Acta mathematica scientia, Series B >
VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION
Received date: 2007-10-19
Online 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)
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.
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 . DOI: 10.1016/S0252-9602(10)60092-6
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