The Jacobi Elliptic Function and Incomplete Elliptic Integral of Second Kind Solutions of the Wazwaz Negative Order Korteweg–de Vries Equation

In this research paper, the negative order Korteweg–de Vries equation expressed as nonlinear partial differential equation, firstly introduced by Wazwaz, is solved for the exact Jacobi elliptic function solution. For this purpose, the Jacobi elliptic function scheme, one of the direct algebraic methods, was used. The obtained exact solutions of the negative-order Korteweg–de Vries equation, a symmetry evolution equation, contains the combination of Jacobi elliptic functions and incomplete elliptic integral of second function. The three unique families of exact solutions are classified and presented. The degeneration of the obtained Jacobi elliptic function solutions into various solitons, periodic and rational solutions, is reported using the modulus transformation of Jacobi elliptic function solutions. The necessary condition existence of certain Jacobi elliptic function solutions is presented. The two-dimensional graphs for certain Jacobi elliptic function solutions are drawn to show the variation in wave propogation with respect to velocity and modulus. The non-existence of certain Jacobi elliptic function solutions for negative-order Korteweg–de Vries equations is reported. Finally, the obtained solutions were compared with the previously obtained solutions of negative-order Korteweg–de Vries equation.