an:07184428
Zbl 1435.53074
Ding, Qing; Zhong, Shiping
The complex 2-sphere in \(\mathbb{C}^3\) and Schr??dinger flows
EN
Sci. China, Math. 63, No. 4, 777-788 (2020).
00448213
2020
j
53E99 53C56 53A04 35Q60 34L40
\(\mathcal{PT}\) symmetry; holomorphic Riemannian manifold; gauge equivalence
Summary: By using holomorphic Riemannian geometry in \(\mathbb{C}^3\), the coupled Landau-Lifshitz equation (CLL) is proved to be exactly the equation of Schr??dinger 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 Schr??dinger 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).