Mathematical studies for models of superconductivity

1.本文研究了环中的一维金兹堡-朗道方程，这是超导导线中的数学模型。当导线均匀时，我们揭示了两个物理参数的全局分岔结构，并确定了哪些解是能量泛函的最小解。我们还研究了非均匀厚度金属丝中金兹堡-朗道模型解的相构型。我们研究了一个非线性方程的解结构如何受到区域几何形状的影响。这种方法将发展到Ginzburg-Ladau方程。研究了时间演化金兹堡-朗道方程的渐近行为。对方程的线性化算子也得到了一些谱结果。提出了一种求解反应扩散方程过渡层问题的变分方法。这种方法可应用于超导性模型。对BEC模型和几种金兹堡-朗道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产生的新的模式动力学。特别是证明了反应扩散方程前波动力学解的存在性。

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.