Geometric realizations of Fordy-Kulish nonlinear Schrodinger systems

A method of Sym and Pohlmeyer, which produces geometric realizations of man y integrable systems, is applied to the Fordy-Kulish generalized non-linear Schrodinger systems associated with Hermitian symmetric spaces. The result ing geometric equations correspond to distinguished arclength-parametrized curves evolving in a Lie algebra, generalizing the localized induction mode l of vortex lament motion. A natural Frenet theory for such curves is formu lated, and the general correspondence between curve evolution and natural c urvature 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) x SO(2) yields evolution equations for curves in Rp+1 and S-p, wit h natural curvatures satisfying a generalized mKdV system. This example is related to recent constructions of Doliwa and Santini and illuminates certa in features of the latter.