Analyzing multi-peak and lump solutions of the variable-coefficient Boiti-Leon-Manna-Pempinelli equation: a comparative study of the Lie classical method and unified method with applications

This research article investigates the (2+1)-dimensional variable-coefficient Boiti-Leon-Manna-Pempinelli equation using the Lie classical method and the unified method. The Lie classical method is employed to deduce the Lie symmetry generators and associated symmetric vectors, shedding light on the symmetries and invariance properties of the equation. Through this method, we deepen our understanding of the (2+1)-dimensional variable-coefficient Boiti-Leon-Manna-Pempinelli equation. Additionally, the unified method is utilized to further explore the equation's properties, aiming to develop a comprehensive understanding of its mathematical properties and solutions. To enhance comprehension, graphical representations such as 3-dimensional plots, 2-dimensional plots, and contour plots are presented using the symbolic computation software Mathematica. Analysis of the graphics reveals various solution profiles, including single-peak, doubly-peaks, multi-peaks, sinusoidal waves, breather solitons, lump solitons, interactions of kinks and solitons, solitary waves, paraboloids, and more. Moreover, this research aims to bridge the gap between mathematical visualization and real-world applications. By advancing knowledge of the (2+1)-dimensional variable-coefficient Boiti-Leon-Manna-Pempinelli equation and its mathematical characteristics, this study contributes to a broader understanding of nonlinear equations and their practical implications. Furthermore, Lumps and multi-peaks also arise in a variety of mathematical and physical fields, including nonlinear dynamics, oceanography, water engineering, optical fibres, and other nonlinear sciences.