Integrable motions of curves in \(S^{1} \times \mathbb R\) (Q2373641)

The motion of non-stretching curves and its connections with integrable systems have been studied since more than 30 years. The authors follow recent papers by the second author and his collaborators and derive in a systematic way equations of motion of non-stretching curves in \(S^1 \times \mathbb R\) equipped with some Klein geometry. The Klein geometries are characterized by associated Lie algebras of vector fields which generate the isometry groups. In particular, for any Klein geometry one can define and compute differential invariants and invariant one-forms. In the considered case these invariants can be reduced to arc-length and curvature. The non-stretching condition means that the arc length parameterization commutes with the time evolution. The authors choose to consider 11 Lie algebras (or rather 10, because one of them seems to be somehow forgotten), all of them containing the field \(\partial_\theta\) (where \(\theta \in S^1\)), i.e., the considered geometries are invariant with respect to translation along \(S^1\). As main result of this paper it is shown that the time evolution of non-stretching curves in the considered Klein geometries is expressed through \(1+1\)-dimensional integrable hierarchies such as: KdV, modKdV, ``defocusing'' modKdV, Sawada-Kotera, Burgers (recursion operators are a natural outcome of the computations). It is interesting that most of the hierarchies appear in more than one geometry. However, this point is not discussed. As an important open problem the authors suggest to perform a complete classification of all Lie algebras of vector fields on \(S^1 \times \mathbb R\).