Bifurcation analysis, phase portrait, and exploring exact traveling wave propagation of M-fractional (3 + 1) dimensional nonlinear equation in the fluid medium

This work studies the bifurcation analysis, phase portrait, and dynamics behavior of exact traveling wave solutions for the time M-fractional (3 + 1)-dimensional Painlevé integrable model. This model is used to describe some complex phenomena in nonlinear science and is crucial for addressing various practical challenges, including key models in areas like quantum mechanics, statistical physics, nonlinear optics, and celestial Mechanics. By bifurcation theory, we find the phase portrait of the proposed model. Bifurcation analysis in the proposed systems helps figure out how small changes in parameters can cause big changes in how the system behaves, like going from stable states to chaotic dynamics. It pinpoints critical thresholds where these transitions occur, aiding in the prediction and control of complex behaviors. We also present some traveling wave solutions according to the phase portrait orbit. Furthermore, the exact traveling wave solutions of the time M-fractional (3 + 1)-dimensional Painlevé integrable model are investigated by using the modified simple equation technique. This method provides the solutions directly without any predefined solutions. Under the condition, the solutions are real and complex valued in terms of exponential, trigonometric, and hyperbolic function form. For the special ideals of free parameters, we will include the bright bell wave, dark bell wave, interaction of kink and periodic lump wave, double periodic wave, double dark bell wave, and so on. By visualizing, the three, two-dimensional and density diagrams, the dynamic properties of the proposed model are examined and illustrated, enhancing the comprehension and application of time M-fractional (3 + 1)-dimensional Painlevé integrable model.