VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

doi:10.1016/S0252-9602(10)60092-6

数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (3): 949-962.doi: 10.1016/S0252-9602(10)60092-6

VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

温焕尧, 丁时进

School of Mathematical Sciences, |South China Normal University, Guangzhou 510631, China

收稿日期:2007-10-19 出版日期:2010-05-20 发布日期:2010-05-20
基金资助:
The project is supported by the National Natural Science Foundation of China (10471050), the National 973 Project of China
(2006CB805902), University Special Research Fund for Ph.D Program (20060574002), and Guangdong Provincial Natural Science Foundation (7005795, 031495)

VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

WEN Huan-Yao, DING Shi-Jin

School of Mathematical Sciences, |South China Normal University, Guangzhou 510631, China

Received:2007-10-19 Online:2010-05-20 Published:2010-05-20
Supported by:
The project is supported by the National Natural Science Foundation of China (10471050), the National 973 Project of China
(2006CB805902), University Special Research Fund for Ph.D Program (20060574002), and Guangdong Provincial Natural Science Foundation (7005795, 031495)

摘要/Abstract

摘要：

In this article, using coordinate transformation and Gronwall inequality, we study the vortex motion law of the anisotropic Ginzburg-Landau equation in a smooth bounded domain $\Omega\subset{\bf R}^2$, that is, $ {\partial_tu_\varepsilon=\sum\limits_{j,k=1}^2(a_{jk}\partial_{x_k}u_\varepsilon)_{x_j}+\frac{b(x)(1-| u_\varepsilon| ^2)u_\varepsilon}{\varepsilon^2},x\in\Omega}$, and conclude that each vortex $ {b_j(t)~(j=1,2,\cdots ,N)}$ satisfies $ \frac{{\rm d}b_j(t)}{{\rm d}t}=-\big(\frac{a_{1k}(b_j(t))\partial_{x_k}a(b_j(t))}{a(b_j(t))},$ $ \frac{a_{2k}(b_j(t))\partial_{x_k}a(b_j(t))}{a(b_j(t))}\big),$ where $ {a(x)=\sqrt{a_{11}a_{22}-a_{12}^2}}$. We prove that all the vortices are pinned together to the critical points of $a(x)$. Furthermore, we prove that these critical points can not be the maximum points.

关键词: Anisotropic Ginzburg-Landau equation, Gronwall inequality, vortex dynamics

Abstract:

Key words: Anisotropic Ginzburg-Landau equation, Gronwall inequality, vortex dynamics

中图分类号:

35A21

温焕尧, 丁时进. VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION[J]. 数学物理学报(英文版), 2010, 30(3): 949-962.

WEN Huan-Yao, DING Shi-Jin. VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION[J]. Acta mathematica scientia,Series B, 2010, 30(3): 949-962.

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VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

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