Novel geometric insights into modulation instability and soliton solutions of the truncated M-fractional $(3+1)$-D generalized Painlevé equation

This research examines soliton solutions of the truncated M-fractional $(3+1)$ -dimensional generalized Painlevé-type equation using the modified exp-function method and the $\exp(-\Phi (\eta ))$ -expansion method for the first time. The $(3+1)$ -dimensional generalized Painlevé-type equation is now frequently utilized in nonlinear wave theory, soliton theory, and plasma physics to investigate the evolution of plasma waves and instabilities. The derived soliton solutions, painstakingly generated utilizing Mathematica, cover an extensive spectrum of mathematical functions including rational hyperbolic, rational trigonometric, rational, and exponential function solutions. These solutions are illustrated by 3D, contour, 2D, and density plots and diverse soliton structures discovered, such as kink, singular kink, dark, bright, combo dark-bright, singular combo dark-bright, antipeakon, peakon bright, peakon kink, compacton, and mixed dark-bright-kink soliton solutions. Our study includes a number of exact solutions with unique wave patterns, such as antipeakon, peakon bright, peakon kink, mixed dark-bright-kink, and compacton. Our comparison demonstrates the originality of the derived solutions, which are the first examples of their emergence for the considered model. The diverse findings showcased that the proposed methods are significant mathematical tools for handling fractional nonlinear partial differential equations, with implications in engineering, plasma physics, ocean physics, and various nonlinear evolution problems in mathematical physics. Additionally, a discussion about the modulation instability of the proposed model is presented. This thorough analysis shows the model’s intricate behavior and its capacity to explain a wide range of nonlinear wave phenomena in many physical contexts.