发表机构
Silesian University in Opava（俄帕瓦西里西亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出基于全形式的新方法，直接计算递归算子的Nijenhuis挠率与Frölicher-Nijenhuis括号，可处理传统方法难应对的高度非局域、多维递归算子，还给出了新的四维通用层级方程递归算子示例。
AI 中文摘要
我们提出了一种计算对称性递归算子的Nijenhuis挠率和Frölicher-Nijenhuis括号的新方法，该方法基于Jahnová和Vojčák（2024）引入的递归算子全形式。与将递归算子表示为给定覆盖内非局域对称性影子之间映射的传统方法不同，我们的方法允许直接从Nijenhuis挠率的定义公式计算它，无需任何额外的数学构造。同一框架还可用于计算递归算子的Frölicher-Nijenhuis括号，并直接在其全形式中验证它们的兼容性。该过程通过几个四独立变量微分方程对称性的全形式递归算子示例进行说明，其中包括一个新的四维通用层级方程的递归算子，据我们所知，该算子此前未在文献中出现。这个新的全形式递归算子的一个显著特征是，其影子分量似乎无法在任何标准传统形式中获得合理表示。尽管如此，我们的方法允许我们直接证明该算子的Nijenhuis挠率为零。对于所考虑的示例，相应的Frölicher-Nijenhuis括号也为零，证实了各对递归算子的兼容性。这表明全形式为研究递归算子的遗传性和兼容性性质提供了有效框架，包括现有方法难以处理的高度非局域和/或多维情况。
英文摘要
We present a novel approach to computing the Nijenhuis torsion and Frölicher-Nijenhuis brackets of recursion operators for symmetries, based on their full-fledged forms introduced by Jahnová and Vojčák (2024). In contrast to the conventional approach, which represents recursion operators as maps between shadows of nonlocal symmetries within a given covering, our method allows the Nijenhuis torsion to be computed directly from its defining formula, without requiring any additional mathematical constructions. The same framework can also be used to compute the Frölicher-Nijenhuis bracket of recursion operators and to verify their compatibility directly in their full-fledged forms. The procedure is illustrated by several examples of full-fledged recursion operators for symmetries of differential equations in four independent variables, including a new recursion operator for the four-dimensional universal hierarchy equation that, to the best of our knowledge, has not previously appeared in the literature. A notable feature of this new full-fledged recursion operator is that its shadow component does not appear to admit a reasonable representation in any of the standard conventional forms. Nevertheless, our approach allows us to prove directly that the Nijenhuis torsion of this operator vanishes. For the examples considered, the corresponding Frölicher-Nijenhuis brackets also vanish, confirming the compatibility of the respective pairs of recursion operators. This demonstrates that full-fledged forms provide an effective framework for studying the hereditary and compatibility properties of recursion operators, including highly nonlocal and/or multidimensional cases that are difficult to handle by existing methods.
Comments17 pages
