Application Of The Function Expansion Method And The Homotopy Analysis Method For Solving Nonlinear Evolution Equations

Recent years, nonlinear problems become a hot topic in modern physics and nonlinear science also contributes greatly in many fields to science and technology. Nonlinear physics is one of the important branches in nonlinear science. Many nonlinear phenomena can be well described by nonlinear equations. Therefore, it is an important and meaningful work to seek the solution of a nonlinear equation. In this thesis, two aspects about solving nonlinear problems are discussed.1. Finding the exact solutions of a nonlinear evolution equation (NEE)The Hyperbola Function Expansion Method (HFEM) is one of the most efficient methods presented recently for seeking the solitary wave solutions of NEEs. This method is investigated in this paper. Based on the HFEM, we find two conditions that the Expansion Function should be satisfied. We then can choose the other functions, which are only need to satisfy the two conditions, as the Expansion Function to construct the exact solutions of a NEE. We also modified this method to solve some non-planar wave equations or NEEs those with variable coefficients.The Jacobi Elliptic Function Expansion Method (JEFEM) is another direct and forward method for solving the periodic solutions of a NEE. Based on this method, we presented a simple principle for choosing the Expansion Function. Under the guidance of this principle, we constructed many kinds of functions, which can be selected as the Expansion Function to find the exact solutions of a NEE.2. Seeking the approximation and analytical solution of a nonlinear evolution equationWhen the exact solutions of a nonlinear problem can not be found or it is very difficult to got, we may seek the approximation solutions. The Homotopy Analysis Method (HAM) is one of the powerful and efficient methods for obtaining the approximation solutions of a nonlinear problem. We applied the HAM to find the periodic solutions of a NEE. Moreover, this approach is firstly applied to solve the solitary wave solutions of many NEEs. The results indicate that the approximation solutions obtained by the HAM are very close to the corresponding exact solutions. We also proposed a simple and practical way to check the validation of the approximation solutions got by the HAM.By the end of this paper, we firstly applied the HAM to solve a non-integrable system, the KdV-Burgers equation. We obtained two kinds of traveling wave solutions of the KdV-Burgers equation with high accuracy. From deterministy analysis we know that the traveling wave solution is either monotone wave solution or oscillation wave solution. The results indicate that this method is still valid for solving some non-integrable systems.All the methods we proposed can be performed partially even completely with the help of Computer Algebraic System, such as Maple or Mathematica.