Propagation of novel traveling wave envelopes of Zhiber-Shabat equation by using Lie analysis

In this paper, we aim to find novel forms of wave structures by employing some innovative ideas. Various solitary wave solutions of the Zhiber-Shabat equation have been extracted using the Lie symmetry analysis and the extended direct algebraic method. In the mathematical community, the considered model has several applications, notably in integral quantum field theory, fluid dynamics, and kink dynamics. First of all, the Lie symmetry has been used to determine the corresponding similarity reductions through similarity variables and wave transformation with the help of optimal systems. Afterward, the method described has been used to create new complex, hyperbolic, rational, and trigonometric forms of solutions to the problem. Depending on the strength of the propagating pulse, these solutions reflect dark, bright, kink-type, and periodic solitary wave envelopes. Further, two-dimensional (2D), three-dimensional (3D), as well as contour 2D graphics of the results have been analyzed by giving some specific values to parameters. At last, sensitivity analysis of the evolution equation has been observed.