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This paper investigates the soliton wave on a free-moving fluid surface by studying the two-fluid model, which are the fourth-order Boussinesq and the modified Liouville equation. This study depends on one of the computational schemes to find exact and soliton wave solutions of these models. These solutions give novel physical properties of these waves, which enable their use in many fluid applications. In order to achieve our goal, the $exp(\pm \mathrm{\Phi }(\xi ))$-expansion method is applied to these two models, and for more explanation of the physical properties of these models, some of the obtained solutions are sketched in different forms (two, three-dimensional and contour plots). Moreover, the obtained results are discussed for its novelty and how it changed from that achieved in the previous work.

This paper investigates novel solitary wave solutions of the unidirectional Dullin–Gottwald–Holm model and employs the modified Khater (MKhat) method for studying the dynamical characterization of the prorogation of waves in shallow water. There are various solution types obtained such as kink, periodic, cone, anti-kink, etc. The accuracy of these solutions is checked by implementing He’s variational iteration technique. The analytical and numerical solutions are numerically simulated through 3D, 2D and contour plots for a visual explanation of the shallow water waves’ propagation and the match between both kinds of solutions. Additionally, the interaction between solutions is explained by some stream plots to show the local direction of the vector field at each point and a roughly uniform density throughout the property, which indicates no background scalar field. The novelty of the study’s solutions is explained by comparing it with the previously published articles.