Potential Symmetries And Exact Solutions To A System Of Diffusion Equation

The nonlinear differential equations (NDES) based on physics are one of the most important aspects in the contemporary study of nonlinear science. Exploring and developing new method to solve the NDES is one of the forefront topics in the studies of nonlinear physics. At present, there are many methods for finding exact solutions of NDES, such as travelling wave solution, homogeneous balanced method, tanh-function method, the Jacobi elliptic method, the ansatz method. In order to discuss questions conveniently we give some basic knowledge in the first chapter of this article. It is well known potential symmetry is connected with Lie symmetry, so we must give some basic koowledge about Lie symmetry. In the second chapter, we introduce the characteristic of potential symmetry,the relationship between potential symmetry and conservation forms,and how to get Lie symmetry from potential symmetry.In the second chapter of this article, we take advantage of potential symmetry to study diffusion equations of this formand prove these equations admit potential symmetry if and only if f=g = h andIn the third chapter of this article, we give some exact solutions. It is worth notice that the solutions here are different form solutions get from Lie symmetry.