Skip to main content Accessibility help
×
Home

Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains

  • Weizhu Bao (a1) and Qinglin Tang (a1)

Abstract

In this paper, we study numerically quantized vortex dynamics and their interaction in the two-dimensional (2D) Ginzburg-Landau equation (GLE) with a dimensionless parameter ε > 0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a review of the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.

Copyright

Corresponding author

Corresponding author.Email:bao@math.nus.edu.sg

References

Hide All
[1]Anderson, B. P., Resource artical: Experiment with vortices in superfluid atomic gases, J. Low Temp. Phys., 161 (2010), 574602.
[2]André, N. and Shafrir, I., Asymptotic behavior for the Ginzburg-Landau functional with weight (I), Arch. Rat. Mech. Anal., 142 (1998), 4573.
[3]Andreé, N. and Shafrir, I., Asymptotic behavior for the Ginzburg-Landau functional with weight (II), Arch. Rat. Mech. Anal., 142 (1998), 7598.
[4]Bao, W., Numerical methods for the nonlinear Schroödinger equation with nonzero far-field conditions, Methods Appl. Anal., 11 (2004), 367388.
[5]Bao, W., Du, Q. and Zhang, Y., Dynamics of rotating Bose-Einstein condensates and their efficient and accurate numerical computation, SIAM J. Appl. Math., 66 (2006), 758786.
[6]Bauman, P., Chen, C. N., Phillips, D. and Sternberg, P., Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems, European J. Appl. Math., 6 (1995), 115126.
[7]Bethuel, F., Brezis, H. and Heélein, F., Ginzburg-Landau Vortices, Brikhaäuser, Boston, 1994.
[8]Chapman, S., Du, Q. and Gunzburger, D., A model for varia ble thickness superconducting thin film, Z. Angew. Math. Phys., 47 (1996), 410431.
[9] M. del Pino, Kowalczyk, M. and Musso, M., Variational reduction for Ginzburg-Landau vortices, J. Funct. Anal., 239 (2006), 497541.
[10]Donnelly, R. J., Quantized Vortices in Helium II, Cambridge Univ., Cambridge, 1991.
[11]Du, Q. and Gunzburger, D., A model for superconduction thin films having variable thickness, Phys. D., 69 (1993), 215231.
[12]Dynamics, W. Eof vortices in Ginzburg-Landau theroties with applications to superconductivity, Phys. D, 77 (1994), 38404.
[13]Glowinski, R. and Tallec, P., Augmented Lagrangian and Operator Splitting Method in Nonlinear Mechanics, SIAM, Philadelphia, PA, 1989.
[14]Gustafson, S. and Sigal, I. M., Effective dynamics of magnetic vortices, Adv. Math., 199 (2006), 448498.
[15]Jerrard, R. and Soner, H., Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech. Anal., 142 (1998), 99125.
[16]Jian, H., The dynamical law of Ginzburg-Landau vortices with a pining effect, Appl. Math. Lett., 13 (2000), 9194.
[17]Jian, H. and Song, B., Vortex dynamics of Ginzburg-Landau equations in inhomogeneous superconductors, J. Diff. Eq., 170 (2001), 123141.
[18]Jian, H. and Wang, Y., Ginzburg-Landau vortices in inhomogeneous superconductors, J. Part. Diff. Eq., 15 (2002), 4560.
[19]Jimbo, S. and Morita, Y., Stability of nonconstant steady-state solutions to a Ginzburg-Landau equation in higer space dimension, Nonlinear Anal.: T.M.A., 22 (1994), 753770.
[20]Jimbo, S. and Morita, Y., Vortex dynamics for the Ginzburg-Landau equation with Neumann condition, Methods App. Anal., 8 (2001), 451477.
[21]Jimbo, S. and Morita, Y., Notes on the limit equation of vortex equation of vortex motion for the Ginzburg-Landau equation with Neumann condition, Japan J. Indust. Appl. Math., 18 (1972), 151200.
[22]Kincaid, D. and Cheney, W., Numerical Analysis, Mathematics of Scientific Computing, Brooks-Cole, 3rd edition, 1999.
[23]Lin, F., Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323359.
[24]Lin, F., A remark on the previous paper “Some dynamical properties of Ginzburg-Landau vortices”, Comm. Pure Appl. Math., 49 (1996), 361364.
[25]Lin, F., Complex Ginzburg-Landau Equations and Dynamics of Vortices, Filaments, and Codimension-2 Submanifolds, Comm. Pure Appl. Math., 51 (1998), 385441.
[26]Lin, F., Mixed vortex-antivortex solutions of Ginzburg-Landu equations, Arch. Rat. Mech. Anal., 133 (1995), 103127.
[27]Lin, F. and Du, Q., Ginzburg-Landau vortices: Dynamics, pining and hysteresis, SIAM J. Math. Anal., 28 (1997), 12651293.
[28]Neu, J., Vortices in complex scalar fields, Phys. D, 43 (1990), 385406.
[29]Peres, L. and Rubinstein, J., Vortex dynamics for U(1)-Ginzburg-Landau models, Phys. D, 64 (1993), 299309.
[30]Rubinstein, J. and Sternberg, P., On the slow motion of vortices in the Ginaburg-Landau heat flow, SIAM J. Appl. Math., 26 (1995), 14521466.
[31]Sandier, E. and Serfaty, S., Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 16271672.
[32]Serfaty, S., Stability in 2D Ginzburg-Landau pass to the limit, Indiana U. Math. J., 54 (2005), 199221.
[33]Serfaty, S., Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow. Part II: the dynamics, J. Euro. Math. Soc., 9 (2007), 383426.
[34]Shen, J. and Tang, T., Spectral and High-Order Method with Applications, Science Press, 2006.
[35]Strang, G., On the construction and comparision of difference schemes, SIAM J. Numer. Anal., 5 (1968), 505517.
[36]Weinstein, M. I. and Xin, J., Dynamic stability of vortex solutions of Ginzburg-Landau and nonlinear Schrödinger equations, Comm. Math. Phys., 180 (1996), 389428.
[37]Zhang, Y., Bao, W. and Du, Q., Numerical simulation of vortex dynamics in Ginzburg-Landau-Schroädinger equation, Euro. J. Appl. Math., 18 (2007), 607630.
[38]Zhang, Y., Bao, W. and Du, Q., The dynamics and interactions of quantized vortices in Ginzburg-Landau-Schroädinger equation, SIAM I. Appl. Math., 67 (2007), 17401775.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed