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The generalized Taylor expansion including a secret auxiliary parameter h which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of h can't be understood in the frame of the homotopy analysis method. This is a serious shortcoming of Liao's method. We solve the problem. Through a detailed study of a simple example, we show that the generalized Taylor expansion is just the usual Taylor's expansion at different point t1. We prove that there is a relationship between h and t1, which reveals the meaning of h and the essence of the homotopy analysis method. As an important example, we study the series solution of the Blasius equation. Using the series expansion method at different points, we obtain the same result with liao's solution given by the homotopy analysis method.

A new method, that is, trial equation method, was given to obtain the exact trav eling wave solutions for nonlinear evolution equations. As an example, a class o f fifth-order nonlinear evolution equations was discussed. Its exact traveling w ave solutions, which included rational form solutions, solitary wave solutions, triangle function periodic solutions, polynomial type Jacobian elliptic function periodic solutions and fractional type Jacobian elliptic function periodic solu tions, were given.

Based on the Taylor expansion, we propose a renormalization method for asymptotic analysis. The standard renormalization group (RG) method for asymptotic analysis can be derived out from this new method, and hence the mathematical essence of the RG method is also recovered. The biggest advantage of the proposed method is that the secular terms in perturbation series are automatically eliminated, but in usual perturbation theory, we need more efforts and tricks to eliminate these terms. At the same time, the mathematical foundation of the method is simple and the logic of the method is very clear, therefore, it is very easy in practice. As application, we obtain the uniform valid asymptotic solutions to some problems including vector field, boundary layer and boundary value problems of nonlinear wave equations. Moreover, we discuss the normal form theory and reduction equations of dynamical systems. Furthermore, by combining the topological deformation and the RG method, a modified method namely the homotopy renormalization method (for simplicity, HTR) was proposed to overcome the weaknesses of the standard RG method. In this HTR method, since there is a freedom to choose the first order approximate solution in perturbation expansion, we can improve the global solution. In particular, for those equations including no a small parameter, the HTR method can also be applied. Some concrete applications including multi-solutions problems, the forced Duffing equation and the Blasius equation are given.