非线性PDE的李对称分析及其实例应用

【摘要】  非线性偏微分方程的精确解在非线性问题的研究中起到了一个重要的作用。李对称分析是研究非线性微分方程精确解的有力工具。李对称可以构造偏微分方程的相似解，从己知的解经过对称群的作用得到新的解，并且通过相似约化可以直接将偏微分方程约化为常微分方程，从而为求非线性偏微分方程的精确解创造条件。本文在实例一中通过广义Burgers方程的例子证明了本论文对李对称分析方法的掌握是正确无误且有意义的。接着在实例二中通过详细的计算纠正了郭玉翠等人在对Gardner方程进行李对称运算中发生的一个错误，然后通过计算得到Gardner方程的三参数李群。最后在两个重要的情况下，提出相应的相似约化。要重点提到的是，本次论文一个非常重要的目的是掌握李对称运算过程，这是本次论文中需要付出许多时间去掌握与运用的。

【关键词】 李对称分析，相似约化，广义Burgers方程，Gardner方程

The Lie symmetry analysis of nonlinear PDE

and its application

【Abstract】   Exact solutions of nonlinear partial differential equations play an impo-

rtant role in the study of nonlinear problems．Lie symmetry analysis is a powerful tool for studying exact solutions of nonlinear differential equations．Similarity solution of the partial differential equations can be constructed by Lie symmetry，new solution is  obtained by the action of symmetric group from a known solution，and the partial diff-

erential equations can be reduced to ordinary differential equations by the similarity re-

duction，so as to create conditions for exact solutions of nonlinear partial differential

equations．On the examples of a generalized Burgers equation，it proves Li symmetry analysis method to master is correct and meaningful．Then through the detailed calcula- tion in other example，we correct an error that Guo et al.calculate the Lie symmetry ana- lysis of Gardner equation．Next we obtain a three parameter Lie group admitted by this Gardner equation．Finally corresponding similarity reduction is proposed under the two important situations．The purpose of the paper is to master the process of the Lie symm- etry analysis，which is the thesis of the need to pay a lot of time to master and use．

【Key Words】   Lie symmetry analysis，similarity reduction，generalized Burgers equation，Gardner equation