Comprehensive symbolic–neural and dynamical framework for the (3+1)-dimensional extended Kadomtsev–Petviashvili equation

This study investigates the (3+1)-dimensional extended Kadomtsev–Petviashvili equation using a symbolic–neural hybrid framework. First, the equation is transformed into its bilinear form, and the bilinear neural network method is employed to construct various analytical wave solutions, including bright, dark, periodic, and interaction waves. These solutions reveal the rich nonlinear behavior of the model and are illustrated through representative graphical visualizations. To further examine the dynamics of the obtained solutions, the governing equation is reduced to a nonlinear ordinary differential system via a wave transformation. The reduced system is analyzed using phase portraits, sensitivity analysis, and Lyapunov exponents to characterize its dynamical behavior. The results demonstrate transitions between stable, periodic, and chaotic states under different initial conditions and parameter perturbations. The proposed framework provides an effective approach for constructing analytical solutions and investigating their dynamical properties, offering new insights into higher-dimensional nonlinear wave phenomena.