Magnetic and electric flowlines of the cmKDV+ motion of curves

Abstract

In this paper, we study the motion of curves governed by the complex modified Korteweg de Vries (cmKdV) equation on the unit 2-sphere S-2. We first introduce novel complex quantities following the Lamb formalism to investigate a moving space curve evolution on S-2 whose Hasimoto transformation yields cmKdV(+) type identities. Then we focus on the geometric features of cmKdV(+) surfaces on S-2 and derive the Gaussian and mean curvatures of cmKdV(+) surfaces in terms of geometric invariants. Next, we define a cmKdV(+) magnetic curve of an associated magnetic field and derive its Killing field thanks to the variations of the complex functions for space curves on S-2. Finally, we determine the rotation of the polarization state plane of the electric field whose flow is governed by the cmKdV+ motion along the optical fiber on S-2.