Narrow Results

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.

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution’s quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated.

In this work, three models of the nonlinear Schrödinger equation are looked at to see if traveling wave solutions have unique structures. “The generalized Korteweg–de Vries, (2+1)-dimensional Ablowitz–Kaup–Newell–Segur, and the Maccari models” are among the examined systems. The kinetic energy operator is affected by where the mass is, how it changes over time, and how the kinetic energy operator is shown. Using the generalized Riccati expansion method, new soliton wave solutions are constructed for the models that have been looked at. Several different graphics depict the numerical simulations of the deduced answers. Verifying and improving these methods (Mathematica 13.1) means checking to see if they are right and re-entering the results into the models.

The problem formulations of models for three-dimensional weakly nonlinear shallow water waves regime in a stratified shear flow with a free surface are studied. Traveling wave solutions are generated by deriving the nonlinear higher order of nonlinear evaluation equations for the free surface displacement. We obtain the velocity potential and pressure fluid in the form of traveling wave solutions of the obtained nonlinear evaluation equation. The obtained solutions and the movement role of the waves of the exact solutions are new travelling wave solutions in different and explicit form such as solutions (bright and dark), solitary wave, periodic solitary wave elliptic function solutions of higher-order nonlinear evaluation equation.

Boussinesq Equations (BEs) are defined to model and determine the nonlinear transformation of shallow-water surface waves induced by shoaling, diffraction, and partial reflection of interior boundaries. In this chapter, one-dimensional (1-D) and two-dimensional (2-D) analytical and numerical solutions of BEs are determined utilizing the concept of shallow-water waves over a variable water depth. The analytical solutions of the 1-D and 2-D Boussinesq-type equations are defined by first-integral method. Linear dispersion properties of BEs are also discussed. The numerical solution of 1-D BEs for regular and irregular waves is estimated by using the finite-difference method with Crank–Nicolson procedure. The numerical model is validated by using the analytical and experimental studies for shallow-water waves over the constant slope. The wave spectrum is also determined by using Pierson–Moskowitz spectra for shallow-water waves. The proposed numerical model is also applicable to any realistic domain for practical applications involving coastal regions.