Mathematical studies for models of superconductivity

1.研究了超导导线中的一维Ginzburg-Landau方程。当导线是均匀的，我们揭示了全局分歧结构的两个物理参数，并确定哪些解决方案是最小的能量泛函。我们还研究了非均匀厚度金属丝中的Ginzburg-Landau模型解的相位结构。2.研究了区域的几何形状对非线性方程解结构的影响。将这种方法推广到Ginzburg-Ladau方程。3.研究了时间发展Ginzburg-Landau方程的一个渐近性态。4.发展了反应扩散方程过渡层问题的变分方法。这种方法将被应用到超导模型中。5.对一个BEC模型和几个Ginzburg-Landau模型进行了数值计算。我们还发现了在这种非线性耗散系统中出现的新的模式动力学。特别地，我们证明了与反应扩散方程的前波动力学有关的解的存在性。

1.We studied a one-dimensional Ginzburg-Landau equation in a ring, which is a mathematical model in a superconducting wire. When the wire is uniform, we revealed the global bifurcation structure for the two physical parameters and determined which solutions are minimizer of the energy functional. We also studied the configuration of the phase of solutions to the Ginzburg-Landau model in the wire with non-uniform thickness.2.We studied how the solution structure of a nonlinear equation is affected by the geometry of a domain. This approach would be developed to the Ginzburg-Ladau equation.3.An asymptotic behavior of the time evolutionary Ginzburg-Landau equations was studied. Some spectral result for the linearized operator of the equations was also obtained4.A variational method to the transition layer problem in reaction-diffusion equations was developed. This approach would be applied to a model of the superconductivity.5.Numerical computations for a BEC model and several Ginzburg-Landau models were achieved. We also discovered new pattern-dynamics arising in such nonlinear dissipative systems. In particular we proved the existence of solutions related to dynamics of front waves to reaction-diffusion equations.