本论文在量子力学的框架下采用解析的方法研究二分量玻色-爱因斯坦凝聚中的拓扑激发。在凝聚态物理中，由于超冷原子之间存在着非线性相互作用，系统中呈现出非常丰富的物理现象，但同样使得相关的量子多体模型的精确求解非常困难。然而精确求解量子多体模型是人们研究量子多体问题以及利用数值方法近似求解许多复杂实际问题的基础，因此，本文的目的就是克服非线性耦合带来的求解困难，分别在准一维、二维、三维的体系中精确求解，并对求得的普适完备解进行多方面尤其拓扑性质的研究。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相关实验，介绍了二分量玻色-爱因斯坦凝聚的相关理论与研究进展，并对冷原子系统中拓扑激发作了介绍，包括相关的拓扑学知识。第二章至第四章则对应于我们的三个主要研究成果，具体介绍如下：（一）我们利用Jacobian椭圆函数的特殊性质对准一维二组分BEC系统进行了精确求解，得到了两类精确解析解：行波孤子解以及复对称破缺解（定态解）。其中，我们在均匀外势系统中求得了解析形式的行波孤子解；而复对称破缺解同样存在于平均势场的体系中，同时要求组分内、组分间所有原子的相互作用长度都相等。两类解都是完备且普适的。并且，复对称破缺解有三种类型，都是周期性的，对每一类型取孤子极限，则可以将周期解转化为周期排列的灰孤子。另外，利用哈密顿量的 $SU(2)$ 幺正对称性，我们还进一步得到了精确的时间演化解（同样具有孤子极限）。这些解都展示出时空上的对称性。在实验中通过合适的调控外加磁场可以平衡两个组分间的化学势差。事实上，这种求解方法可以应用于其他类型的非线性方程的求解，而且，雅可比椭圆函数可以用双曲函数替代，从而得到精确的单孤子解，具有非常广泛的应用前景。（二）我们巧妙的将二维二组分 BECs 系统的两个维度分开并成功解耦，将原耦合 GP 方程组转化为六个独立方程。我们在没有采用任何形式的近似以及任何省略的情形下自洽求解得到了六组普适且完备的周期解（由雅可比椭圆函数组合而成的复数-实数解），并对它们的拓扑性质及稳定性进行了分析。从拓扑结构来看，六组解可以分为三类：Skyrmion 晶体、（反）Skyrmion 晶体以及偶极子状晶体。1. 形成Skyrmion 晶体的自洽解有两组，一组在椭圆模$m$定义域内都可以呈现为二维正方形晶格排列的 Skyrmion 晶体，普适度非常高；另一组则在孤子极限$m \rightarrow 1$ 时，自旋结构表现为 Skyrmion 晶体。Skyrmion 晶体中每个单位元计算其拓扑不变量——拓扑荷都为$1$。2. 呈现（反）Skyrmion 晶体结构的是一组排斥相互作用自洽解，晶体中每个单位元拓扑荷为$-1$，同样椭圆模$m$ 在定义域可以任意取值。3. 所谓的偶极子状晶体结构其实是周期性排列的 Skyrmion-anti-Skyrmion 对。这种结构存在于其他三组自洽解的孤子极限情况。经过时间演化（傅里叶频谱法），我们发现，偶极子状的晶格结构的稳定性要远远大于其他两种结构。（三） 我们采用与二维系统中类似的方法给出了三维空间中具有自旋轨道耦合的非线性定态 GP 方程的精确解析解，求解过程利用了一个规范变换，没有采用任何省略或近似。我们发现二组分的凝聚体中可以存在很多类型孤子状扭结，它们都具有整数拓扑荷。我们从二维和三维两个方面探究了该态的拓扑结构。二维方面，在每一个$z$值确定的$x-y$平面中，构成了一个（反）skyrmion，其相应的二维拓扑荷为$Q_{2D}=-(n+1)$。在三维空间来看，自旋矢量在$z$方向上还存在着螺旋状的扭转，量子数为$m$。并且，通过一个关于坐标系的映射，我们发现二组分的凝聚体中可以存在多种扭结状的拓扑结构。利用ribbon理论分析，这些扭结结构具有的扭转数$Tw=m$，缠绕数为$Wr=mn$，于是环绕数$Lk=m(n+1)$，与C$\check{a}$lug$\check{a}$reanu–White–Fuller 定理符合。Stern-Gerlach 的实验中已经在处于外磁场中的旋量凝聚体中实现了扭结。该技术可以进一步实现更多类型的自旋结构。我们的态在$V(\textbf{r})=V_0[\tanh ^2(kx)+\tanh ^2(ky)]$ 的外势场中是可以稳定的，该外势与其总密度形状恰好是一致的。另外要说明的是，我们的解析方法可以应用于多种其他耦合非线性方程中，具有非常好的应用前景。最后在第五章中，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。

The analytical method is used in this paper to research topological excitation of two-component Bose-Einstein condensation(BEC) in the framework of quantum theory.In condensed matter physics, because of the nonlinear interaction between ultracold atoms, the system presents a very rich physical phenomena. And the exact solution of quantum multi-body model is that the basis of studying quantum many-body problem and the use of approximate numerical methods for solving practical problems. Therefore, the purpose of this paper is to overcome the difficulties in solving the nonlinear coupling caused by the nonlinear equation, and obtain the exact solutions.In this paper, we make a brief review of BEC phenomena theoretical background and related experiments in the first chapter, as well as an introduction of the topology in cold atom excitation system, including the associated topology knowledge.The second chapter to chapter IV corresponding to our three main research results are detailed below:Firstly, we solve 1D two-component BEC system exactly by useing the special nature of the Jacobian elliptic functions, and obtain two types of exact analytical solution: the traveling wave soliton solutions and the complex symmetry breaking solutions. Both are obtained in analytical form of uniform external potential system while second one requires all of the interaction length between atoms equal.All solutions are completely and universal. There are three types of the complex periodically solutions, all of which have a soliton limit.In addition, making use of the $ SU(2) $ unitary symmetry of Hamiltonian, we further obtain the time-involving solutions with the periodicity of space and time. In the experiment, we could balance the chemical potential difference between the two components Through appropriate regulation applied magnetic field. As a matter of fact, this method can be applied to solve other types nonlinear equations and the Jacobi elliptic functions in former solutions can be replaced by hyperbolic functions to obtain precise single soliton solution.Secondly, the chapter III, we successfully decouple the GP equations for the 2D two-component BECs system into independently equations, resulting in a series of universal and complete exact solution.The equations are solved to obtain the six groups of universal periodic solutions (a combination of Jacobi elliptic functions) without using any form of approximation and omiting none of the terms. And their topological properties and stability were analyzed. From the topological view, the solutions can be divided into six groups of three categories: Skyrmion crystal, (anti-) Skyrmion crystals and dipole-like crystals.1. There are two forms of sulutions performance the Skyrmion crystals. One of them was arranged as Skyrmion crystal in the oval domain of mold $m$, and for the other one, its' spin texture performance Skyrmion crystals in the soliton limit.The solution requires two atomic systems is attractive interaction. The topological invariants in each unit of Skyrmion crystal element could calculate to be $ 1 $.2. The solutions performance as (anti-) Skyrmion crystal structure is a set of self-consistent solution of repulsive interactions. The crystals per unit per topological charge is $ -1 $, with the mold $ m $ can be any value in the domain.3. The so-called dipole-like crystal structure is actually a periodic array of Skyrmion-anti-Skyrmion pairs. This structure is present in the other three groups soliton self-consistent solution of the limiting case. After a time evolution (numerical methods), we found that the stability of the dipole-like lattice structure is much larger than the other two structures.Thirdly, the chapter IV, we use a similar approach with the third chapter, and construct exact analytical solutions in 3D space with spin-orbit coupling system by utilizing a gauge transformation. We found that there exist many types of soliton -like kinks.We explore the topological structure of the state both from the 2D and 3D.In aspects of 2D, the state constitutes an anti-skyrmion in every $ xy $ plane, with the corresponding 2D topological charge to be $Q_{2D}=-(n+1)$.In the 3D space , the spin vector in the $ z $ direction has a spiral twist with the quantum number $ m $.Furthermore, by a map of coordinate, we found two components may be present in a variety of knots. Using the ribbon theoretical analysis, the twist number is $ Tw = m $, and the writhe number is $ Wr = mn $, then linking number is $Lk = m (n +1) $, which is the C$\check{a}$lug$\check{a}$reanu–White–Fuller fomula.The Stern-Gerlach's experiment in fourth external magnetic field in spinor condensates achieved the kinks. This technique can be further achieved more types of spin textures. Our state in $V(\textbf{r})=V_0[\tanh ^2(kx)+\tanh ^2(ky)]$ external potential field is stable, the overall density of the outer shape of its potential is exactly the same.Finally, in chapter V, we summarize the results of this paper in detail and outlook for the future research work.