Link between travelling waves and first order nonlinear ordinary differential equation with a sixth-degree nonlinear term

Many travelling wave solutions of nonlinear evolution equations can be written as a polynomial in several elementary or special functions which satisfy a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. From that property, we deduce an algebraic method for constructing those solutions by determining only a finite number of coefficients. Being concise and straightforward, the method is applied to three nonlinear evolution equations. As a result, many exact travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.