论文标题
$ \ MATHCAL {C}^{\ infty} $ - 分布的对称性和集成性
$\mathcal{C}^{\infty}$-symmetries of distributions and integrability
论文作者
论文摘要
引入了可解的结构概念的扩展，以进行矢量场的参与分布。新结构基于对向量场分布的对称概念的概括，灵感来自于将lie Point对称性扩展到$ \ Mathcal {C}^{c}^{\ infty} $ - 近年来开发的ODES的对称性。 这些新对象，名为$ \ Mathcal {C}^{\ Infty} $ - 结构，在分布的集成性中起着基本作用：$ \ Mathcal {C}^{c}^{\ infty} $结构的知识，用于corank $ k $ wissitage wissition允许其整体c $ k $ kssactive contression允许求解其集成差异。这些结果对微分方程的整合性具有重要的后果。特别是，我们得出了一个新的程序，通过将问题分成$ m $完全可以集成的Pfaffian方程来整合$ m $ thord的普通微分方程。此逐步集成过程被应用以集成几个无法通过标准过程解决的方程式。
An extension of the notion of solvable structure for involutive distributions of vector fields is introduced. The new structures are based on a generalization of the concept of symmetry of a distribution of vector fields, inspired in the extension of Lie point symmetries to $\mathcal{C}^{\infty}$-symmetries for ODEs developed in the recent years. These new objects, named $\mathcal{C}^{\infty}$-structures, play a fundamental role in the integrability of the distribution: the knowledge of a $\mathcal{C}^{\infty}$-structure for a corank $k$ involutive distribution permits to find its integral manifolds by solving $k$ successive completely integrable Pfaffian equations. These results have important consequences for the integrability of differential equations. In particular, we derive a new procedure to integrate an $m$th-order ordinary differential equation by splitting the problem into $m$ completely integrable Pfaffian equations. This step-by-step integration procedure is applied to integrate completely several equations that cannot be solved by standard procedures.
