On the Existence of Superposed and Superposed-type Real and Complex Elliptic Periodic Waves of KdV Equation

The KdV equation serves as a mathematical model for ion acoustic waves in a plasma, long internal waves in a density-stratified ocean, acoustic waves on a crystal lattice, and shallow-water waves with weakly non-linear restoring forces. Owing to its significance and numerous uses, a variety of solutions have been developed and examined in literature. The unexplained superposition of elliptic functions as solutions is one of them. In this article, certain theorems and a corollary on the existence of superposed and superposed-type solutions for KdV equations are discussed. The KdV equation’s six sets of superposed solutions are derived by applying the theorem and corollary on the existence of superposed solutions. It was demonstrated that superposed solutions of the KdV equation could be created by combining two elementary solutions that contained reciprocal Jacobi elliptic functions. Moreover, to explain the existing claimed superposed solutions of this equation in the literature, we proposed a few theorems and corollaries on the existence of these superposed solutions. Among them, the splitting technique theorem is the most significant and captivating. We obtained several superposed-type solutions of KdV equation in terms of the Jacobi elliptic function by using the splitting technique. It is further established that the splitting technique consists of the generalised Miura transformation. This represents an additional modification to the generalised Miura transformation. These theorems provide an explanation for the emergence of various seemingly strange superposition-type solutions to several recently published nonlinear equations. Additionally, the KdV equation was solved using the splitting technique and earlier theorems, which produced a large number of additional superposed-type solutions. It is additionally shown that solutions generated by the generalised Miura transformation are particular instances of solutions that are acquired by using the splitting technique. All of the derived solution characteristics have been demonstrated utilising 3D, and 2D plots.