Ovchinnikov, Yu.N., Sigal, I.M.
Ginzburg-Landau Equation I. Static Vortices
(254K, PS-version)
ABSTRACT. We consider radially symmetric
solutions of the Ginzburg-Landau equation (without magnetic
field) in dimension 2.
Such solutions are called vortices and are specified
by their winding number at infinity (vorticity).
For a given vorticity $n$ we prove existence and uniqueness
(modulo symmetry transformations) of an $n$-vortex and
show that for $n=0,\pm 1$ such vortices are stable
while for $|n|\ge 2$, unstable.
We introduce the renormalized Ginzburg-Landau energy
and use it for the existence and uniqueness proof.
Our stability proof is novel and uses the concept
of symmetry breaking and its consequence in the form of zero
modes of the linearized equation.