Stable Vortex Solutions to the Ginzburg–Landau Equation with a Variable Coefficient in a Disk

Abstract

This paper deals with stable solutions with a single vortex to the Ginzburg–Landau equation having a variable coefficient subject to the Neumann boundary condition in a planar disk. The equation has a positive parameter, say λ, which will play an important role for the stability of the solution. We consider the equation with a radially symmetric coefficient in the disk and suppose that the coefficient is monotone increasing in a radial direction. Then the equation possesses a pair of solutions with a single vortex for large λ. Although these solutions for the constant coefficient are unstable, they can be stable for a suitable variable coefficient and large λ. The purpose of this article is to give a sufficient condition for the coefficient to allow those solutions being stable for any sufficiently large λ. As an application we show an example of the coefficient enjoying the condition, which has an arbitrarily small total variation.