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The complex 2-sphere in $$\mathbb{C}^3$$ and Schrödinger flows. (English) Zbl 1435.53074
Summary: By using holomorphic Riemannian geometry in $$\mathbb{C}^3$$, the coupled Landau-Lifshitz equation (CLL) is proved to be exactly the equation of Schrödinger flows from $$\mathbb{R}^1$$ to the complex 2-sphere $$\mathbb{C} S^2(1) \hookrightarrow \mathbb{C}^3$$. Furthermore, regarded as a model of moving complex curves in $$\mathbb{C}^3$$, CLL is shown to preserve the $$\mathcal{PT}$$ symmetry if the initial data is of the $$\mathcal{P}$$ symmetry. As a consequence, the nonlocal nonlinear Schrödinger equation (NNLS) proposed recently by Ablowitz and Musslimani is proved to be gauge equivalent to CLL with initial data being restricted by the $$\mathcal{P}$$ symmetry. This gives an accurate characterization of the gauge-equivalent magnetic structure of NNLS described roughly by Gadzhimuradov and Agalarov (2016).
##### MSC:
 53E99 Geometric evolution equations 53C56 Other complex differential geometry 53A04 Curves in Euclidean and related spaces 35Q60 PDEs in connection with optics and electromagnetic theory 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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