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 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 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.

Authors