Perline R., Geometric realizations of Fordy–Kulish nonlinear Schrödinger systems

A method of Sym and Pohlmeyer, which produces geo-metric realizations of many integrable systems, is applied to the Fordy–Kulish generalized non-linear Schrödinger systems as-sociated with Hermitian symmetric spaces. The resulting geometric equations correspond to distinguished arclength-parametrized curves evolving in a Lie algebra, generalizing the localized induction model of vortex filament motion. A natu-ral 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 symmet-ric space SO(p+2)/SO(p)×SO(2) yields evolution equations for curves in Rp+1 and Sp, with natural curvatures satisfying a generalized mKdV system. This example is related to recent constructions of Doliwa and Santini and illuminates certai