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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. [ABSTRACT FROM AUTHOR]

In this paper, we use two efficient mathematical approaches to obtain a variety of soliton solutions to the (3+1)-dimensional Schrödinger equation: the generalized Riccati equation mapping method and the newly proposed modified generalized exponential rational function method. These techniques extracted standard, illustrative, rich dynamical structures, and further comprehensive soliton solutions and traveling wave solutions involving hyperbolic form, trigonometric form, and exponential form. The obtained results have been verified by placing them back into the mentioned nonlinear partial differential equation via symbolic computation in Mathematica. Thereafter, the graphical demonstrations of some attained solutions are discussed for a better understanding of the physical phenomenon. We have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the generalized Riccati equation mapping method and modified generalized exponential rational function techniques for extracting soliton solutions to nonlinear partial differential equations are efficient, compatible, and reliable in nonlinear sciences, optical fibers, and engineering. [ABSTRACT FROM AUTHOR]