Abstract
This article investigates the existence theory, exact solutions, and the unique solutions of physical problems. In this study the well-known Selkov-Schnakenberg system of coupled nonlinear unidirectional PDEs is analyzed. This is a simple chemical reaction system that admits periodic solutions. The existence of the system is extracted by applying contraction and self-mapping conditions. The new families of exact solutions which represent the concentration and chemical reactants in different forms are formulated. The periodic, singular periodic, shock wave, singular wave, and complex solitary-shock, shock-singular, double singular, and periodic-singular solutions are successfully extracted by using the new modified extended direct algebraic (MEDA) technique. Additionally the unique problems with corresponding proposed auxiliary data are discussed. The trends of these solutions are also sketched in different plots.