标题： 一种一致的方法来近似微分方程的李对称性 显示英文标题

标题： A Consistent Approach to Approximate Lie Symmetries of Differential Equations

摘要： 李理论提供了处理微分方程的统一而有力的方法。 不幸的是，方程的任何小扰动通常会破坏一些重要的对称性，这限制了李群方法在实际应用中出现的微分方程中的适用性。 另一方面，包含\emph{小项}的微分方程通常通过摄动技术被广泛且成功地研究。 因此，将李群方法与摄动分析结合，\emph{即}，建立近似对称性理论是很有必要的。 近似对称性的两种常用方法：一种是1988年由Baikov、Gazizov和Ibragimov提出的，另一种是1989年由Fushchich和Shtelen引入的。 此外，为了减少计算量，已经提出了一些Fushchich--Shtelen方法的变体。 在这里，我们提出了一种与摄动理论一致的方法，允许将李群分析的所有相关特性扩展到近似情况下。 还展示了一些应用。 显示英文摘要

摘要： Lie theory of continuous transformations provides a unified and powerful approach for handling differential equations. Unfortunately, any small perturbation of an equation usually destroys some important symmetries, and this reduces the applicability of Lie group methods to differential equations arising in concrete applications. On the other hand, differential equations containing \emph{small terms} are commonly and successfully investigated by means of perturbative techniques. Therefore, it is desirable to combine Lie group methods with perturbation analysis, \emph{i.e.}, to establish an approximate symmetry theory. There are two widely used approaches to approximate symmetries: the one proposed in 1988 by Baikov, Gazizov and Ibragimov, and the one introduced in 1989 by Fushchich and Shtelen. Moreover, some variations of the Fushchich--Shtelen method have been proposed with the aim of reducing the length of computations. Here, we propose a new approach that is consistent with perturbation theory and allows to extend all the relevant features of Lie group analysis to an approximate context. Some applications are also presented.