GEOMETRIC KdV FLOWS, MOTIONS OF CURVES AND THE THIRD-ORDER SYSTEM OF THE AKNS HIERARCHY

Abstract

By applying the concept of geometric KdV flows into Kähler and para-Kähler manifolds, we give a unified geometric interpretation for the four typical integrable equations in the third-order system of the AKNS hierarchy. That is: the KdV equation, the mKdV equation, the complex mKdV- equation and the complex mKdV+ equation are, respectively, equivalent to the first kind reduction, the second kind reduction of the geometric KdV flow of maps from R1 to the de Sitter two-space S1,1 ↪ R2,1 (regarded as a para-Kähler manifold), the geometric KdV flow of maps from R1 to the hyperbolic two-space H2 ↪ R2,1 (regarded as a Kähler manifold of noncompact type) and the geometric KdV flow of maps from R1 to the two-sphere S2 ↪ R3 (regarded as a Kähler manifold of compact type). This is an application of the general geometric KdV flows.