The Unification Of Function Expansion Methods To Solve Nonlinear Evolution Equation

It is a class of important problem in mathematics and physics to solve nonlinear evolution equations, especially to obtain the exact solutions of them. In recent years, based on the homogeneous balance principle and computer algebra, function expansion method is developed as an effective method to derive exact solutions of nonlinear evolution equations by using the known solutions of nonlinear ordinary differential equations, and meanwhile solitary solutions and periodic solutions can be obtained easily. At the same time, the related modified extended methods are also developed. Because of the complexity of nonlinear equations, many new forms of solutions in the results are mistaken as new solutions.In this paper, firstly, three generalized variable coefficients nonlinear evolution equations are proposed. Through the uniform transformation of equations, the transformation, property and solution of the three equations are discussed and classic equations show the efficiency of our method proposed in this paper. The solution and property of variable coefficients equation are easily obtained from the results of ordinary coefficient equation, especially as variable coefficients equation is transformed to ordinary coefficient equation. Secondly, using homogeneous balance principle, scale transformation and the new form of solution in the extended function expansion method, the sub-equation is studied and we can know that the results obtained by modified extended tanh method and extended F expansion method are only variety forms of the solutions of sub-equation. Many equations such as Burgers type equations, Broer-kaup-Kupershmid equations, new Hamiltonian amplitude equation and generalized Zakharov equations in the reference are illustrated and show that this method in this paper can be applied to other extended methods.