In this paper, we attempt to obtain exact and novel solutions for Date-Jimbo-Kashiwara-Miwa equation (DJKM) via two different techniques: Lie symmetry analysis and generalized Kudryashov method (GKM). This equation has applications in plasma physics, fluid mechanics, and other fields. The Lie symmetry method is applied to reduce the governing equation to five different ordinary differential equations (ODEs). GKM is used to obtain general and various periodic solutions. These solutions have different behaviors such as kink wave, anti-kink wave, double soliton, and single wave solution. The physical behavior of the solutions was reviewed through 2-D and 3-D graphs.