Dynamics of the (3+1)-Dimensional Nonlinear Extended Zakharov-Kuznetsov Equation and its Applications

The (3+1)-dimensional nonlinear extended Zakharov-Kuznetsov (3-dEZK) equation is a fundamental model in plasma physics and nonlinear wave dynamics, playing a key role in describing shallow water waves, ion-acoustic waves in magnetized plasma, and related physical systems. This dynamic model is also a powerful tool in ocean engineering for modeling complex wave interactions, predicting hazardous wave events, and enhancing coastal and offshore infrastructure design. In this work, we employ two analytical techniques, namely the new modified simple equation (nMSE) method and the generalized Arnous (GA) method, to construct soliton and other wave solutions, including bright and dark interactional waves and multi-peak solitons, for the 3-dEZK equation. These results are significant for the physical interpretation of this dynamic model and have applications in fluid dynamics, optical waveguides, plasma physics, etc. We also use bifurcation analysis to reveal state transitions under different parameter settings, identifying critical thresholds where wave propagation changes qualitatively. Furthermore, sensitivity and chaotic analyses are performed to examine the impact of perturbation terms. The novelty lies in deriving new soliton solutions of the 3-dEZK equation via nMSE and GA methods, along with extended bifurcation, sensitivity, and chaos analyses. The graphical analytical solutions highlight the approach’s effectiveness and practical relevance in applied sciences and engineering.