Abstract
Abstract In this manuscript, several types of exact solutions including trigonometric, hyperbolic, exponential, and rational function are successfully constructed with the implementation of two modified mathematical methods, namely called extended simple equation and modified F-expansion methods on the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa and the combined sinh–cosh-Gordon equations. Diverse form of solitary wave solutions is achieved from exact solutions by passing the special values to the parameters. Some solution are plotted in the form of 3D and 2D by assigning the specific values to parameters under the constrain condition to the solutions. These approaches yield the new solutions that we think other researchers have missed in the field of nonlinear sciences. Hence the searched wave’s results are loyal to the researchers and also have imperious applications in applied sciences.