Mathematical studies for models of superconductivity

超导模型的数学研究

• 批准号: 15340037
• 负责人: MORITA Yoshihisa
• 金额: $ 5.38万
• 依托单位: Ryukoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340037/
• 关键词: Superconductivity; Ginzburg-Landau equation; Bifurcation structure; Variational method; Numerical simulation; Pattern dynamics; 反応拡散方程式; 大域的分岐構造; 漸近挙動; 時間依存ギンツブルグ・ランダウ方程式; 渦糸; 領域の摂動; 非線形楕円型方程式; 解の安定性; 解の分岐; 安定性解析

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.

1.研究了环中的一维Ginzburg-Landau方程，它是超导导线中的一个数学模型。当导线均匀时，我们揭示了两个物理参数的全局分叉结构，并确定了哪些解是能量泛函的最小化。我们还研究了具有非均匀厚度的导线中Ginzburg-Landau模型解的位形。2.研究了区域的几何形状对非线性方程的解结构的影响。这一方法将推广到Ginzburg-Ladau方程。3.研究了时间发展的Ginzburg-Landau方程的渐近行为。得到了方程线性化算子的一些谱结果。4.发展了求解反应扩散方程过渡层问题的变分方法。将这种方法应用于超导模型。5.对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，我们证明了反应扩散方程与前波动力学有关的解的存在性。

项目成果

Instability in a geometric parabolic equation on convex domain

凸域上几何抛物线方程的不稳定性

• 期刊: J. Differential Equations
• 作者: S. Jimbo;翟健
• 通讯作者: 翟健

Singular perturbation of domains and semilinear elliptic equations III

域的奇异摄动和半线性椭圆方程 III

• 发表时间: 2004
• 期刊: Hokkaido Math.J. 33
• 作者: Hitoshi Isozaki;G.Uhlmann;Shuichi Jimbo
• 通讯作者: Shuichi Jimbo

Y.Morita: "Stable Solutions to the Ginzburg-Landau Equation with Magnetic Effect in a Thin Domain"Japan Journal of Industrial and Applied Mathematics. Vol.21-2(掲載予定). (2004)

Y.Morita：“薄域中具有磁效应的 Ginzburg-Landau 方程的稳定解”《日本工业与应用数学杂志》第 21-2 卷（即将出版）。

Phase pattern in a Ginzburg-Landau model with a discontinuous coefficient in a ring

环中具有不连续系数的 Ginzburg-Landau 模型中的相位模式

• 发表时间: 2006
• 期刊: Discrete Conti.Dynam.Systems 14
• 作者: S.Kosugi;Y.Morita
• 通讯作者: Y.Morita

Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain

• DOI: 10.1007/bf03167468
• 发表时间: 2004-06
• 期刊: Japan Journal of Industrial and Applied Mathematics
• 影响因子: 0.9
• 作者: Y. Morita