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

If a partial differential equation is reduced to an ordinary differential equation in the form image under the traveling wave transformation, where image are parameters, its solutions can be written as an integral form image. Therefore, the key steps are to determine the parametersʹ 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 parameterʹs scopes can be determined easily. The complete discrimination system for polynomial is a natural generalization of the discrimination image of the second degree polynomial image. For example, the complete discrimination system for the third degree polynomial image is given by image and image. 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 Fanʹs expansion method.