A direct method for producing Hamiltonian structure of nonlinear evolution equations

The equation hierarchy presented in this paper contains the KdV equation and the mKdV equation. By use of the concept of characteristic number, an undetermined-constant method is proposed by us, for which the polynomial Hamiltonian functions are constructed. By employing the method, the Hamiltonian structure of the equation hierarchy is established. The approach presented in the paper shares extensive applications. In addition, four explicit expressions of the travelling wave solutions to the above equation hierarchy are obtained. One of them is regular, the other three are singular.