本论文采用解析和数值的方法对多组分玻色爱因斯坦凝聚主要是spin-1 BEC系统和两组分 BEC系统的拓扑激发进行研究。当玻色爱因斯坦凝聚囚禁在光势阱里，自旋的方向由于粒子间相互作用可以动态改变，因而产生了丰富变化的自旋格局。解析解对于研究拓扑激发具有很好的指导意义，但是由于旋量系统非线性密度-密度相互作用以及自旋-自旋相互作用的存在，精确求解BEC的耦合GP方程，尤其是二维和三维系统将会碰到很多问题。我们研究了一种方法来具体求解spin-1 BEC非线性GP方程的严格解，从而分析其拓扑结构。由于大部分系统是不能够给出非线性GP方程的严格解，我们利用数值虚时演化的方法探索了拓扑结构(knot和winding hopfion)在两组分玻色爱因斯坦凝聚系统的稳定性问题，并设计和研究了一种新型的三维拓扑结构3D dimeron。本文在第一章中简要的回顾了玻色-爱因斯坦凝聚现象的理论背景与最新实验，介绍了多组分玻色-爱因斯坦凝聚的相关理论与研究进展，并对冷原子系统中常见的拓扑激发作了简单介绍，包括利用拓扑学主要是同伦群理论进行拓扑分类。第二章至第五章则对应于我们的几个主要研究成果，具体介绍如下：（一）我们主要研究一维自旋轨道耦合作用对spin-1 BEC凝聚态波函数的影响和调制。从含有自旋轨道耦合的GP方程出发，我们导出了严格的定态解。孤子解在自旋轨道耦合的影响下变成了d矢量随空间变化的polar孤子解。虽然自旋轨道耦合体系违反伽利略不变性，但是我们给出了一种新奇的运动孤子解。我们通过数值的虚时演化发现这定态解是系统的基态解。这特殊的结构源于自旋轨道耦合下参数空间的螺旋调制。在本部分里，我们最主要的结论是得出自旋轨道耦合下的polar孤子解进行实时演化，将会变成FM和polar相的混合演化。而之前关于spin-1一维得出的都是d矢量平庸的解，因此没有混合演化。而自旋轨道耦合能产生d矢量随空间变化的解，根据d矢量的演化公式一般会产生铁磁项F矢量，d矢量和F矢量的相互竞争，产生了FM 和polar相的混合演化。因此，自旋轨道耦合的spin-1 BEC 体系为观察混合演化提供了一个很好平台。（二）我们利用分离变量的方法将二维和三维的spin-1 BEC的耦合方程解耦为一系列一维方程组，然后利用雅可比椭圆函数或双曲函数的性质，给出了满足非线性方程的孤子解。我们详细的研究了这些解对应的拓扑结构及其性质。这里我们主要研究了3种情况：1. 忽略非线性自旋-自旋相互作用下$|\textbf{F}|\neq 0$的二维解，对应的拓扑结构有FM-core 涡旋，half-skyrmion。2. 利用spin-1 polar态的性质解耦非线性自旋-自旋相互作用得到的二维polar解，对应的拓扑结构有多种类型的half-skyrmion.3. 在二维polar解的基础上，得出三维polar解。据我们所知这是第一次在笛卡尔坐标系下给出spin-1 BEC 体系的三维孤子解(在z方向上是周期的)。我们发现这类孤子解具有特殊的拓扑结构，这种结构向半球映射，具有半整数的hopf 数，因此我们称之为half-knot。（三）在两组分玻色-爱因斯坦凝聚，我们首先在不考虑非线性相互作用下给出了knot和孤子(winding hopfion)的解析解，然后我们加上非线性相互作用，利用数值虚时演化的方法去探究在什么条件下这些结构是稳定的。（四）在两组分玻色-爱因斯坦凝聚，我们设想构建了一种新型的三维拓扑结构，它不同于之前人们广泛研究的3D skyrmion。这种结构由两个涡旋环组成，并且涡旋环由一条奇异的弦来连接，从而组成了三维涡旋分子。因为在每个二维的竖平面都是2D bimeron结构，因此，我们将这种新型结构称为 3D dimeron。通过在不同体系下，利用数值方法对这种结构的稳定性研究，我们发现在耦合非阿贝尔规范体系下，这种拓扑结构可以自发形成。最后在第六章中，我们对本论文的研究结果做了详细的总结和对后续研究工作的做了展望。

In this paper we use both analytical and numerical methods to explore topological excitations of multi-component Bose-Einstein condensations (BECs), mainly about the spin-1 BEC and the two-component BEC. When the BEC is confined in an optical trap, the direction of the spin can change dynamically due to the inter-particle interactions, leading to a rich variety of spin textures. The analytical solution have a good guide for the topological excitations. However, due to the nonlinear density-density interactions and spin-spin interactions in spinor systems, it is difficult to analytically solve the coupled Gross-Pitaevskii equations(GPE), especially in the two-dimensional and three-dimensional systems. We invent a method to solve the non-linear GPE of the spin-1 BEC exactly, from which we can analyze the topological structures. As it is impossible to analytically solve the non-linear GPEs for most systems, we employ the imaginary time evolving method to examine the stability of the knot and winding hopfion in the two-component BEC. We also design and study a new kind of three-dimensional topological structure named 3D dimeron.In this paper, we make a brief review of theoretical background of BEC phenomena and new experiments in the first chapter, as well as an introduction of related theories and progress in multi-component BEC.We also study the general topological excitations in the cold atom system, including the homotopy theory of the topology to classify the topological excitations.The second chapter to chapter V corresponding to our several main research results are detailed below:Firstly, we study the condensed wave function of the spin-1 BEC under the effects and modulation of one-dimensional spin-orbit coupling (SOC). From the GPE with SOC, we derive the exact solution. The soliton solutions become a polar soliton with the d vector changed under the effects of SOC. Galilean invariance is violated for a BEC with SOC, but we get a novel moving soliton. The static polar soliton is shown to be the ground state by the imaginary-time evolution method. This special structure is due to the helical modulation of the order parameter in the presence of SOC. In this section, our main conclusion is the polar state will evolve into a state of mixed FM and polar manifolds. For a conventional BEC without SOC, the one-dimensional solution correspond to a trivial solution for d vector and there are no mixed manifolds. According to the evolution equation, the d vector can be changed under the effects of SOC and the polar state with well-defined d-vector will induce F-vector. The dynamical evolution of F-vector forms a competitionwith the d-vector, producing the mixed FM and polar manifolds. The spin-1 BEC with SOC is the ideal testing ground for investigating manifold mixing dynamics.Secondly, we use the method of separation of variables to decouple the GPE of the 2D and 3D spin-1 BEC into independently 1D equations. Depending on the properties of Jacobi elliptic functions or hyperbolic functions, we can get the soliton solutions. We detailed study the topological structure and properties of these solutions, mainly including the following three cases:1. Ignore the spin-spin interactions, we get the 2D solutions corresponding to $|\textbf{F}|\neq 0$. These solutions have the structure of FM-core vortex and half-skyrmion.2. Using the properties of polar states, we decouple the spin-spin interactions and get 2D polar solutions. These solutions have the structure of various half-skyrmion.3. On the basis of the 2D polar solution, we get the 3D polar solution. To our knowledge, this is the first analytical result to coupled nonlinear equations in 3D Cartesian coordinates systems(periodical in the z direction). We demonstrate it is a special topological structure with half-integer hopf number, so we call it half-knot.Thirdly, we first give the knot and winding hopfion solution without considering the nonlinear interactions in the two-component BEC. Then add the nonlinear interactions, and adopt the numerical method to explore the stable condition for these structures.Fourth, we design a new type of 3D topological structure. It is different from the conventional 3D skyrmion. The structure host two interlocked vortex-rings connected by a singular string and the complexity constitutes a 3D vortex-molecule. Generally, there is a 2D bimeron in all of the vertical half-planes. Based on this analogy, we call it 3D dimeron. The stability isinvestigated through numerically evolving in different systems, we find the 3D dimeron spontaneously created with the non-Abelian gauge field.Finally, in chapter VI, we summarize the results of this paper in detail and outlook for the future research work.