Classical double, R-operators and negative flows of integrable hierarchies

TL;DR

This paper explores the use of classical doubles and R-operators to construct commuting functions and zero-curvature equations, including negative flows, in integrable hierarchies, with applications to Toda field equations.

Contribution

It introduces a method to generate mutually commuting functions and zero-curvature equations using classical doubles and R-operators, extending integrable hierarchies.

Findings

Constructed two sets of commuting functions for Lie algebras with R-operators.

Derived zero-curvature equations including negative flows.

Applied framework to Toda field equations, both abelian and non-abelian.

Abstract

Using classical double G of a Lie algebra g equipped with a classical R-operator we define two sets of mutually commuting functions with respect to the initial Lie-Poisson bracket on g* and its extensions. We consider in details examples of the Lie algebras g with the "Adler--Kostant--Symes" R-operators and the corresponding two sets of mutually commuting functions. Using the constructed commutative hamiltonian flows on different extensions of g we obtain zero-curvature equations with g-valued U-V pairs. Among such the equations are so-called "negative flows" of soliton hierarchies. We illlustrate our approach by examples of abelian and non-abelian Toda field equations.