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Published in:
01-11-2023
Magnetic and electric flowlines of the cmKDV+ motion of curves
Published in: Optical and Quantum Electronics | Issue 12/2023
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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 \({\mathbb {S}}^{2}.\) We first introduce novel complex quantities following the Lamb formalism to investigate a moving space curve evolution on \({\mathbb {S}} ^{2}\) whose Hasimoto transformation yields cmKdV\(^{+}\) type identities. Then we focus on the geometric features of cmKdV\(^{+}\) surfaces on \({\mathbb {S}}^{2}\) 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 \({\mathbb {S}}^{2}.\) 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 \({\mathbb {S}}^{2}.\)