Levi方程的Painleve分析和精确解

【作者】 邓勇；

【导师】 张金顺；

【作者基本信息】 华侨大学 ， 基础数学， 2009， 硕士

【摘要】 非线性微分方程的可积性与求解是非线性科学中一个重要的研究课题.而Painleve分析方法是判定其可积性和求解的一个有力工具.本文针对两个高阶Levi方程,做了以下工作:(1).对(1+1)维Levi方程,利用WTC方法对其进行Painleve测试,仿照Painleve求解常微分方程(组)调谐因子的方法得到(1+1)维Levi方程的调谐因子,并通过Painleve测试证明了方程具有Painleve性质,在Painleve意义下可积(即P-可积).然后通过相容性分析得出了相容方程,从而将方程组的解的奇异流形展式截断为有限项的形式,利用Schwarz导数及其性质得到了方程的一类精确解.同时得到了方程的一个自Backlund变换.(2).对(2+1)维Levi方程,利用WTC方法对其进行Painleve测试,对特殊情况下的耦合(2+1)维Levi方程进行分析求解,得到了方程的单孤子解,双孤子解及多孤子解.(3).对于求得的Levi方程的精确解,选择适当的参数,利用Matlab作出了解的图形,并根据图形分析解的性质.在对耦合(2+1)维Levi方程双孤子解图像的分析中,讨论了孤立波的裂变现象.(4).对(1+1)维Levi方程主导项系数的其他情况进行介绍,并进行Painleve测试,得到了这些情况下的相容方程组.更多还原

【Abstract】 The integrable property for nonlinear differencial equations is important in nonlinear science. In this thesis, two higher order Levi equation are studied :(1). We make use of WTC method to the (1+1)-dimensional Levi equation. It is proved that the equation possess Painleve property (P-integrable). The resonances are obtained by the WTC method. The compatible equations are obtained and the expansions of solution are truncated through the analysis of compatibility. Thereby, the exact solutions and Daboux-Backlund Transformation of the Levi equation are obtained by means of Schwarzian derivative.(2). As to the (2+1)-dimensional Levi equation, we make use of WTC method to do the Painleve Test. Furthermore, One-soliton solution, Two-soliton solution, and N-soliton solution are obtained in some particular conditions.(3). We use Matlab to get graphics of Levi equation, and do some analysis about the solutions. Furthermore, the fission phenomenon in Two-soliton solution of the coupled (2+1)-dimensional Levi equation is discussed.(4). We introduce some special condition of the (1+1)-dimensional Levi equation in Painleve test, The compatible equations are obtained.更多还原