Nonlinear behavior of dispersive solitary wave solutions for the propagation of shock waves in the nonlinear coupled system of equations.

In this paper, the nonlinear coupled system of partial differential equations named the Wu-Zhang system investigated by applying the new auxiliary equation method. The Wu-Zhang system help us to investigate the various nonlinear wave propagation phenomena physically including the width and amplitude of solitons, physically form of shock, traveling and solitary wave structures in fiber optics, fluid dynamics, plasma physics, nonlinear optics, these nonlinear wave equations play significant role in these phenomenas. In this regard, the dispersive long wave is described by the Wu-Zhang system, from which a number of solitons and solitary wave structure are formally extracted as an accomplishment. On the basis of the computational program Mathematica, soliton and many other solitary wave results have been obtained with the ability to use an analytical approach. Consequently, various solutions in solitons and solitary waves are generated in rational, trigonometric, and hyperbolic functions and displayed within contour, two-dimension and three-dimension plotting by using the numerical simulation. The soliton solutions are obtained including bright and dark solitons, anti-kink wave solitons, peakon bright and dark solitons, kink wave soliton, periodic wave solitons, solitary wave structure, and other mixed solitons. In order to comprehend the significance of investigating various nonlinear wave phenomena in engineering and science, including soliton theory, nonlinear optics, fluid mechanics, material energy, water wave mechanics, mathematical physics, signal transmission, and optical fibers, all research outcomes are required. With precise analytical results, shed light that the applied approach to be more powerful, dependable, and accurate.