非线性薛定谔方程组是量子物理中的重要方程组，是近年来应用数学领域的热点问题。研究重点包括：基态解、爆破解、整体解。虽然单个方程的结果比较丰富，但由于方程组和单个方程结构上的差异，会带来新的困难。这篇论文集中探讨薛定谔方程组的性质。全文系统研究了耦合的非线性薛定谔方程组。第一，给出了一类用于判定解是全局存在还是有限时间爆破的判定法则。所探讨的解都在标准的Sobolev空间中，方程则定义在欧式空间、双曲空间和球面上，对应于平坦流形、负曲率流形，和正曲率紧流形。对不同的流形，判据差异很大。结果是针对三种流形陈述的，但分析方法适用于一般流形。这一部分的创新点有二，一是将前人引进的泛函扩展成两大类，丰富了之前的结果。二是给出了流形上薛定谔方程（组）爆破现象新刻画，这一现象一直为物理学家关心。第二，给出了解在能量空间中的稳定性。所得定理指出，对于一大类指标（包括超临界指标）的薛定谔方程组，任意两个不同初值的解，在后续时刻的能量差，能够被初始的能量差控制。这部分发展了Struwe的方法，这一方法同样可用来证明Hartree方程正解的稳定性。第三，对于只具有两个分量的薛定谔方程组，在某些参数条件下，给出了基态解的唯一性。大家知道，处理单个方程解的唯一性的方法，在处理方程组时失效，因此方程组解的唯一性在文献中不多见。本文证明方法目前只适用于两个分支的方程组，多个分支的情况有待探讨。得到基态解的唯一性后，文章给出了向量值的Gagliardo-Nirenberg不等式的最佳常数。这个最佳常数对应于爆破解的临界质量。值得一提的是，采用积分形式的Moving-plane方法及相关的方程组思想，可以证明Choquard方程正解的唯一性, 从而完善Lieb的结果。第四，给出了一个不满足有限能量条件的弱解（Rough解）的整体存在性定理。这个定理，是对Colliander和Tao及其合作者关于单个薛定谔方程的结果在方程组上的推广。
Nonlinear Schrodinger system is one of the most important and fundamental partial differential equations appearing in Quantum Physics. It has applications in many physical branches, especially in nonlinear optics. It is also a hot topic in the field of applied mathematics in the latest five years. The main concerns from physicians and mathematicians are ground states (existence, uniqueness, stability), blow-up solutions (criterion, blow-up rate),global existence (functional space, regularity, scattering), etc. Although the related quantitative results on the single Schrodinger equations are rich, it's also challenging to gain an accurate understanding on the Sch\"{o}dinger system, due to the structural differences between the coupled system and the single equations. These structural differences also imply new difficulties and new consequences. The focus of my PhD thesis is to explore and investigate various properties and phenomena on Schrodingersystems.In this thesis, we study systematically the coupled Nonlinear Schrodinger system. Firstly, we give the sharp thresholds of blow-up and global existence in the standard Sobolev space. We assume the Schrodinger system to be posed on Euclidean space, hyperbolic space and spheres, corresponding to the flat manifold, the compact manifold with positive Ricci curvature and the manifold with negative Ricci curvature respectively. Our results rely heavily on the geometry of the manifolds and our method is applicable to other Riemannian surfaces. The contributions of these results are twofold. They extend the two types of classic functionals to two broad classes as well as give a new characterization of blow-up phenomenon on manifolds, and such phenomenon is a great concern ofPhysicians and Mathematicians. Secondly, we study the stability ofthe coupled nonlinear Schrodinger system in the energy space. Our result points out that for a large class of exponents, includingthe supercritical ones, the associated energy gap of two solutionswith different initial values can be controlled along time by theinitial energy gap, which implies stability. We develop the methodintroduced by Struwe, and our method can also be applied to study the energy stability of Hartree equation. Thirdly, we give a uniqueness result on the ground states of the two-component Schrodinger system, under some parameter restrictions. To our knowledge, uniqueness of solutions to partial differential system is not widely studied, due to the invalidity of the arguments which work for the single equations. Our result can't be extended to the Schrodinger system with components more than two and we leave this case open for further research. Having obtained uniqueness, we present a natural application to determine the sharp constant in a vector-valued Gagliardo-Nirenberg inequality, which corresponds to the critical mass of blowing up. It is worthy to note that applying the Moving-Plane method in integral form, in spirit of our analysis, we can prove the uniqueness of positive solutions of Choquard equation and thus enhance a result of Lieb. Fourthly, we study the global existence of solutions for which the initial energy finiteness is not satisfied, which are called ``rough solutions" in the literature. Our theorem generalizes the seminal result for the single Schrodinger equation by Colliander, Tao and his collaborators to the Schrodinger system. The results can help us to gain a deeper understanding on the relationship of blowing up and initial energy.
