The current study examines the special class of a generalized reaction-advection-diffusion dynamical model that is called the system of coupled Burger's equations. This system plays a vital role in the essential areas of physics, including fluid dynamics and acoustics. Moreover, two promising analytical integration schemes are employed for the study; in addition to the deployment of an efficient variant of the eminent Adomian decomposition method. Three sets of analytical wave solutions are revealed, including exponential, periodic, and dark-singular wave solutions; while an amazed rapidly convergent approximate solution is acquired on the other hand. At the end, certain graphical illustrations and tables are provided to support the reported analytical and numerical results. No doubt, the present study is set to bridge the existing gap between the analytical and numerical approaches with regard to the solution validity of various models of mathematical physics.