Emergence of Complex Wave Structures and Stability Analysis for (3 + 1)-Dimensional Generalized <i>B</i>-Type Kadomtsev-Petviashvili Equation with <i>M</i>-Fractional Derivative Using Advanced Technique

This study investigates the fractional (3 + 1)-dimensional Generalized B-type Kadomtsev-Petviashvili Equation (GBKPE) using the Modified Extended Mapping Method (MEMM). The model plays a fundamental role in describing nonlinear wave propagation in fluid dynamics and other complex media, particularly the evolution of three-dimensional surfaces, shallow water waves, and diverse physical phenomena. By incorporating the local M-fractional derivative, the equation captures non-local interactions and memory effects—features inaccessible to classical derivatives—making it ideal for modeling long-range disturbances and hereditary properties. The primary objective is to derive novel exact solutions exhibiting complex dynamics in higher dimensions. Through MEMM, we obtain a wide range of solutions, including dark and singular solitons, Jacobi elliptic functions, hyperbolic, exponential, and singular periodic waves. Notably, some solutions exhibit previously unreported characteristics, underscoring the method’s innovation. We analyze the impact of fractional parameters on wave profiles, supported by 2D, 3D, and contour plots to visualize their dynamic behavior. A linear stability analysis further confirms the robustness of key solutions under small perturbations, ensuring their physical relevance. The results demonstrate the efficacy of MEMM in solving fractional GBKPE, significantly expanding the known analytical solutions. This work not only advances the understanding of multidimensional nonlinear equations but also provides a foundation for future studies in wave dynamics, stability, and applications to real-world systems like plasma physics and nonlinear optics.