Quantization and Motion Law for Ginzburg–Landau Vortices

@article{Smets2007QuantizationAM,
  title={Quantization and Motion Law for Ginzburg–Landau Vortices},
  author={Didier Smets and Fabrice B{\'e}thuel and Giandomenico Orlandi},
  journal={Archive for Rational Mechanics and Analysis},
  year={2007},
  volume={183},
  pages={315-370}
}
We study the vortex trajectories for the two-dimensional complex parabolic Ginzburg–Landau equation without a well-preparedness assumption. We prove that the trajectory set is rectifiable, and satisfies a weak motion law. In the case of degree  ±  1 vortices, the motion law is satisfied in the classical sense. Moreover, dissipation occurs only at a finite number of times. Away from these times, possible collisions and splittings of vortices are constrained by algebraic equations involving their… 
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References

SHOWING 1-10 OF 26 REFERENCES
Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics
In this paper we describe a natural framework for the vortex dynamics in the parabolic complex Ginzburg-Landau equation in R. This general setting does not rely on any assumption of well-preparedness
On the slow motion of vortices in the Ginzburg-Landau heat flow
We study vortex motion in the Ginzburg–Landau flow. We consider this flow in the limit of large Ginzburg–Landau parameter. It is shown that when this parameter tends to infinity, the vortex mobility
Vortex annihilation in nonlinear heat flow for Ginzburg–Landau systems
We consider the Cauchy problem for the system where . Let e ∈ ℝ2 with |e| = 1. If u(x, 0) is smooth, bounded and we prove u → e uniformly in x as t → ∞. Of particular interest is the motion of the
Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations
In the Ginzburg‐Landau model for superconductivity a large Ginzburg‐Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located exactly where an applied
Vortices in complex scalar fields
  • J. Neu
  • Mathematics, Physics
  • 1990
Ginzburg-Landau Vortices
The mathematics in this book apply directly to classical problems in superconductors, superfluids and liquid crystals. It should be of interest to mathematicians, physicists and engineers working on
Symmetry-breaking solutions of the Ginzburg-Landau equation
We consider the question of the existence of nonradial solutions of the Ginzburg-Landau equation. We present results indicating that such solutions exist. We seek such solutions as saddle points of
Gamma‐convergence of gradient flows with applications to Ginzburg‐Landau
We present a method to prove convergence of gradient flows of families of energies that Γ‐converge to a limiting energy. It provides lower‐bound criteria to obtain the convergence that correspond to
Travelling Waves for the Gross-Pitaevskii Equation II
The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based
...
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