Modulation instability, bifurcation analysis, and ion-acoustic wave solutions of generalized perturbed KdV equation with M-fractional derivative

The perturbed Korteweg-de Vries (PKdV) equation is essential for describing ion-acoustic waves in plasma physics, accounting for higher-order effects such as electron temperature variations and magnetic field influences, which impact their propagation and stability. This work looks at the generalized PKdV (gPKdV) equation with an M-fractional operator. It uses bifurcation theory to look at critical points and phase portraits, showing system changes such as shifts in stability and the start of chaos. Figures 1, 2 and 3 provide detailed analyses of static soliton formation through saddle-node bifurcation. We also use the modified simple equation (MSE) method to look for ion-acoustic wave solutions directly, without having to first define them. This lets us find shapes like hyperbolic, exponential, and trigonometric waves. These solutions reveal complex phenomena, including double periodic waves, periodic lump waves, bright bell-shaped waves, and singular soliton waves. Additionally, we analyze modulation instability in the gPKdV equation, which signifies chaotic transitions and is crucial for understanding nonlinear wave dynamics. Those methods demonstrate their value in generating precise soliton solutions relevant to nonlinear science and mathematical physics. This research illustrates how theoretical mathematics and physics can support solutions to practical world issues, especially in energy and technological advancement.