Influence of Fractional and Integer Derivatives on the (4+1)-Dimensional Fractional Davey–Stewartson–Kadomtsev–Petviashvili Equation

Fractional-order models are capable of capturing memory, nonlocality, and anomalous dispersion, making them powerful tools for describing nonlinear wave propagation in complex media. This work investigates the influence of fractional and integer-order derivatives on the multidimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili (FDSKP) equation, a higher-dimensional model widely used to characterize internal waves, optical pulses, and fluid–structure interactions. Here, the term fractional parameters refers explicitly to the orders of the fractional time derivatives appearing in the governing equations no spatial fractional derivatives are considered. Physically, fractional time derivatives introduce temporal nonlocality through memory effects, implying that the system evolution depends on its entire past history and can capture long-memory dynamics beyond the capability of classical integer-order models. Fractional beta, M-truncated, and classical integer-order derivatives are incorporated, and the resulting modifications in wave structures under different operator definitions are systematically analyzed. The FDSKP equation is reduced to an ordinary differential equation via appropriate traveling-wave transformations corresponding to each derivative type, and exact analytical solutions are obtained using the Modified Auxiliary Equation (MAE) and Jacobi Elliptic Function (JEF) methods. The proposed framework generates a rich family of wave structures, including bell-shaped, W-shaped, composite dark–bright, and periodic waves. The results demonstrate that the fractional parameters effectively regulate amplitude, steepness, and overall wave dynamics with greater flexibility than classical integer-order derivatives. Furthermore, two- and three-dimensional visualizations reveal the dependence of dispersion characteristics and symmetry on the fractional order. Overall, the findings confirm that the choice of fractional operator significantly influences the physical interpretation of the model, highlighting the importance of fractional calculus for accurately representing complex wave dynamics in higher-dimensional nonlinear systems.