In this article, new (G′/G)-expansion method and new generalized (G′/G)-expansion method is proposed to generate more general and abundant new exact traveling wave solutions of nonlinear evolution equations. The novelty and advantages of these methods is exemplified by its implementation to the KdV equation. The results emphasize the power of proposed methods in providing distinct solutions of different physical structures in nonlinear science. Moreover, these methods could be more effectively used to deal with higher dimensional and higher order nonlinear evolution equations which frequently arise in many scientific real time application fields.

We construct new analytical solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.

The generalized and improved -expansion method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. In this article, we investigate the higher dimensional nonlinear evolution equation, namely, the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation via this powerful method. The solutions are found in hyperbolic, trigonometric and rational function form involving more parameters and some of our constructed solutions are identical with results obtained by other authors if certain parameters take special values and some are new. The numerical results described in the figures were obtained with the aid of commercial software Maple.

We construct the traveling wave solutions of the fifth‐order Caudrey‐Dodd‐Gibbon (CDG) equation by the (G^′/G)‐expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the (G^′/G)‐expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.

In this article, the generalized Riccati equation mapping together with the basic (G′/G)-expansion method is implemented which is advance mathematical tool to investigate nonlinear partial differential equations. Moreover, the auxiliary equation G′(ϕ) = h + f G(ϕ) + g G2(ϕ) is used with arbitrary constant coefficients and called the generalized Riccati equation. By applying this method, we have constructed abundant traveling wave solutions in a uniform way for the Sawada-Kotera equation. The obtained solutions of this equation have vital and noteworthy explanations for some practical physical phenomena.

The generalized Riccati equation mapping is extended with the basic (G^′/G)‐expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty‐seven traveling wave solutions for the (2+1)‐dimensional modified Zakharov‐Kuznetsov equation by applying this method. Further, the auxiliary equationG^′(η) =w+uG(η) +vG²(η) is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple.