Dynamics for Ginzburg-Landau vortices under a mixed flow

Abstract

We consider a complex Ginzburg-Landau equation that contains a Schrödinger term and a damping term that is proportional to the time derivative. Given well-prepared initial conditions that correspond to quantized vortices, we establish the vortex motion law until collision time.

Vortex Liquids and the Ginzburg–landau Equation

“…On the one hand, it provides quantitative bounds that are useful in type II superconductors without going to the ε → 0 limit of "extreme" type II superconductivity. More importantly, as our approach does not rely on the gradient flow structure, it can be adapted to yield results for more general situations, such as the mixed flows studied in [26] and [37] for the ungauged problem and in [27] and [48] for the gauged problem. Such motion laws have physical importance, as they can be used to explain the sign change in the Hall effect of type II superconductors; see [11], [24].…”

“…The proof of Theorem 3 follows along the same lines as the proof of Theorem 2. In particular, we use assumptions (23)- (26) in order to satisfy the hypotheses of Theorem 1. Next, assumptions (25)-(26) ensure the long-time existence of the vortex dynamics via Proposition 9.…”

Vortex dynamics for nonlinear Klein–Gordon equation

“…Moreover, by setting ξ ≡ 1 in Proposition 3.3 and in light of Proposition 2.1 in [7], we conclude that…”

“…Motivated from [3][4][5][6][7][10][11][12], the proofs for the various dynamical laws of Ginzburg-Landau (G.-L. for short) vortices rely on the establishments of:…”

“…The corresponding vortex dynamical motion laws were shown in [3][4][5][6][10][11][12] and the references therein. When combining the (PE) and the (SE) together, in [7], the authors studied a mixed flow problem defined by:…”