【作者】 吕丹；

【导师】 张玉峰；

【作者基本信息】 辽宁师范大学 ， 应用数学， 2008， 硕士

【摘要】 孤立子理论是非线性科学的一个重要组成部分,许多理论和应用科学中的数学模型导出的非线性方程具有孤立子特性。因此,孤立子方程的求解在理论和应用中都具有极其重要的意义。本文根据数学机械化思想,以符号计算软件为工具,研究了非线性发展方程的求解问题,利用符号计算系统Maple,并应用改进的求解方法解得一些方程的新的精确解。第一章是绪论,介绍了孤立子研究的历史和发展的概况,包括孤立子理论的起源,非线性发展方程的求解方法的发展过程,还介绍了数学机械化和符号计算的概念和应用。第二章内容是应用F-展开法求解广义Hirota-Satsuma耦合方程。首先介绍F-展开法的步骤,接着应用F-展开法来求解Hirota-Satsuma耦合方程,得到了很多文献[14]中没有给出的新的方程的精确解。第三章内容是利用文献[13]提出的改进的耦合的Riccati方程组来求解(2+1)维Burgers方程。首先介绍了利用改进的耦合的Riccati方程组来求解的方法步骤,接着应用此方法,得到了(2+1)维Burgers方程的更多的新的精确解。第四章首先利用改进的Riccati方程求得KdV-MKdV方程的一些新解,其次求解2+1维广义浅水波方程的类孤子解和周期解,求得的解带有变系数,由于系数的可变性,可获得更多的方程的类孤子解和周期解.第五章首先简要的介绍了求解非线性发展方程的一种有效的方法——达布变换法,其次提出了一种新的达布变换,并加以证明,最后用新的达布变换对于BK系统进行应用,求得了方程的新的解.更多还原

【Abstract】 The soliton theory is an important part of the nonlinear science. There are many nonlinear partial differential equations that have solution properties in the pure and applied science. Therefore, solving soliton Equations is very important in the theory and in the application. This dissertation, under the guidance of mathematics mechanization and by means of symbolic computation software, considers the exact solutions to the nonlinear partial differential equations. With the help of symbolic computation system Maple, by the application of improved methods of solving equations, some new exact solutions are presented.Chapter1 is to introduce the history and development of the soliton theory, including the origin of the soliton theory, development of the method solving nonlinear equations, also introduced mathematical mechanization and symbolic computation and application.Chapter2 is the application of the F-expansion method to the generalized Hirota -Satsuma coupled equations. First on the F-expansion method steps, and then by the F-expansion method , we get a lot of the new exacted solutions that can not be find in reference .Chapter 3 is to introduce the solving of (2+1)-dimensional Burgers equation by the improved coupling Riccati equations raised by reference [13]. First of all, we describe the steps of the F-expansion method , and then apply the method, many of the new exact solutions of (2+1)-dimensional Burgers equation are obtained.Chapter 4 is to firstly introduce the solving of KdV-MKdV equation by the improved Riccati equation . Secondly, we obtained the like soliton solutions and the periodic solutions of the (2+1)-dimensional generalized shallow-water wave equation which are with variable coefficient, because of the variable coefficient ,the equation will be have more like soliton solutions and periodic solutions.Chapter 5 begins with a brief introduction to solving the nonlinear development equation by an effective method - Darboux transformation method, followed by a new Darboux transformation is presented with the proof, finally, the new Darboux transformation is applied to BK system, then we obtained the new solutions of BK system.更多还原