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

Abstract

It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually all known today recursion operators and are much easier to verify than those found in earlier work.We also give explicit formulas for the nonlocal parts of higher recursion, Hamiltonian and symplectic operators of integrable systems in (1+1) dimensions.Using these two results we prove, under some natural assumptions, the Maltsev-Novikov conjecture stating that higher Hamiltonian, symplectic and recursion operators of integrable systems in (1+1) dimensions are weakly nonlocal, i.e., the coefficients of these operators are local and these operators contain at most one integration operator in each term.

Curve flows in Lagrange–Finsler geometry, bi-Hamiltonian structures and solitons

“…For N-anholonomic spaces with dimensions n = m, we have T(γ τ , γ l ) = 0 and R(γ τ , γ l )e defined by constant, or vanishing, d-curvature coefficients (see discussions related to formulas (75) and (70)). For such cases, we can consider the h-and v-components of (34) and (35) in a similar manner as for symmetric Riemannian spaces but with the canonical d-connection instead of the Levi Civita one. One obtains, respectively,…”

Invertible Coupled KdV and Coupled Harry Dym Hierarchies

“…Moreover, all symplectic operators of the invertible cHD hierarchy are nonlocal of first or third order. The main tool for our considerations was a generalization of formulae (7) and (8) from [9], that is, Theorems 2 and 3 above.…”

“…Here and below I N −1 stands for the (N − 1) × (N − 1) unit matrix. Note also that (9) implies that the operator R † adjoint to R has the form…”

“…One proves this theorem by induction. A special case of this theorem (for n = 1) can be found in [9], but it includes the erroneous coefficients ( s − 1 j ) that are not present in the correct version of the formula.…”

Recursion Operator of the Narita–Itoh–Bogoyavlensky Lattice

“…One important question is whether the operator is guaranteed to generate local symmetries starting from a proper seed. Sufficient conditions for weakly nonlocal [31] Nijenhuis differential operators are formulated in [23, 32], which are also valid for weakly nonlocal Nijenhuis difference operators [12]. This result is generalized to Nijenhuis operators, which are the product of weakly nonlocal Hamiltonian and symplectic operators [20].…”