非线性Gross-Pitaevskii(GP)方程组是描述自旋轨道耦合玻色-爱因斯坦凝聚(BEC: Bose-Einstein condensate)的重要数学模型。对自旋轨道耦合的非线性GP方程组的数值方法研究，不仅对冷原子物理研究有着重要意义，而且在非线性光学、量子计算机、新型材料、量子相对论等领域都具有重要的应用价值。项目研究自旋轨道耦合的非线性GP方程组的数值求解方法以及探索自旋轨道耦合BEC的基态和动力学行为。结合自旋轨道耦合BEC的物理背景与有限元方法、网格自适应技术等，发展强健、高效、快速的有限元数值方法，使得这类数值方法具有高精度和很好的稳定性，以便更好地适应复杂计算区域和非光滑外势作用下的数值模拟。开展数值方法守恒性、收敛性的研究。基于数值模拟和理论分析，研究粒子自旋、外势、原子间相互作用等因素对自旋轨道耦合凝聚体的影响和量子效应。

The nonlinear Gross-Pitaevskii (GP) system is an important mathematical model to describe the spin-orbit coupled Bose Einstein condensate (BEC: Bose-Einstein condensate). Research on numerical methods for the nonlinear GP system with spin-orbit coupling not only has the important significance to cold atom physics, but also has the valuable application in nonlinear optics, quantum computer, new material, quantum-relativistic field. In this project, numerical methods for solving the nonlinear GP system with spin-orbit coupling are researched, then ground states and dynamical behavior of the spin-orbit coupled BEC are explored. Combining the physical background of spin-orbit coupled BEC with the finite element method and the adaptive technique, we will propose fast, robust and efficient numerical methods. The properties of the numerical method such as high accuracy, the better stability and conservation of physical quantity can facilitate the adaptation to the complex geometry and non-smooth external potential. We also carries out the research on convergence and conservation law for numerical methods. Based on the numerical simulation and theoretical analysis, we can analyze influences on condensates by the external potential, the particle spin, the interaction between atoms, and explore quantum effects of the spin-orbit coupled BEC.