Notes on the limit equation of vortex motion for the Ginzburg-Landau equation with Neumann condition

  • Shuichi Jimbo
  • Yoshihisa Morita
Article

Abstract

This paper deals with the motion law of vortices in the limit asε → 0 of the Ginzburg-Landau equationu t = Δu+ (1/ε2)(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 ε → 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.

Key words

Ginzburg-Landau equation Neumann condition vortex limit equation equilibrium state 

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Copyright information

© JJIAM Publishing Committee 2001

Authors and Affiliations

  • Shuichi Jimbo
    • 1
  • Yoshihisa Morita
    • 2
  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan
  2. 2.Department of Applied Mathematics and InformaticsFaculty of Science and Technology, Ryukoku UniversitySeta, OtsuJapan

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