Classical Poisson structures and r-matrices from constrained flows

Details Classical Poisson structures and r-matrices from constrained flows Classical Poisson structures and r-matrices from constrained flows

1995-09-14

arxiv.org/abs/solv-int/9509005v1

Mathematics

Quantum Algebra

16 pages in LaTeX

Scientific paper

10.1088/0305-4470/29/16/038

We construct the classical Poisson structure and $r$-matrix for some finite dimensional integrable Hamiltonian systems obtained by constraining the flows of soliton equations in a certain way. This approach allows one to produce new kinds of classical, dynamical Yang-Baxter structures. To illustrate the method we present the $r$-matrices associated with the constrained flows of the Kaup-Newell, KdV, AKNS, WKI and TG hierarchies, all generated by a 2-dimensional eigenvalue problem. Some of the obtained $r$-matrices depend only on the spectral parameters, but others depend also on the dynamical variables. For consistency they have to obey a classical Yang-Baxter-type equation, possibly with dynamical extra terms.

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