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Dynamics for Ginzburg-Landau vortices under a mixed flow. (English) Zbl 1191.35261

The authors consider a complex Ginzburg-Landau equation in two dimentions of the following “mixed” form, which combines conservative and gradient (dissipative) flows:

(α ε +i)u=Δu+ε -2 u(1-|u| 2 ),

with the boundary conditions of an either Dirichlet or Neumann type (Δ is the Laplacian, and ε is a formal a small parameter). The equation may have some applications in the theory of supercondictivity, and to the description of the magnetization dynamics in ferromagnets. It admits stable solution in the form of vortices with the unitary topological charge (vortices with multiple charges are unstable). The authors consider the interaction between far separated vortices, aiming to develop effective particle-like equations of motion for centers of the interacting vortices, in the regime corresponding to α ε α 0 /|lnε|. The main result is that the equation of motion valid for each (n-th) vortex with charge d n =±1, whose pivot (central point, at which the local amplitude vanishes) is located at a point with coordinates a x,y , is a straightforward combination of previously derived equations of motion for vortices in the conservative and purely dissipative versions of the equation.

35Q56Ginzburg-Landau equations
35Q55NLS-like (nonlinear Schrödinger) equations
35B20Perturbations (PDE)
35B40Asymptotic behavior of solutions of PDE