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This study presents two novel analytical methodologies for the derivation of solitary wave solutions within the framework of the Benney-Luke ( B L ) model. The B L equation, characterized as a nonlinear evolution equation, plays a pivotal role in the description of long-wavelength, weakly nonlinear, and dispersive waves in shallow water. It holds particular importance in comprehending the behaviors of water waves in shallow channels and across the surfaces of shallow water bodies. By encompassing nonlinearity, dispersion, and dissipation, this equation encapsulates fundamental factors that significantly impact water wave dynamics, providing invaluable insights into attributes such as wave amplitude, frequency, and propagation speed. The soliton wave solutions obtained are rigorously validated using a contemporary numerical scheme, demonstrating a remarkable congruence with the analytical results. Further analysis of these solutions is carried out through comprehensive graphical representations, effectively highlighting the distinctive and innovative characteristics inherent in the B L model. This investigation makes a substantial contribution to the understanding of the intricate dynamics within the B L equation, thereby expanding the repertoire of available analytical and numerical techniques for the study of similar nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This study introduces recent computational advancements targeting the discovery of precise and novel solitary wave solutions in the (1 + 1)-dimensional Mikhailov-Novikov-Wang (MNW ) integrable equation. Solitary wave solutions are of paramount importance for comprehending complex nonlinear phenomena and have practical implications in diverse scientific fields such as fluid dynamics, nonlinear optics, and plasma physics. The proposed computational techniques encompass powerful numerical methods, including the Khater II (KHA - II) method, modified Rational method (MRE ) method, and He’s variational iteration (HVI ) method. These techniques offer robust and accurate approaches for computing solitary wave solutions by leveraging the inherent integrability properties of the MNW equation. Notably, these approaches demonstrate substantial improvements in efficiency, accuracy, and the ability to explore a broader range of solitary wave solutions compared to traditional computational methods. The efficacy of these techniques is effectively demonstrated through extensive numerical simulations and comprehensive comparisons with other computational approaches. The results highlight the superior performance of the proposed methods in identifying new solitary wave solutions, precisely characterizing their properties, and providing deeper insights into the model’s behavior. Overall, these computational techniques significantly contribute to advancing our understanding of the underlying dynamics of the MNW equation and hold promising potential for applications across multiple scientific disciplines. [ABSTRACT FROM AUTHOR]