Jacobi elliptic function expansion method for the improved modified kortwedge-de vries equation

In this paper, the Jacobi elliptic function expansion technique is put forward for the first time to build the exact solution of the improved modified Kortwedge-de varies (mKdV) equation. The goal of this work is to explore more new exact solutions to the improved mKdV equation by using a finite series of degree n in terms of Jacobi elliptic functions. More Jacobi elliptic function solutions are obtained including the single function solutions and the combined function solutions. Jacobi elliptic function expansion (JEFE) method is an effective method, and it is extensively used for the exact analytical solution of nonlinear partial differential equations (NPDEs). Moreover, this technique has multiple versions depending on its intrinsic and essential qualities, therefore, they provided exact solutions to such types of nonlinear problems. This method is based on the Jacobi elliptic functions. We use the traveling wave variable to transform the NPDEs into nonlinear ordinary differential equations (ODEs) with integer order. After adopting, the most general and explicit form of the JEFE method is suggested to produce the exact and more accurate solution to the improved mKdV equation. It is also confirmed that the solutions of periodic nature obtained by the JEFE method comprise certain solitary and shock wave solutions. Additionally, direct-viewing analysis is explicated by showing some figures of partial solutions. It is concluded from the results that the suggested technique is quite a helpful technique for solving a large variety of NPDEs analytically. In mathematical physics, many other kinds of nonlinear evaluation equations can be solved by this method.