Quasi-rational function solutions of an elliptic equation and its application to solve some nonlinear evolution equations

Using the differential transformation method and the homogeneous balance method, some new solutions of an auxiliary elliptic equation are obtained. These solutions possess the forms of rational functions in terms of trigonometric functions, hyperbolic functions, exponential functions, power functions, elliptic functions and their operation and composite functions and so on, which are so-called quasi-rational function solutions. Based on these new quasi-rational functions solutions, a direct method is proposed to construct the exact solutions of some nonlinear evolution equations with the aid of symbolic computation. The coupled KdV–mKdV equation and Broer–Kaup equations are chosen to illustrate the effectiveness and convenience of the suggested method for obtaining quasi-rational function solutions of nonlinear evolution equations.