An application of the $(G'/G)$-expansion method to search for exact solutions of nonlinear partial differential equations is analysed. This method is used for Burgers, Fisher, Huxley equations and combined forms of these equations. The $(G'/G)$-expansion method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the $(G'/G)$-expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G'/G )-expansion method to solve partial differentialequations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.

In this paper, we use the Hirota bilinear method. With the help of symbolic calculation and applying this method, we solve the (2 + 1)-dimensional bidirectional Sawada–Kotera (bSK) equation to obtain some new lump-kink, lump-solitons, periodic kink-wave, periodic soliton and periodic wave solutions.