本项目旨在研究旋转偶极化旋量玻色-爱因斯坦凝聚体的基态解与动力学的数学理论及其相关数值方法，并借此发现一些新的物理现象。由于旋转角动量、长程且有奇异性的偶极作用项(DDI)和自旋作用项三项耦合在一起，使对应的模型变得复杂，给理论分析和数值方法的设计都带来新的挑战。如何处理旋转项和DDI是算法设计成功与否的关键所在。本项目将基于Gross-Pitaevskii方程组，拟从以下三方面展开研究：1.利用PDE相关分析工具来研究基态解与动力学的性质。2.设计一类卷积核截断方法(KTM)来正则化DDI的奇性，进而结合卷积定理与快速傅立叶变换来求解DDI。更进一步，结合KTM与黎曼流形上的优化方法，构造一类预处理共轭梯度流--KTM方法来计算CGPE的基态解。3.设计相应的旋转拉格朗日坐标变换技巧来消去CGPE中的旋转项，并结合KTM方法，构造一类两步紧时间分裂谱方法来模拟CGPE的动力学演化。

This project is to investigate the mathematical theory and numerical methods for the ground states and dynamics of rotating dipolar spinor Bose-Einstein condensates. Due to the coupling of the rotating angular momentum, the long-range singular dipole-dipole interaction (DDI) and the spin-spin interaction, the governed model becomes complicated, which brings significant challenge for both mathematical analysis and design of numerical methods. The key points of the numerical methods lie on the proper treatments of the rotational term and DDI. Based on the coupled Gross-Pitaevskii equations (CGPE), we will carry out studies via the following three aspects: (1). Utilizing PDE analyzing tools to study the properties of the ground states and dynamics. (2). Develop a kernel truncation method to regularize the DDI, based on which the DDI can be solved out efficiently via the convolution theory and fast Fourier transform. Then, combine KTM with the optimization method on Riemannian manifolds, we design a preconditioned conjugated gradient--KTM method to compute the ground states. (3). We will develop a rotating Lagrangian coordinate transform technique to eliminate the rotating term. Based on this and the KTM, we propose a two-step compact time splitting spectral method to simulate the dynamics of the rotating dipolar spinor BEC.