Analyzing diverse soliton wave profiles and bifurcation analysis of the (3 + 1)-dimensional mKdV–ZK model via two analytical schemes

The (3 + 1)-dimensional modified Korteweg–deVries–Zakharov–Kuznetsov model is widely used in the study of nonlinear wave phenomena. These forms of wave phenomena are more useful in science and engineering. This work will analyze the model to identify the processes of the obtained traveling wave solutions through the modified version of the new Kudryashov and extended hyperbolic function schemes, as well as evaluate the solidity of the solitons at numerous equilibrium points using bifurcation analysis in conjunction with the Hamiltonian planar system. In addition, bifurcations are used to display the shifting framework and to test for the presence of different traveling wave solutions. Moreover, we show the balance point in the photographic form to examine the signal’s stability by specifying the saddle point and system center. Thus, the originality of this study is that the obtained traveling wave solutions of the mentioned governing model produce a variety of waves, including dark, kink, bell, and cospon bright soliton, depending on spacetime and wave propagation variables, which are illustrated in two-dimension, three-dimension, and contour charts. Furthermore, this research examines the wave’s nature using the governing model’s ion acoustic parameters and characterizes the outcome of these factors on the wave structure. As can be seen, the bifurcation exploration and the stated method are extremely valuable and instructive for describing the mathematical structure in later studies such as this one and many others.