Solitary-wave solutions of the Klein–Gordon equation with quintic nonlinearity

In this paper, the $(G'/G)$-expansion method is used to obtain exact solitary-wave and periodic-wave solutions for nonlinear evolution equations arising in mathematical physics with the aid of symbolic computations, namely, the Klein–Gordon equation with quintic nonlinearity. Our work is motivated by the fact that the $(G'/G)$-expansion method provides not only more general forms of solutions, but also periodic and solitary waves. As a result, hyperbolic function solutions and trigonometric function solutions with parameters are obtained. The method is straightforward and concise, and its application is promising for other nonlinear evolution equations in mathematical physics.