Using the classical double G of a Lie algebra gequipped with the classical R-operator, we define two sets of functions commuting with respect to the initial Lie-Poisson bracket on g* and its extensions. We consider examples of Lie algebras 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 g, we obtain zero-curvature equations with 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.

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

Dubrovin, Boris;
2012

Abstract

Using the classical double G of a Lie algebra gequipped with the classical R-operator, we define two sets of functions commuting with respect to the initial Lie-Poisson bracket on g* and its extensions. We consider examples of Lie algebras 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 g, we obtain zero-curvature equations with 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.
172
1
911
931
https://arxiv.org/abs/1011.4894
Dubrovin, Boris; Skrypnyk, T.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/14319
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