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

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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.