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Vortex Solutions of the Ginzburg-Landau Equation in a Thin Domain

薄域中Ginzburg-Landau方程的涡解

基本信息

批准号：
13640142
负责人：
MORITA Yoshihisa
金额：
$ 1.73万
依托单位：
Ryukoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640142/
关键词：
Ginzburg-Landau equation vortex solution thin domain superconductivity stability of solution nonlinear partial differential equation dynamical systems nonlinear elliptic equation

项目摘要

The Ginzburg-Landau equation is a macroscopic model which describes superconducting phenomena. This equation is derived by taking the first variation of the Ginzburg-Landau energy functional and it has the unknown variables of a complex-valued order parameter and a vector potential of magnetic field. We studied the Ginzburg-Landau equation in a 3-dimensional thin domain without an applied magnetic field. We assume that the thickness can be controlled and consider the limiting behavior as the thickness vanishes. The formal reduction tells that in the limit the equation can be reduced to a simpler one without the magnetic effect. We proved by using a perturbation method that if the reduced equation has a non-degenerate stable solution, then the original equation in the thin domain has also stable solution. We also give an explicit example of the domain allowing a non-degenerate stable vortex solution.We also studied the motion law of vortices arising in a gradient system of a simplified Ginzburg-Landau functional in a simply connected 2-dimensional bounded domain. That is a semilinear heat equation of only the order parameter. We derive an explicit form of a singular limit equation as the parameter goes to infinity. By virtue of this explicit form we revealed some dynamical properties of vortices.

金兹堡-朗道方程是描述超导现象的宏观模型。该方程是利用金兹堡-朗道能量泛函的一阶变分推导出来的，它具有一个复值阶参数和一个磁场矢量势的未知变量。研究了无外加磁场条件下三维薄域中的金兹堡-朗道方程。我们假设厚度是可以控制的，并考虑厚度消失时的极限行为。形式化简表明，在极限情况下，方程可以简化为不存在磁效应的简单方程。利用摄动方法证明了如果简化方程具有非简并稳定解，则原方程在薄域内也具有稳定解。我们还给出了一个允许非简并稳定涡解的显式例子。研究了二维单连通有界区域上简化金兹堡-朗道泛函梯度系统中涡的运动规律。这是一个只有阶参量的半线性热方程。我们导出了当参数趋于无穷时奇异极限方程的显式形式。利用这种显式形式，我们揭示了涡旋的一些动力学性质。