VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

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 of R(2), that is, partial derivative(t)u(epsilon) = Sigma(2)(j,k=1) (a(jk)partial derivative(xk)u(epsilon))(xj) + b(x)(1-vertical bar u(epsilon)vertical bar(2))u(epsilon)/epsilon(2), x is an element of Omega, and conclude that each vortex b(j)(t) (j = 1, 2,..., N) satisfied db(j)(t)/dt = -(a(1k)(b(j)(t))partial derivative(xk)a(b(j)(t))/a(b(j)(t)), a(2k)(b(j)(t))partial derivative(xk)a(b(j)(t))/a(b(j)(t)), where a(x) = root 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.