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This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail. Full article

The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the $(2+1)$-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the $(3+1)$-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the $(G^{\prime }/G^2)$-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the $(G^{\prime }/G^2)$-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations. Full article

This paper is based on finding the exact solutions for Burger’s equation, Zakharov-Kuznetsov (ZK) equation and Kortewegde vries (KdV) equation by utilizing exponential function method that depends on the series of exponential functions. The exponential function method utilizes the homogeneous balancing principle to find the solutions of nonlinear equations. This method is simple, wide-reaching and helpful for finding the exact solution of nonlinear conformable PDEs. Full article