Wave dynamics for the new generalized (3+1)-D Painlevé-type nonlinear evolution equation using efficient techniques

In this paper, diverse wave solutions for the newly introduced (3+1)-dimensional Painlevé-type evolution equation were derived using the improved generalized Riccati equation and generalized Kudryashov methods. This equation is now widely used in soliton theory, nonlinear wave theory, and plasma physics to study instabilities and the evolution of plasma waves. Using these methods, combined with wave transformation and homogeneous balancing techniques, we obtained concise and general wave solutions for the Painlevé-type equation. These solutions included rational exponential, trigonometric, and hyperbolic function solutions. Some of the obtained solutions for the Painlevé-type equation were plotted in terms of 3D, 2D, and contour graphs to depict the various exciting wave patterns that can occur. As the value of the amplitude increased in the investigated solutions, we observed the evolution of dark and bright solutions into rogue waves in the forms of Kuztnetsov-Ma breather and Peregrine-like solitons. Other exciting wave patterns observed in this work included the evolution of kink and multiple wave solitons at different time levels. We believe that the solutions obtained in this paper were concise and more general than existing ones and will be of great use in the study of solitons, nonlinear waves, and plasma physics.