常微分方程约化中的局部和非局部可解结构
Local and nonlocal solvable structures in ODEs reduction
摘要：
可解结构，类似于局部对称的可解代数，可以用来通过求积积分标量常微分方程。然而，可解结构特别适合于缺乏局部对称性的常微分方程的积分。事实上，在正则性假设下，任何给定的常微分方程总是承认可解结构，即使找到它们通常是一个非常困难的任务。在实践中，一个值得注意的简化可能来自计算可解结构，这些结构适用于某些公认的对称代数。在本文中，我们考虑适用于任何阶的局部和非局部对称代数的可解结构（即，古典和更高）。特别是，我们引入了非局部可解结构的概念。
英文摘要：
Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order (i.e., classical and higher). In particular we introduce the notion of nonlocal solvable structure.
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