论文标题
重新审视热方程的点和广义对称性
Point and generalized symmetries of the heat equation revisited
论文作者
论文摘要
我们得出了（1+1）维线性热方程的点对称转换的不错表示，并正确解释了它们。这使我们能够证明这些转换的伪群正好具有两个连接的组件。也就是说，热方程式允许单个独立的离散对称性，可以选择将因变量交替交替。我们介绍了一个谎言组的伪二散元素的概念，并表明将空间变量的迹象交替出现，这很长一段时间以来被误解为热方程的离散对称性，实际上是其必不可少的点对称群体的伪折线元素。加强了热方程的基本谎言不变性代数的亚代代代数的分类，并且对该方程的广义对称性的描述也得到了完善。我们还考虑了汉堡方程式，因为它与热方程式有关，并证明它不承认离散点对称性。在某些变量中具有线性分数的点对称组的开发方法可以直接扩展到许多其他线性和非线性微分方程。
We derive a nice representation for point symmetry transformations of the (1+1)-dimensional linear heat equation and properly interpret them. This allows us to prove that the pseudogroup of these transformations has exactly two connected components. That is, the heat equation admits a single independent discrete symmetry, which can be chosen to be alternating the sign of the dependent variable. We introduce the notion of pseudo-discrete elements of a Lie group and show that alternating the sign of the space variable, which was for a long time misinterpreted as a discrete symmetry of the heat equation, is in fact a pseudo-discrete element of its essential point symmetry group. The classification of subalgebras of the essential Lie invariance algebra of the heat equation is enhanced and the description of generalized symmetries of this equation is refined as well. We also consider the Burgers equation because of its relation to the heat equation and prove that it admits no discrete point symmetries. The developed approach to point-symmetry groups whose elements have components that are linear fractional in some variables can directly be extended to many other linear and nonlinear differential equations.
