广义薄膜方程的Lie对称性与精确解
Lie Symmetries and Exact Solutions of the Generalized Thin Film Equation
摘要：
考虑二阶和四阶扩散项，对四阶反应扩散方程进行了对称群分类。四阶方程的处理，首先，作为系统的二阶方程，承担一些相似的系统的耦合反应扩散方程与交叉扩散，其次，作为系统的二阶方程和两个一阶方程。本文推广了这些方程的特殊情况下，李对称性分析的结果。利用反应扩散系统到常微分方程的Lie对称约化构造了各种精确解。这些解包括一些不寻常的结构以及对称性约化中经常出现的熟悉类型，即自相似解、减速和衰减行波以及稳态。
英文摘要：
A symmetry group classification for fourth-order reaction-diffusion equations, allowing for both second-order and fourth-order diffusion terms, is carried out. The fourth order equations are treated, firstly, as systems of second-order equations that bear some resemblance to systems of coupled reaction-diffusion equations with cross diffusion, secondly, as systems of a second-order equation and two first-order equations. The paper generalizes the results of Lie symmetry analysis derived earlier for particular cases of these equations. Various exact solutions are constructed using Lie symmetry reductions of the reaction-diffusion systems to ordinary differential equations. The solutions include some unusual structures as well as the familiar types that regularly occur in symmetry reductions, namely self-similar solutions, decelerating and decaying traveling waves, and steady states.
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