97-214 Ovchinnikov, Yu.N., Sigal, I.M.

Ginzburg-Landau Equation I. Static Vortices (254K, PS-version) Apr 14, 97

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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.

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