Vortex motion law of an evolutionary Ginzburg-Landau equation in 2 dimensions (Q5890205)

The paper considers a two-dimensional Ginzburg-Landau (GL) equation with real coefficients and external one-dimensional potential, NEWLINE\[NEWLINE u_t = \Delta u + \epsilon^{-2} [\beta^2(x) - | u| ^2]u,\tag{1} NEWLINE\]NEWLINE where \(\Delta\) is the Laplace operator. The equation is supplemented by the Neumann boundary conditions. The GL coefficient in the homogeneous space has a stable solution in the form of a vortex, at the center of which \(| u(x,y)\) vanishes. It is known that the motion of the vortex' center in an external potential is governed by an overdamped equation of the first order, driven by the potential force. The paper presents rigorous proof of the weak and strong convergence of the vortex solution to its limit form corresponding to \(\epsilon = 0\) in Eq. (1).