Bifurcation analysis, modulation instability and dynamical analysis of soliton solutions for generalized (3 + 1)-dimensional nonlinear wave equation with m-fractional operator.

This research investigates the bifurcation theory of a generalized (3 + 1)-dimensional P-type nonlinear wave equation with an M-fractional derivative (M-fGP-NWE) and develops its soliton solutions. The model is initially transformed into an ordinary differential equation form using a wave variable. By employing a Galilean transformation, a dynamical system of equations is obtained. The phase portrait and Hamiltonian function are examined under different parametric conditions, facilitating the identification of homoclinic and heteroclinic orbits. These orbits illustrate solitary, bell-shaped, periodic wave solutions for certain parameter values. The modified simple equation (MSE) method is employed to derive soliton solutions for the M-fractional generalized (3 + 1)-dimensional P-type nonlinear wave equation. The resultant solutions are articulated in hyperbolic, trigonometric, and exponential forms under parametric circumstances. Complex wave phenomena are further exemplified through detailed 3D, 2D, and density graphs for certain parameter values. Additionally, we also analyse the modulation Instability of the proposed model. The computational results and visual depictions validate the efficiency and dependability of the MSE method, highlighting its efficacy as a flexible solution for complex fractional differential equations.