一直以来，物理中的Ginzburg-Landua理论与数学中的非线性椭圆偏微分方程有着密切的联系。Ginzburg-Landau模型刻画了在高温超导体，超流体，液态晶体中的相变现象和涡旋现象。涡旋的存在性，数量，位置以及涡旋的局部结构，一直都是物理学家和数学家共同关注的对象。因此，理解涡旋的结构不仅对于数学理论的完善，而且对于实际应用都具有很重要的意义。本项目将分别对一般的耦合Ginzbug-Landau系统的涡旋解在不同边界条件下的稳定性，以及一类带有高阶扰动项的单一Ginzburg-Landau模型涡旋解的结构进行探讨。希望通过本项目的开展，我们可以深入挖掘数学与物理的相互联系，以及进一步发展偏微分方程、变分方法和非线性泛函分析中的新方法。

It is well known that Ginzburg-Landau theory in physics has intimate relation with nonlinear elliptic partial differential equations. A Ginzburg–Landau model describes phase transitions and vortices in high-temperature superconductors, superfluids, and liquid crystals. The presence, number, location and local structure of vortices is an area of intensive mathematical activity which attracts most attention from physicians and mathematicians. Therefore, understanding the structure of vortices will not only help us improving the mathematical analysis, but also help interpretating the physical applications. This project will consider the stability of vortices to a general coupled Ginzburg-Landau system with different boundary conditions, and the structure of vortices to a single Ginzburg-Landau model with higher order perturbation. The goal of this proposed research is to further discover this exciting interplay between mathematics and physics, and to introduce new techniques in partial differential equations, the calculus of variations, and nonlinear functional analysis which are both inspired by and shed new light upon these phenomena.