This article is cited in 2 papers

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

B. A. Dubrovin^ab, T. V. Skrypnik<sup>acd

a</sup> Lomonosov Moscow State University, Moscow, Russia
^b International School for Advanced Studies, Trieste, Italy
^c Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
^d Universita di Milano Bicocca, Milan, Italy

Abstract: Using the classical double $\mathcal G$ of a Lie algebra $\mathfrak g$ equipped with the classical $R$-operator, we define two sets of functions commuting with respect to the initial Lie–Poisson bracket on $\mathfrak g^*$ and its extensions. We consider examples of Lie algebras $\mathfrak g$ with the “Adler–Kostant–Symes” $R$-operators and the two corresponding sets of mutually commuting functions in detail. Using the constructed commutative Hamiltonian flows on different extensions of $\mathfrak g$, we obtain zero-curvature equations with $\mathfrak g$-valued $U$–$V$ pairs. The so-called negative flows of soliton hierarchies are among such equations. We illustrate the proposed approach with examples of two-dimensional Abelian and non-Abelian Toda field equations.

Keywords: classical $R$-operator, integrable hierarchy.

Received: 28.04.2011
Revised: 13.11.2011

DOI: 10.4213/tmf6903

 English version: Theoretical and Mathematical Physics, 2012, 172:1, 911–931

Bibliographic databases:

• 
• 
• 
• 
• 
•