Integrable Equations Arising from Motions of Plane Curves. II

Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces

“…That is, if a flow is invariant under the action of the group (G = O(n) in the Euclidean case and G = PSL(n, R) in the projective case) the question of integrability could be shifted to a question of integrability of the flow of the differential invariants (what we call the invariantization of the flow) and, with it, to the construction and study of compatible Hamiltonian structures in this space. Several integrable systems have been obtained this way by looking at the evolution of the differential invariants of curves in different geometries and by trying to find known completely integrable systems among them (see [SW], [KQ1], [KQ2] and the references within) although there was no study of the Hamiltonian structures themselves or of the relationship of the Hamiltonian structures to geometry.…”

Poisson structures for geometric curve flows in semi-simple homogeneous spaces

“…serve as a basis for the Lie algebra g = sl (2). Using the commutation relations, we find that every ad * -invariant symmetric 2 tensor is a scalar multiple of the Killing form:…”

Geometric Hamiltonian Structures on Flat Semisimple Homogeneous Manifolds

“…As far as the author knowns, except for the planar case, conformal and the general case are still largely open. Integrable systems in the planar case have been classified in [CQ1], [CQ2], although they have not been linked to this Hamiltonian structures yet. One of the main obstacles to the swift application of these results is the identification of possible reductions and non-degenerate brackets among the reduced ones.…”

“…All best known Poisson brackets used in the integration of PDEs can be obtained in this geometric way. Except for the planar case ([CQ1], [CQ2]), integrable systems associated to conformal and other geometries are largely unexplored. Their study is likely to shed light on the geometry of curves itself.…”