Nijenhuis torsion and Fr\"olicher-Nijenhuis brackets of recursion operators via their full-fledged forms

Abstract

We present a novel approach to computing the Nijenhuis torsion and Fr\"olicher-Nijenhuis brackets of recursion operators for symmetries, based on their full-fledged forms introduced by Jahnov\'a and Voj\v{c}\'ak (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 Fr\"olicher-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 Fr\"olicher-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.