A direct trial function approach for solving nonlinear evolution equations

The Klein equation, Infeld equation, and Sivashinsky equation not only start from realistic physical phenomena but can also be widely used in many physically significant fields such as plasma physics, fluid dynamics, crystal lattice theory, nonlinear circuit theory, and astrophysics. As a consequence, it is a very significant and challenging topic to research the explicit and accurate travelling wave solutions to these three equations. In this paper, in order to solve these three nonlinear partial differential equations (NPDEs), we have made some modifications to the trial function technique proposed by Xie and Tang by bringing in an ansatz solution containing two E-exponential functions. On this basis, we have developed a direct trial function technique to seek the explicit and accurate travelling wave solutions of nonlinear evolution equations (NEEs). We have illustrated its feasibility by applying it to the Klein equation, Infeld equation, and Sivashinsky equation. As a result, a lot of more general explicit and accurate travelling wave solutions of these three equations, including the solitary wave solutions and the singular travelling wave solutions, are successfully constructed in a straightforward and simple manner. The obtained solutions are quite equivalent to those given in the existing references. In addition, compared with the proposed approaches in the existing references, the approach described herein appears to be less calculative. Our technique may provide a novel way of thinking for solving NEEs. It is our firm conviction that the procedure used herein may also be utilized to explore the explicit and accurate travelling wave solutions of other NEEs. We try to generalize this approach to search for the explicit and accurate traveling wave solutions of other NEEs.