In-depth analysis and exploration of rogue wave, lump wave, and different solitonic patterns to the Painlevé-integrable (3+1)D nonlinear evolution equations using a new extended methodology

This paper proposes a novel extended methodology, establishing the new extended Generalized Exponential Rational Function Method (GERFM), which is known as a “new extended GERFM” for solving the Painleve-integrable [Formula: see text]D generalized nonlinear evolution equation. We enhance GERFM by incorporating additional terms that combine functions and derivatives, making it a more generalized and efficient approach for solving nonlinear partial differential equations. Moreover, the unified method is applied to thoroughly delve deeper into the equation’s properties to understand its mathematical characteristics and potential solutions. This approach aims to systematically investigate and elucidate the equation’s inherent features, providing a more complete insight into its behavior and the nature of its solutions. Utilizing these methods, we have derived new exact solutions that involve exponential, trigonometric, hyperbolic, polynomial, and rational functions with additional parameters to the model. To emphasize the physical importance of these solutions, we present three-dimensional (3D) and contour plots, investigating different parameter choices. The graphs illustrate various wave patterns such as irregular periodic solitons, rogue waves, lumps, and the interaction of solitary and lump waves based on different parameter values. This visualization method enriches our comprehension of the solutions and facilitates a thorough exploration of how they could be applied to the generation of prolonged waves with small magnitudes in plasma physics, fluid dynamics, and other media with weak dispersion.