Geometric realizations of the Fordy–Kulish nonlinear Schrödinger systems

A method of Sym and Pohlmeyer, which produces geometric re-alizations 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 arclength-parametrized curves evolving in a Lie alge-bra, generalizing the localized induction model of vortex filament mo-tion. A natural Frenet theory for such curves is formulated, and the general correspondence between curve evolution and natural curva-ture 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 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 certain features of the latter