几个非线性Schrodinger方程组模型及相关问题研究

N-components nonlinear Schrodinger systems has many importment applications in nonlinear optical、Bose-Einstein condensates、 Plasma physics and so on. The first, we will consider some models on the nonlinear Schrodinger systems, .research the existence and uniqueness of their ground states，orbital stabiity of bound states and related properties of solution. In particular, we forcus the super-critical case and orbital stability of dark soliton solution. In addition, we study the difference between N-components nonlinear Schrodinger systems and single Schrodinger equation. Secondly, we change the nonlinear Schrodinger system to equivalent integral systems, and consider the following three objectives: The first objective is to establish the existence of solutions in the super-critical case; The second objective is to show that the `ground states are the only positive solutions with finite energy; The final objective is to study the nonexistence of locally bounded solutions in the sub-critical case.

多组份非线性Schrodinger方程组在非线性光学、多组份Bose-Einstein 凝聚(BEC)以及等离子物理等领域有极其重要的应用。第一方面：我们研究几个不同模型多组份非线性Schrodinger方程组的基态解的存在唯一性，束缚态解的轨道稳定性以及解的相应性质等。特别是超临界情形的结果以及暗孤子解的稳定性等。另外着重研究多组份Schrodinger方程组与单个方程之间的区别。第二方面：我们将非线性Schrodinger方程组转化为等价的积分方程组来研究。考虑以下三个目标：1. 建立超临界情形下积分方程组解的存在性; 2.证明基态解是具有有限能量的正解; 3.在次临界情形下局部有界解的不存在性。

多组份非线性Schrodinger方程组在非线性光学、多组份Bose-Einstein 凝聚(BEC)以及等离子物理等领域有极其重要的应用。第一方面：我们研究几个不同模型多组份非线性Schrodinger方程组的基态解的存在唯一性，以及解的相应性质,特别是超临界情形的结果等。另外着重研究多组份Schrodinger方程组与单个方程之间的区别。第二方面：我们将非线性Schrodinger方程组转化为等价的积分方程组来研究。考虑以下三个目标：1. 建立超临界情形下积分方程组解的存在性; 2.证明基态解是具有有限能量的正解; 3.在次临界情形下局部有界解的不存在性。