F-expansion method and its application for finding new exact solutions to the sine-Gordon and sinh-Gordon equations

In this paper, based on the F-expansion method and the extended version of F-expansion method we investigate the exact solutions of the sine-Gordon equation and sinh-Gordon equation, respectively. Firstly, several special transformations are introduced to change the sine-Gordon and sinh-Gordon equations into polynomial nonlinear evolution equations. Then these polynomial nonlinear evolution equations are solved by the F-expansion method and its extended version. Finally, more new exact Jacobian elliptic function solutions and soliton solutions are constructed explicitly.