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Nonlinear evolution equations are typically used to model and analyse the majority of natural processes. These equations are extremely important for comprehending many intricate situations involving wave propagation, including heat transfer, fluid dynamics, optical fibres, electrodynamics, physics, chemistry, biology, condensed matter physics, ocean engineering, and many other nonlinear science fields. In this study, the anticipation of a bilinear Kdv model known as the (1+1)-dimensional integro-differential Ito equation is considered which depicts the oceanic shallow water wave dynamics. Firstly, we determine the bilinear form of the model and then we develop lump waves and analyse their interactions with periodic waves using symbolic computation approaches. Later, the interactions between single, double-kink and lump waves solutions, as well as interactions between periodic, lump and soliton solutions is considered. Furthermore, the ( G ′ / G 2 ) expansion approach and the Adomian technique are implemented to acquire a variety of novel configurations for the governing dynamical model. Later, the corresponding absolute accuracy is evaluated through a comparison of several types of observed solutions. Additionally, the stability of the solitary travelling wave solutions to the specified model is also established. The 3D, 2D, and contour plots have been employed to demonstrate the characteristics of the nonlinear model and aid in determine the proper set of parameters. As a result, a collection of options for bright, dark, periodic, rational, and elliptic functions is established. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the integro-differential equation appearing in shallow water waves. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research. • The interactions between single, double-kink and lump waves solutions is considered. • The (G'/G2) expansion and the Adomian techniques are implemented for the dynamical model. • We have established a comparison between analytical and semi-analytical solutions. • The 3D, 2D, and contour plots have been employed in determine the proper set of parameters. [ABSTRACT FROM AUTHOR]

In this article, the Sharma–Tasso–Olver model is investigated which characterizes the nonlinear double-dispersive evolution dispersive waves' dynamical propagation in heterogeneous mediums. To study the dynamical nature of moving waves with a periodic and isolated nature analytically, the governing model is converted into an ordinary differential equation via a wave transformation to employ the extended modified auxiliary equation mapping approach and the G ′ G 2 expansion method. In order to validate the computations, the stability of the obtained results is also established. It has been found that the model supports nonlinear solitary waves, periodic waves, shock waves and stable oscillatory waves. For a set of appropriate parameters, Mathematica is used to represent the dynamics of various wave structures as 3D, 2D and contour visualizations. Additionally, we may remark that the results discussed here are novel and innovative. • We conduct an analytical investigation of the Sharma–Tasso–Olver model. • The governing model represents a nonlinear double-dispersive evolution dispersive waves' dynamical propagation. • It has been found that the model supports nonlinear solitary, periodic, shock and stable oscillatory waves. • To represent the dynamics of wave structures as 3D, 2D, and contour visualizations are plotted. [ABSTRACT FROM AUTHOR]

In this study, the coupled nonlinear (1＋1)-dimensional Drinfel'd–Sokolov–Wilson (DSW) equation in dispersive water waves is investigated. The proposed structure is very important nonlinear evolution model in mathematical physics and engineering, which is used to describe nonlinear surface gravity wave propagating over horizontal sea bed. By using three analytical approaches namely, the new extended hyperbolic function method, the modified extended t a n hyperbolic function method and the new ( G ′ G 2 ) -expansion method, some new traveling wave solutions for the suggested nonlinear model are derived. As a result, a variety of multi soliton solutions structures in the form of dark, multi-bell shaped, singular bell shaped, trigonometric, hyperbolic and rational functions are obtained via proposed methods. By choosing appropriate set of parameters, the 3D, contour and 2D plots for some solutions are also demonstrated to visualize their nature in an efficient way. The modulation instability and stability analysis for the observed solutions are also addressed. The results obtained demonstrate the effectiveness, conciseness, and efficiency of the applied integration techniques, which may be applied to investigate more complex models arising in contemporary science and engineering. • The coupled nonlinear Drinfel'd–Sokolov–Wilson model is studied analytically. • The new extended hyperbolic function method, the modified extended t a n hyperbolic function method and the new ( G ′ G 2 ) -expansion method, are employed to construct some new traveling wave solutions. • Stability analysis and modulation instability are investigated. [ABSTRACT FROM AUTHOR]