Qualitative Dynamics Wave Phenomena Arising in the (2+1)‐Dimensional Chaffee–Infante and Zakharov Equations: Travelling Wave Solutions, Bifurcations, and Chaos

ABSTRACT This paper explores the qualitative dynamics wave phenomena that arise in the ‐dimensional Chaffee–Infante equation and the Zakharov equation; both models have key physical applications in nonlinear dynamical systems. The Chaffee–Infante equation is extensively utilized to describe gas diffusion and reaction activity in diverse physical media. At the same time, the Zakharov equation governs ion‐acoustic waves in plasma fluid dynamics, with applications in plasma physics, signal processing, and electromagnetic wave theory. By utilizing the modified ‐expansion method, we attained a range of soliton solutions for both equations, including singular periodic, kink, anti‐kink, and dark solitons. The physical connection of these solutions is examined through 2D and 3D visualizations, representing the wave dynamics and the transitions between different wave structures. All symbolic computations and visualizations were carried out by utilizing Wolfram Mathematica 11. Furthermore, bifurcation and chaos phenomena are studied by changing key system parameters, revealing complex dynamical behaviors. The attained results show that the modified ‐expansion method yields superior accuracy and reliability compared to previous methods for solving nonlinear partial differential equations (NLPDEs). This study emphasizes the potential of this approach in advancing the understanding of nonlinear wave phenomena, with remarkable applications in areas such as gas diffusion modeling, plasma wave dynamics, and electromagnetic wave propagation.