On recursion operators and nonlocal symmetries of evolution equations

TL;DR

This paper studies recursion operators with nonlocal terms for (1+1)-dimensional evolution equations, extending them to act on nonlocal symmetries and applying these results to integrable systems.

Contribution

It introduces a method to extend recursion operators to nonlocal symmetries for evolution equations, broadening their applicability to integrable systems.

Findings

Extended recursion operators preserve the space of nonlocal symmetries.

Results apply to most known (1+1)-dimensional integrable evolution systems.

Examples demonstrate the practical use of the extended operators.

Abstract

We consider the recursion operators with nonlocal terms of special form for evolution systems in (1+1) dimensions, and extend them to well-defined operators on the space of nonlocal symmetries associated with the so-called universal Abelian coverings over these systems. The extended recursion operators are shown to leave this space invariant. These results apply, in particular, to the recursion operators of the majority of known today (1+1)-dimensional integrable evolution systems. We also present some related results and describe the extension of them and of the above results to (1+1)-dimensional systems of PDEs transformable into the evolutionary form. Some examples and applications are given.