Curve motion inducing modified Korteweg-de Vries systems

Abstract

A straightforward vector generalization of the modified Korteweg-deVries equation is shown to be intimately related to the geometry of curves in n -dimensional Euclidean spaces and spheres. This mKdV system, which is coupled in a particularly a simple way, describes the dynamics of the natural curvature vector of a unit speed curve subject to an elementary geometric evolution equation. The underlying structure of these equations is related to generalizations of the nonlinear Schrödinger and localized induction equations in the context of Hermitian symmetric Lie algebras.