A sinh-Gordon equation expansion method to construct doubly periodic solutions for nonlinear differential equations

Based on a sinh-Gordon equation, a transformation method (called the sinh-Gordon equation expansion method) and its algorithm are proposed, which are powerful and simple. We choose the mkdV equation, the NLS+ equation and the KdV equation to illustrate our method. Consequently, many doubly periodic (Jacobian elliptic function) solutions are obtained. When the module m→1 or 0, the corresponding solitary waves and singly periodic solutions are also found. This approach can be also applied to solve other nonlinear differential equations. Moreover we further improve the method to be a generalized form at the end of the paper.