The complex 2-sphere in ℂ3 and Schrödinger flows

Abstract

By using holomorphic Riemannian geometry in $\mathbb~C^3$, the coupled Landau-Lifshitz (CLL) equation is proved to be exactly the equation of Schrodinger flows from $\mathbb~R^1$ to the complex 2-sphere $\mathbb~C~\mathbb~S^2(1)\hookrightarrow\mathbb~C^3$. Furthermore, regarded as a model of moving complexcurves in $\mathbb~C^3$, the CLL equation is shown to preserve the $\mathcal{P}\mathcal{T}$ symmetry if the initial data is of the $\mathcal{P}$ symmetry. As a consequence, the nonlocal nonlinear Schrodinger (NNLS) equation proposed recently by Ablowitz and Musslimani is proved to be gauge equivalent to the CLL equation with initial data being restricted by the $\mathcal{P}$ symmetry. This gives an accurate characterization of the gauge-equivalent magnetic structure of the NNLS equation described roughly by Gadzhimuradov and Agalarov (2016).