【作者】 胡晓瑞；

【导师】 陈勇；

【作者基本信息】 宁波大学 ， 基础数学， 2009， 硕士

【摘要】 对称方法及达布变换是求解非线性数学物理方程精确解的强有力工具。首先我们学习和研究了对称群和约化的理论和方法,其中包括:经典李群方法、非经典李群方法、CK直接约化法及改进的CK方法。以这些方法为基础,研究了一个非等谱的KP方程。其次,微分方程的李对称群理论的重要应用之一是其群不变解的构造和分类,这就涉及到优化系统的问题。在这方面,我们研究了一般的含参数的(2+1)维的非线性Klein-Gordon方程。最后,学习和研究了达布变换的相关知识,构造了修正KP方程的一种二元达布变换。论文安排如下:第一章绪论:简要介绍了求解非线性数学物理方程精确解的几种方法。第二章对称约化:介绍了对称和相似约化的知识,分别利用CK直接约化法、经典李群方法和改进的CK方法研究了一个非等谱的KP方程的对称及其相似约化,从中可以看出这三种方法的相似和不同之处。第三章对称优化:利用对称优化方法构造群不变解及其分类,研究了一个一般的(2+1)维的非线性Klein-Gordon方程,根据其李代数维数的不同将其分成了三种类型,分别构造了其群不变解及相应的一维优化系统。第四章二元达布变换:介绍了达布变换的相关知识,构造了修正KP方程的一种二元达布变换,并且给出了它的n次迭代形式。更多还原

【Abstract】 The symmetry method and Darboux transformation are the powerful tools to solve the equations in nonlinear mathematical physics. Firstly, we studyed the theories and methods of the symmetry group and reductions, including classical Lie group method, nonclassical Lie group method, CK direct method and the modified CK method. Based on these methods, we investigated a nonispectral KP equation. Secondly, one of the main applications of Lie theory of symmetry groups for differential equations is the construction and classification of group invariant solutions, which is in relation to the optimal system. In this respect, we considered a general (2+1) dimensional nonlinear Klein-Gordon equation. Lastly, after studying the related knowledge of Darboux transformation, we constructed a binary Darboux transformation for the modified KP equation.The thesis is arranged as follows:Chapter 1 Introduction. Several methods for obtaining exact solutions of the nonlinear mathematical physics equations are introduced in this chapter.Chapter 2 Symmetry and Reductions. This chapter introduces some relevant knowledge about the symmetry and Reductions. We make use of CK direct method, the classical Lie group method and the modified CK method to investigate the symmetry of a nonispectral KP equation, respectively. From the results, one can see the similarities and differences among the above three methods.Chapter 3 Symmetry Optimization. One can make use of symmetry optimization to construct group invariant solutions and its classification. In this chapter, we investigate a general (2+1) dimensional nonlinear Klein-Gordon equation. In accordance with the dimensions of its Lie algebra, it is divided into three types. For every type, we construct its group invariant solutions and the corresponding one dimentional optimal system, respectively.Chapter 4 Binary Darboux Transformation. This chapter firstly introduces the relevant knowledge about Darboux transformation. Then we construct a binary Darboux transformation for the modified KP equation and give out its N-step iteration.更多还原