Stabilization of Vortices in the Ginzburg--Landau Equation with a Variable Diffusion Coefficient
Xu-Yan Chen, Shuichi Jimbo, Yoshihisa Morita · SIAM Journal on Mathematical Analysis · 1998
We study equilibria of the Ginzburg--Landau equation with a variable diffusion coefficient on a bounded planar domain subject to the Neumann boundary condition. It has been previously shown that if the diffusion coefficient is constant and the ambient domain is convex, the system does not carry stable vortices in the sense that any stable equilibrium solution is a constant of modulus 1. In this article we shall prove that arbitrarily given a domain, an appropriate choice of inhomogeneous diffusion coefficient yields a stable equilibrium solution having vortices. We can even manage to make the configuration of stable vortices close to prescribed locations. Our method is to minimize the free energy functional in suitably constructed positive invariant regions for the time-dependent Ginzburg--Landau equation.