The complex 2-sphere in ℂ3 and Schrödinge flows

  • Qing DingEmail author
  • Shiping Zhong


By using holomorphic Riemannian geometry in ℂ3, the coupled Landau-Lifshitz equation (CLL) is proved to be exactly the equation of Schrödinger flows from ℝ1 to the complex 2-sphere ℂS2(1) ↪ ℂ3. Furthermore, regarded as a model of moving complex curves in ℂ3, 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).


\(\mathcal{PT}\) symmetry holomorphic Riemannian manifold gauge equivalence 


53C44 53C56 53A04 35Q60 


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This work was supported by National Natural Science Foundation of China (Grant No. 11271073).


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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiChina

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