Negative flows of integrable hierarchies, dual $R$-matrix integrability and classical double.

Abstract:

We present a general approach of the construction of the so-called ``negative flows'' of the hierarchies of 1+1 integrable equations basing on the infinite-dimensional Lie algebras \mathfrak{g} possessing classical R-operators. For this purpose we construct infinite number of commuting hamiltonian flows on \mathfrak{g}^* and its extensions using the theory of classical double. We illustrate this approach by several examples, including Thirring equation, finite and infinite-component Toda equations.