Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations

If a partial differential equation is reduced to an ordinary differential equation in the form u′(ξ)=G(u,θ1,…,θm)u′(ξ)=G(u,θ1,…,θm) under the traveling wave transformation, where θ1,…,θmθ1,…,θm are parameters, its solutions can be written as an integral form ξ−ξ0=∫duG(u,θ1,…,θm). Therefore, the key steps are to determine the parameters' scopes and to solve the corresponding integral. When G   is related to a polynomial, a mathematical tool named complete discrimination system for polynomial is applied to this problem so that the parameter's scopes can be determined easily. The complete discrimination system for polynomial is a natural generalization of the discrimination △=b2−4ac△=b2−4ac of the second degree polynomial ax2+bx+cax2+bx+c. For example, the complete discrimination system for the third degree polynomial F(w)=w3+d2w2+d1w+d0F(w)=w3+d2w2+d1w+d0 is given by △=−27(2d2327+d0−d1d23)2−4(d1−d223)3 and D1=d1−d223. In the paper, we give some new applications of the complete discrimination system for polynomial, that is, we give the classifications of traveling wave solutions to some nonlinear differential equations through solving the corresponding integrals. In finally, as a result, we give a partial answer to a problem on Fan's expansion method.