An elliptic equation method and its applications in nonlinear evolution equations

An elliptic equation method is presented for constructing new types of elliptic function solutions of nonlinear evolution equations. The key idea of this method is to use solutions of an elliptic equation involving four real distinct roots to construct solutions of nonlinear evolution equations. The (3+1)-dimensional modified KdV–ZK equation and Whitham–Broer–Kaup equation are chosen to illustrate the application of the elliptic equation method. Consequently, new elliptic function solutions of rational forms are derived that are not obtained by the previously known methods.