Stable vortex solutions to the Ginzburg-Landau equation with a variable coefficient in a disk

This paper deals with stable solutions with a single vortex to the Ginzburg Landau equation having a variable coefficient subject to the Neumann bound ary condition in a planar dish. The equation has a positive parameter, say lambda, which will play an important role for the stability of the solution . We consider the equation with a radially symmetric coefficient in the dis k and suppose that the coefficient is monotone increasing in a radial direc tion. Then the equation possesses a pair of solutions with a single vortex for large lambda. Although these solutions for the constant coefficient are unstable, they can be stable fur a suitable variable coefficient and large lambda. The purpose of this article is to give a sufficient condition for the coefficient to allow those solutions being stable for any sufficiently large lambda. As an application we show an example of the coefficient enjoy ing the condition, which has an arbitrarily small total variation. (C) 1999 Academic Press.