Complex hyperbolic-function method and its applications to nonlinear equations

Based on computerized symbolic computation, a complex hyperbolic-function method is proposed for the general nonlinear equations of mathematical physics in a unified way. In this method, we assume that exact solutions for a given general nonlinear equations be the superposition of different powers of the sech-function, tanh-function and/or their combinations. After finishing some direct calculations, we can finally obtain the exact solutions expressed by the complex hyperbolic function. The characteristic feature of this method is that we can derive exact solutions to the general nonlinear equations directly without transformation. Some illustrative equations, such as the (1+11+1)-dimensional coupled Schrödinger–KdV equation, (2+12+1)-dimensional Davey–Stewartson equation and Hirota–Maccari equation, are investigated by this means and new exact solutions are found.