Exact solutions of the (2+1)-dimensional cubic Klein–Gordon equation and the (3+1)-dimensional Zakharov–Kuznetsov equation using the modified simple equation method

Abstract

Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this article, we implemented the modified simple equation (MSE) method for finding the exact solutions of NLEEs via the (2+1)-dimensional cubic Klein–Gordon (cKG) equation and the (3+1)-dimensional Zakharov–Kuznetsov (ZK) equation and achieve exact solutions involving parameters. When the parameters are assigned special values, solitary wave solutions are originated from the exact solutions. It is established that the MSE method offers a further influential mathematical tool for constructing exact solutions of NLEEs in mathematical physics.