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.

Ginzburg-Landau方程是一个描述超导现象的宏观模型。该方程是通过采用Ginzburg-Landau能量函数的第一个变化来得出的，并且具有复杂值阶的参数的未知变量和磁场的矢量电位。我们在没有施加磁场的3维薄域中研究了金茨堡 - 兰道方程。我们假设可以控制厚度，并将限制行为视为厚度消失。正式还原说，在极限中，方程可以简化为更简单的，而无需磁效应。我们通过使用扰动方法证明，如果还原方程具有非分化稳定的溶液，则薄域中的原始方程也具有稳定的解。我们还提供了一个明确的示例，说明了该域允许非排优体稳定的涡流解决方案。我们还研究了在简化的Ginzburg-landau在简单连接的二维有限域中功能的梯度系统中产生的涡度运动定律。这是仅订单参数的半线性热方程。随着参数为无穷大，我们得出了一个奇异极限方程的明确形式。由于这种明确的形式，我们揭示了涡流的一些动力学特性。