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arXiv · 1011.4894

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

B. Dubrovin·T. Skrypnyk
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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.

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B. Dubrovin, T. Skrypnyk. 2010-11-22. Classical double, R-operators and negative flows of integrable hierarchies. https://arxiv.org/abs/1011.4894

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