Temporal wave dynamics, phase portrait and qualitative analysis of the time-dependent (2+1)-dimensional Zakharov-Kuznetsov equation

Abstract In this article, we analyze the effect of time-dependent coefficients and the complex wave dynamics of the (2+1)-dimensional Zakharov-Kuznetsov (ZK) equation. This equation provides a detailed, insightful, and realistic description of space physics, plasma physics, controlled fusion, and nonlinear sciences. The wave solutions are established using the generalized Kudryashov, modified simple equation, and modified sine-Gordon expansion techniques and are illustrated by graphical depictions, which provide valuable insight into understanding the complex dynamics of waves across different physical systems. Exact solitary wave solutions offer a dependable approach to investigating the behavior of a system under particular conditions and facilitating a comprehensive understanding of its dynamics. We also conduct a stability analysis and present the phase portrait of the solutions, which are useful in various fields, including physics, plasma physics, chemistry, biology, economics, and sociology. We ascertain that the profiles of 3D and 2D soliton-shaped waves are significantly affected by dynamic changes in coefficients, wave velocity, and associated model parameters. This research could help clarify the dynamics of intricate systems, paving to a better understanding and analysis of the temporal aspects of various phenomena.