Dynamic stability of vortex solutions of Ginzburg-Landau and nonlinear Schrödinger equations

Abstract

The dynamic stability of vortex solutions to the Ginzburg-Landau and nonlinear Schrödinger equations is the basic assumption of the asymptotic particle plus field description of interacting vortices. For the Ginzburg-Landau dynamics we prove that all vortices are asymptotically nonlinearly stable relative to small radial perturbations. Initially finite energy perturbations of vortices decay to zero in L p (ℝ 2 ) spaces with an algebraic rate as time tends to infinity. We also prove that under general (nonradial) perturbations, the plus and minus one-vortices are linearly dynamically stable in L 2 ; the linearized operator has spectrum equal to (−∞, 0] and generates a C 0 semigroup of contractions on L 2 (ℝ 2 ). The nature of the zero energy point is clarified; it is resonance , a property related to the infinite energy of planar vortices. Our results on the linearized operator are also used to show that the plus and minus one-vortices for the Schrödinger (Hamiltonian) dynamics are spectrally stable, i.e. the linearized operator about these vortices has ( L 2 ) spectrum equal to the imaginary axis. The key ingredients of our analysis are the Nash-Aronson estimates for obtaining Gaussian upper bounds for fundamental solutions of parabolic operator, and a combination of variational and maximum principles.