The extended first kind elliptic sub-equation method and its application to the generalized reaction Duffing model

In this Letter, an extended first kind elliptic sub-equation method and its algorithm are proposed by studying a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. Being concise and straightforward, the method is applied to the generalized reaction Duffing model. As a consequence, more types of exact travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions. The method can be also extended to many other nonlinear wave equations.