This paper considers the extreme type-II Ginzburg–Landau equations that model vortex patterns in superconductors. The nonlinear PDEs are solved using Newton's method, and properties of the Jacobian operator are highlighted. Specifically, this paper illustrates how the operator can be regularized using an appropriate phase condition. For a two-dimensional square sample, the numerical results are based on a finite-difference discretization with link variables that preserves the gauge invariance. For two exemplary sample sizes, a thorough bifurcation analysis is performed using the strength of the applied magnetic field as a bifurcation parameter and focusing on the symmetries of this system. The analysis gives new insight into the transitions between stable and unstable states, as well as the connections between stable solution branches.

MSC codes

  1. 82D55
  2. 35Q56
  3. 81R40
  4. 65F22


  1. superconductors
  2. Ginzburg–Landau system
  3. symmetry-breaking bifurcations
  4. vortices
  5. regularization

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Information & Authors


Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 447 - 477
ISSN (online): 1536-0040


Submitted: 31 August 2010
Accepted: 2 January 2012
Published online: 22 March 2012

MSC codes

  1. 82D55
  2. 35Q56
  3. 81R40
  4. 65F22


  1. superconductors
  2. Ginzburg–Landau system
  3. symmetry-breaking bifurcations
  4. vortices
  5. regularization



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