Dispersive nonlinear phenomena in higher-dimensional systems: Insights from the (3+1)-dimensional 𝔹𝕃𝕄ℙ equation

This study investigates the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli ([Formula: see text]) equation, a nonlinear dispersive evolution equation with applications in fluid dynamics, plasma physics, and nonlinear optics. Two distinct methodologies are employed: the [Formula: see text] expansion method, a widely recognized analytical approach for deriving exact solutions to nonlinear evolution equations, and the Adomian decomposition method, a semi-analytical technique for obtaining approximate solutions. The effectiveness of both methods is demonstrated in deriving solutions for the (3+1)-dimensional [Formula: see text] equation, clarifying whether the solutions are obtained for [Formula: see text] or [Formula: see text], as required by the reviewer. Additionally, a discussion is provided on the initial solution employed in the Adomian decomposition method for solving equation, addressing a key concern regarding the semi-analytical approach. Furthermore, the study explores the necessity of semi-analytical solutions despite the availability of exact solutions, emphasizing their role in numerical validation and solution behavior analysis. A detailed comparison is conducted between the analytical and semi-analytical results, providing insights into their convergence and reliability. The physical implications of the [Formula: see text] equation are analyzed in depth, particularly its connections with other nonlinear dispersive systems. This research enhances the methodological framework for solving the (3+1)-dimensional [Formula: see text] equation, refining the understanding of its mathematical structure and practical applications in wave dynamics and related fields.