Superposed and Superposed-type Double Periodic Jacobi Elliptic Function Solutions of Variable Coefficients KdV Equation

Acoustic waves on a crystal lattice, long internal waves in a density-stratified ocean, ion acoustic waves in a plasma, and shallow-water waves with weakly non-linear restoring forces are all represented mathematically by the KdV equation. Its importance and wide range of applications have led to the development and analysis of multiple solutions in the scientific community. Beside those in this article we prove the existence of superposed solutions of KdV equation. Some theorems and corollary on the existence of superposed and superposed-type solutions for KdV equations with variable coefficients are presented in this article. The six sets of superposed solutions to the variable coefficient KdV equation are obtained by using the corollary and theorem on the existence of superposed solutions. It was demonstrated that superposed solutions of the KdV problem with variable coefficients can be constructed by combining two elementary solutions that contain reciprocal Jacobi elliptic functions. Additionally, we present a few theorems and corollaries about the existence of superposed-type solutions for this equation in the literature. The most significant and fascinating of them all is the splitting technique theorem. We obtained many superposed-type solutions of KdV equations with variable coefficients in terms of the Jacobi elliptic function by using the splitting technique. It is additionally confirmed that the generalised Miura transformation is a sub-case of the splitting procedure. This represents an additional modification to the generalised Miura transformation. These theorems explain why a number of seemingly bizarre superposition-type solutions to a number of newly published nonlinear equations have appeared. It is further demonstrated that the solutions produced by the generalised Miura transformation are specific examples of solutions obtained through the application of the splitting technique. Plots in two dimensions, three dimensions, contour, and density have all been used to illustrate the features of the derived solutions.