Bi-Hamiltonian flows and their realizations as curves in real semisimple homogeneous manifolds

Abstract

We describe a reduction process that allows us to define Hamiltonian structures on the manifold of differential invariants of parametrized curves for any homogeneous manifold of the form G/H, with G semisimple. We also prove that equations that are Hamiltonian with respect to the first of these reduced brackets automatically have a geometric realization as an invariant flow of curves in G/H. This result applies to some well-known completely integrable systems. We study in detail the Hamiltonian structures associated to the sphere SO(n + 1)/SO(n).

Integrable Flows for Starlike Curves in Centroaffine Space

“…The proof of the reduction of these brackets can be found in [8], where P is linked to the Adler-Gel'fand-Dikii bracket for sl (3) and Q is associated to its companion. The brackets were later linked to curve evolutions and differential invariants in [15], where more details are available.…”

Integrable systems associated to curves in flat Galilean and Minkowski spaces

“…Results in [5] show that, if dim(M) ¼ m, then there exists a generating set A with m functionally independent differential invariants. If G is a matrix group, we can write the horizontal component of the moving coframe as K ¼ À1 x .…”

“…The reduction is constructive and can be found explicitly, as we will see below. For more information see [4] or [5]. The reduction method applies to the Minkowski case but not the Galilean case, because G is not semisimple.…”

“…[1][2][3][4][5][6][7][8] and references within). These are evolutions of curves on a homogeneous geometric manifold, invariant under the action of a group of geometric transformations, and such that it becomes the integrable system when written in terms of the invariants of the flow.…”

Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

“…The Hasimoto transformation has been generalized in [51] to the Riemannian manifold with constant curvature, which is used to obtain the corresponding integrable equations associated with the invariant non-stretching curve flows. The parallel frames and other kinds of frames are also used to derive bi-Hamiltonian operators and associated hierarchies of multi-component soliton equations from non-stretching curve flows on Lie group manifolds [3,4,31,39]. The KdV equation, the modified KdV equation, the Sawada-Kotera equation and the Kaup-Kuperschmidt equation were shown to arise from the invariant curve flows respectively in centro-equiaffine geometry [7,9,48], Euclidean geometry [21], special affine geometry [11,35] and projective geometries [11,30,41].…”