Abstract: From the definition of Legendre elliptic integration and Jacobi elliptic function, new transformations are obtained and applied to construct the exact solutions of nonlinear wave equations. The nonlinear Klein-Gordon equation, Boussinesq equation and the coupled mKdV equations are taken as three examples to illustrate the detailed steps in obtaining exact solutions. There new analytical solutions such as periodic solutions and soliton solutions are derived for these nonlinear evolution equation (or equations).

Keywords: 
• Jacobi elliptic function /
• nonlinear wave equation /
• periodic solution /
• soliton solution