VORTEX DYNAMICS FOR THE

GINZBURG-LANDAU-SCHRÖDINGER EQUATION

JAMES ELLIS COLLIANDER ROBERT L. JERRARD

Abstract. The initial value problem for the Ginzburg-Landau-Schrödinger equation is examined in the ϵ → 0 limit under two main assumptions on the initial data ϕ^{ϵ}. The first assumption is that ϕ^{ϵ} exhibits m distinct vortices of degree ±1; these are described as points of concentration of the Jacobian [Jϕ^{ϵ}] of ϕ^{ϵ}. Second, we assume energy bounds consistent with vortices at the points of concentration. Under these assumptions, we identify “vortex structures” in the ϵ → 0 limit of ϕ^{ϵ} and show that these structures persist in the solution u^{ϵ}(t) of GLS_{ϵ}. We derive ordinary differential equations which govern the motion of the vortices in the ϵ → 0 limit. The limiting system of ordinary differential equations is a Hamitonian flow governed by the renormalized energy of Bethuel, Brezis and Hélein. Our arguments rely on results about the structural stability of vortices which are proved in a separate paper.