Geometric realizations of Fordy–Kulish nonlinear Schrödinger systems

Abstract

A method of Sym and Pohlmeyer, which produces geometric realizations of many integrable systems, is applied to the Fordy-Kulish generalized non-linear Schrodinger 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. ∗Dept. of Mathematics, Case Western Reserve University, Cleveland OH 44106 email: jxl6@po.cwru.edu †Dept. of Mathematics and Comp. Sci., Drexel University, Philadelphia PA 19104 email: rperline@mcs.drexel.edu Mathematics Subject Classification: 58F07, 35Q55