Study of two soliton and shock wave structures by weighted residual method and Hirota bilinear approach

The dissipation effect in the viscous plasma is expressed in the current study by a Burgers term. The Weighted Residual Method is used to produce a solitary type progressive wave solution for very small values of the Burgers term. However, the strong dissipation may cause the origination of a shock solution. An approximate analytical solution is also explored by means of simplified Hirota’s bilinear method, through which the evolution of shock waves is determined. Finally, under the variation of various physical parameters the oscillatory shock yield the asymptotic solutions of the nonlinear models. The approach mentioned above is employed to provide diverse solutions of Korteweg-de Vries-Burgers equation and Zakharov-Kuznetsov-Burgers equation involving 2-soliton and shock solution, and others. The Hirota’s bilinear approach enhances our comprehension of nonlinear processes, offers precise solutions to nonlinear equations, facilitates the investigation of solitons, propels the development of mathematical tools, and is applicable in many scientific and technical fields. The solutions are graphically shown in three-dimensional (3D) surface, density and two-dimensional (2D) plots using MATLAB software. All screens display the absolute wave configurations in the resolutions of the equation with the proper parameters. Furthermore, it can be deduced that the physical properties of the found solutions and their characteristics may help us comprehend how shallow water waves move in nonlinear dynamics.