New non-radial solutions and concentration phenomena in the magnetic Ginzburg-Landau equations of superconductivity

本论文的主要研究内容是全平面上和有界区域上的磁Ginzburg-Landau方程。 它们是描述超导体在低温和高温下宏观行为的耦合协变非线性薛定谔方程和麦克斯韦方程以及粒子物理学的U（1）Higgs模型。 我将找到新的非径向磁涡旋解Ginzburg-Landau方程在整个平面上和涡旋解集中在一个子集的有界区域。 这些新的非径向磁涡旋解的发现将引发一个问题，即这些配置是否可以在超导体中通过实验观察到。 在寻找这些解决方案中使用的分析技术也将产生找到其他椭圆和非线性偏微分方程的其他新的解决方案的方法。因此，这项研究对物理学家和数学家都很有意义。 这项工作将探讨椭圆型偏微分方程，分析和几何之间的密切关系。

The proposed research will be on the magnetic Ginzburg-Landau equations on the whole plane and on bounded domains. These are coupled covariant nonlinear Schrodinger and Maxwell equations which describe (macroscopically) the behavior of superconductors at low and high temperatures and U(1) Higgs model of particle physics. I will find new non-radial magnetic vortex solutions to Ginzburg-Landau equations on the whole plane and vortex solutions concentrating on a subsets of bounded domains. The discovery of these new non-radial magnetic vortex solutions will give rise to questions on whether these configurations can be observed experimentally in superconductors. The analytical techniques used in finding these solutions will also generate ways to find other new solutions to other elliptic and non-linear partial differential equations. Therefore, this research will be of interest to both Physicists and Mathematicians. This work will explore the intimate relationship between elliptic partial differential equations, analysis and geometry.