Abstract
This paper deals with the motion law of vortices in the limit ase → 0 of the Ginzburg-Landau equationu t = Δu+ (1/e2)(1 − ¦u¦2),u = (u1,u2)T in a planar contractible domain with Neumann boundary condition, where the vortices are meant by zeros of a solution. As e → 0, applying the argument by Jerrard-Soner to the Neumann case yields an ordinary differential equation, called a limit equation, describing the dynamics of the vortices. We show that the limit equation can be written by using the Green function with Dirichlet condition and the Robin function of it. With this nice form we discuss the dynamics of a single or two vortices together with equilibrium states of the limit equation. In addition for the disk domain an explicit form of the equation is proposed and the dynamics for multi-vortices is investigated.
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