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Two-dimensional anisotropic vortex–bright soliton and its dynamics in dipolar Bose–Einstein condensates in optical lattice

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Abstract

We study construction and dynamics of two-dimensional (2D) anisotropic vortex–bright (VB) soliton in spinor dipolar Bose–Einstein condensates confined in a 2D optical lattice (OL), with two localized components linearly mixed by the spin–orbit coupling and long-range dipole–dipole interaction (DDI). It is found that the OL and DDI can support stable anisotropic VB soliton in the present setting for arbitrarily small value of norm N. We then present a new method via examining the mean square error of norm share of bright component to implement stability analysis. It is revealed that one can control the stability of anisotropic VB soliton only by adjusting OL depth for a fixed DDI. In addition, the dynamics of the anisotropic VB soliton was studied by applying the kick to them. The mobility of the single kicked VB soliton is Rabbi-like oscillation. However, for the collision dynamics of two kicked anisotropic VB solitons, their properties mainly depend on their initial distance and OL, and they can realize the transition from the bright component to the vortex component. Our work may provide a convenient way to prepare and manipulate anisotropic VB soliton in high-dimensional space.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No.12175315, the Hunan Provincial Natural Science Foundation under Grants No. 2019JJ30044, the Scientific Research Fund of Hunan Provincial Education Department under Grant No. 19A510 and the Talent project of Central South University of Forestry and Technology under Grant No. 2017YJ035.

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  1. School of Life science and technology, Central South University of Forestry and Technology, Changsha, 410004, China

    Zhi Tan, Huilin Gong & Honghua Zhong

  2. Institute of Mathematics and Physics, Central South University of Forestry and Technology, Changsha, 410004, China

    Zhi Tan, Huilin Gong, Bo Zhu, Honghua Zhong & Shufang Hu

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Appendix A: Equivalence of the mean square errors of \(F_{1}\) and \(F_{2}\)

Appendix A: Equivalence of the mean square errors of \(F_{1}\) and \(F_{2}\)

In Eq. (13), we use the mean square error of norm share of bright component as a criterion to implement stability analysis. Here, we will prove the mean square errors of \(F_{1}\) and \(F_{2}\) are equivalent. According to the definition, the mean square error of norm share of vortex component defined as

$$\begin{aligned} \langle \Delta ^{2} F_{2}(t)\rangle =\langle F_{2}^{2}(t) \rangle -\overline{F}_{2}^{2}, \end{aligned}$$
(A1)

where the time-dependent norm share of the vortex component \(F_{2}(t)={\int \int |\psi _{2}(x, y, t)|^{2}dxdy}/N\), the average value of the norm share of the vortex component in the evolution \(\overline{F}_{2}=\langle F_{2}(t) \rangle \) and the variance \(\Delta F_{2}(t)=F_{2}(t)-\overline{F}_{2}\). Because our system is close, we have \(F_{1}(t)+F_{2}(t)= 1\) and \(\overline{F}_{1}+\overline{F}_{2}=1\). Combining the Eq. (13), we can obtain

$$\begin{aligned} \Delta F_{1}(t)= & {} F_{1}(t)-\overline{F}_{1} \nonumber \\= & {} 1-F_{2}(t)-(1-\overline{F}_{2}) \nonumber \\= & {} \overline{F}_{2}-F_{2}(t) \nonumber \\= & {} -\Delta F_{2}(t) \end{aligned}$$
(A2)

Obviously, the mean square errors of \(F_{1}\) and \(F_{2}\) are equivalent, \(\langle \Delta ^{2}F_{1}(t) \rangle = \langle \Delta ^{2}F_{2}(t) \rangle \).

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Tan, Z., Gong, H., Zhu, B. et al. Two-dimensional anisotropic vortex–bright soliton and its dynamics in dipolar Bose–Einstein condensates in optical lattice. Nonlinear Dyn 111, 9467–9476 (2023). https://doi.org/10.1007/s11071-023-08335-8

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