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 arclength-parametrized 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 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.
