Exact solutions of the Kudryashov–Sinelshchikov equation and nonlinear telegraph equation via the first integral method

Nonlinear evolution equations are widely used to describe complex phenomena in various sciences such as fluid physics, condensed matter, biophysics, plasma physics, nonlinear optics, quantum field theory and particle physics, etc. In recent years, various powerful methods have been presented for finding exact solutions of the nonlinear evolution equations in mathematical physics, such as, tanh method [1–3], multiple exp-function method [4], transformed rational function method [5], Hirotas direct method [6, 7], extended tanh-function method [8] and so on. The first integral method, which is based on the ring theory of commutative algebra, was first proposed by Feng [9]. This method was further developed by the same author in [10–13]. The aim of this work is to find new exact solutions of Kudryashov–Sinelshchikov equation by using the first integral method. The rest of this paper is organized as follows. In Section 2, we give the description of the first integral method. In Sections 3 and 4, we apply this method to nonlinear telegraph equation and Kudryashov–Sinelshchikov equation.