Geometric realizations of Fordy–Kulish nonlinear Schrödinger systems

Abstract

A method of Sym and Pohlmeyer, which produces geometric realizations of many integrable systems, is applied to the Fordy-Kulish generalized non-linear Schrödinger systems associated with Hermitian symmetric spaces. The resulting geometric equations correspond to distinguished arclengthparametrized curves evolving in a Lie algebra, generalizing the localized induction model of vortex filament motion. A natural Frenet theory for such curves is formulated, and the general correspondence between curve evolution and natural curvature evolution is analyzed by means of a geometric recursion operator. An appropriate specialization in the context of the symmetric space SO(p + 2)/SO(p) × SO(2) yields evolution equations for curves in R p+1 and S p , with natural curvatures satisfying a generalized mKdV system. This example is related to recent constructions of Doliwa and Santini and illuminates certain features of the latter. 157

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

“…Furthermore, it is was pointed out ( [LP1], [LP2]) that one of the Hamiltonian structures used to integrate the NLS flow could be defined using the Euclidean geometry of the flow itself. This situation was also known to be true for a family of projective flows linked to KdV type equations ( [DS], [M4]).…”

Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems

“…A similar case can perhaps be made for Schrödinger flows, mKdV and sine-Gordon flows as linked to Riemannian geometry. As we said before, several authors [1,25,26,28,29,38,46,48,49], have described geometric realizations of these evolutions on manifolds that have what amounts to be a classical natural moving frame, i.e., a frame whose derivatives of non tangential vectors have a tangential direction. This frame appears in Riemannian manifolds and is generated by the action of the group in first order frames.…”

“…In the last years many examples of geometric realizations for most known completely integrable systems have been appearing in the literature. Some are linked to the geometric invariants of the flow (see for example [1,2,11,13,17,25,26,28,29,31,36,38,39,46,47,48,49,51]). This list is, by no means, exhaustive as this paper is not meant to be an exhaustive review of the subject.…”

Geometric Hamiltonian Structures on Flat Semisimple Homogeneous Manifolds

“…In particular, the evolution of curvature and torsion was biHamiltonian. Langer and Perline pointed out in their papers on the subject (see [LP1], [LP2]) that the Hamiltonian structures that were used to integrate some of these systems were defined directly from the Euclidean geometry of the flow. This situation was known to exist not only in Riemannian geometry but also in projective geometry.…”