Integrable systems and their recursion operators

Abstract

In this paper we discuss the structure of recursion operators. We show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries. This enables us to compute recursion operators more systematically. Under certain conditions (which hold for all examples known to us) Nijenhuis operators are well defined, i.e., they give rise to hierarchies of infinitely many commuting symmetries of the operator. Moreover, the nonlocal part of a Nijenhuis operator contains the candidates of roots and coroots.

Рекурсионные Операторы, Законы Сохранения И Условия Интегрируемости Для Разностных Уравнений

“…Иерархии симметрий можно построить, последовательно применяя ре-курсионный оператор R к затравочным или корневым симметриям K (1) и K (2) -начальным точкам для иерархии симметрий [32]. Коммутативность векторных по-лей, отвечающих K (1) и K (2) , была установлена в работе [33].…”

Why nonlocal recursion operators produce local symmetries: new results and applications

“…For γ ∈ V * we define [2,3,11,28] its Lie derivative along Q ∈ V as L Q (γ) = γ ′ (Q) + Q ′ † (γ), see [3,28] for more details and for the related complex of formal calculus of variations. For Q ∈ V and γ ∈ V * we have, see e.g.…”

“….. Notice that, unlike e.g. [11,16], we do not require the hierarchy in question to be time-independent, and our Proposition 1 and Theorem 1 can be successfully employed for proving locality of the so-called variable coefficients hierarchies, including for instance those constructed in [17,18] and [2], cf. Example 2 below.…”

“…It is therefore natural to ask [11] whether a weakly nonlocal hereditary operator will always produce a local hierarchy, as in the earlier work [3], [11]- [16] one always had to require the existence of some nontrivial additional structures (e.g., the scaling symmetry [11,15,16] or bihamiltonian structure [3,12]) in order to get the proof of locality through. We show that this is not necessary: Theorem 1 below states that if for a normal weakly nonlocal hereditary recursion operator R the Lie derivative L Q (R) of R along a local symmetry Q vanishes 1 and R(Q) is local then R j (Q) are local for all j = 2, 3, .…”

“…Amazingly enough, the existence of a scaling symmetry shared by R and Q enables us to avoid the cumbersome direct verification of whether R is hereditary and allows to prove locality and commutativity of the corresponding hierarchy in a very simple and straightforward manner, as shown in Proposition 3 and Corollary 3 below. This is in a sense reminiscent of the construction of compatible Hamiltonian operators via infinitesimal deformations in Smirnov [19] (see also [20] and references therein) and is quite different from the approach of [11], where both R being hereditary and existence of scaling symmetry were required ab initio.…”