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

Published online by Cambridge University Press:  03 June 2015

Weizhu Bao  and
Qinglin Tang
Weizhu Bao*
Affiliation:
Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 119076
Qinglin Tang*
Affiliation:
Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 119076
*
Corresponding author.Email:bao@math.nus.edu.sg
Email:tqltql2010@gmail.com
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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.

Keywords

35Q5665M0665M7082D55Ginzburg-Landau equationquantized vortexDirichlet boundary conditionhomogeneous Neumann boundary conditionreduced dynamical lawstime splittingcompact finite difference methodfinite element method
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 3 , September 2013 , pp. 819 - 850
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Anderson, B. P., Resource artical: Experiment with vortices in superfluid atomic gases, J. Low Temp. Phys., 161 (2010), 574–602.Google Scholar
[2]André, N. and Shafrir, I., Asymptotic behavior for the Ginzburg-Landau functional with weight (I), Arch. Rat. Mech. Anal., 142 (1998), 45–73.Google Scholar
[3]Andreé, N. and Shafrir, I., Asymptotic behavior for the Ginzburg-Landau functional with weight (II), Arch. Rat. Mech. Anal., 142 (1998), 75–98.Google Scholar
[4]Bao, W., Numerical methods for the nonlinear Schroödinger equation with nonzero far-field conditions, Methods Appl. Anal., 11 (2004), 367–388.CrossRefGoogle Scholar
[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), 758–786.Google Scholar
[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), 115–126.Google Scholar
[7]Bethuel, F., Brezis, H. and Heélein, F., Ginzburg-Landau Vortices, Brikhaäuser, Boston, 1994.Google Scholar
[8]Chapman, S., Du, Q. and Gunzburger, D., A model for varia ble thickness superconducting thin film, Z. Angew. Math. Phys., 47 (1996), 410–431.Google Scholar
[9] M. del Pino, Kowalczyk, M. and Musso, M., Variational reduction for Ginzburg-Landau vortices, J. Funct. Anal., 239 (2006), 497–541.Google Scholar
[10]Donnelly, R. J., Quantized Vortices in Helium II, Cambridge Univ., Cambridge, 1991.Google Scholar
[11]Du, Q. and Gunzburger, D., A model for superconduction thin films having variable thickness, Phys. D., 69 (1993), 215–231.Google Scholar
[12]Dynamics, W. Eof vortices in Ginzburg-Landau theroties with applications to superconductivity, Phys. D, 77 (1994), 38–404.Google Scholar
[13]Glowinski, R. and Tallec, P., Augmented Lagrangian and Operator Splitting Method in Nonlinear Mechanics, SIAM, Philadelphia, PA, 1989.Google Scholar
[14]Gustafson, S. and Sigal, I. M., Effective dynamics of magnetic vortices, Adv. Math., 199 (2006), 448–498.Google Scholar
[15]Jerrard, R. and Soner, H., Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech. Anal., 142 (1998), 99–125.Google Scholar
[16]Jian, H., The dynamical law of Ginzburg-Landau vortices with a pining effect, Appl. Math. Lett., 13 (2000), 91–94.Google Scholar
[17]Jian, H. and Song, B., Vortex dynamics of Ginzburg-Landau equations in inhomogeneous superconductors, J. Diff. Eq., 170 (2001), 123–141.Google Scholar
[18]Jian, H. and Wang, Y., Ginzburg-Landau vortices in inhomogeneous superconductors, J. Part. Diff. Eq., 15 (2002), 45–60.Google Scholar
[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), 753–770.Google Scholar
[20]Jimbo, S. and Morita, Y., Vortex dynamics for the Ginzburg-Landau equation with Neumann condition, Methods App. Anal., 8 (2001), 451–477.Google Scholar
[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), 151–200.Google Scholar
[22]Kincaid, D. and Cheney, W., Numerical Analysis, Mathematics of Scientific Computing, Brooks-Cole, 3rd edition, 1999.Google Scholar
[23]Lin, F., Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323–359.Google Scholar
[24]Lin, F., A remark on the previous paper “Some dynamical properties of Ginzburg-Landau vortices”, Comm. Pure Appl. Math., 49 (1996), 361–364.Google Scholar
[25]Lin, F., Complex Ginzburg-Landau Equations and Dynamics of Vortices, Filaments, and Codimension-2 Submanifolds, Comm. Pure Appl. Math., 51 (1998), 385–441.Google Scholar
[26]Lin, F., Mixed vortex-antivortex solutions of Ginzburg-Landu equations, Arch. Rat. Mech. Anal., 133 (1995), 103–127.Google Scholar
[27]Lin, F. and Du, Q., Ginzburg-Landau vortices: Dynamics, pining and hysteresis, SIAM J. Math. Anal., 28 (1997), 1265–1293.Google Scholar
[28]Neu, J., Vortices in complex scalar fields, Phys. D, 43 (1990), 385–406.Google Scholar
[29]Peres, L. and Rubinstein, J., Vortex dynamics for U(1)-Ginzburg-Landau models, Phys. D, 64 (1993), 299–309.Google Scholar
[30]Rubinstein, J. and Sternberg, P., On the slow motion of vortices in the Ginaburg-Landau heat flow, SIAM J. Appl. Math., 26 (1995), 1452–1466.Google Scholar
[31]Sandier, E. and Serfaty, S., Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627–1672.Google Scholar
[32]Serfaty, S., Stability in 2D Ginzburg-Landau pass to the limit, Indiana U. Math. J., 54 (2005), 199–221.Google Scholar
[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), 383–426.Google Scholar
[34]Shen, J. and Tang, T., Spectral and High-Order Method with Applications, Science Press, 2006.Google Scholar
[35]Strang, G., On the construction and comparision of difference schemes, SIAM J. Numer. Anal., 5 (1968), 505–517.Google Scholar
[36]Weinstein, M. I. and Xin, J., Dynamic stability of vortex solutions of Ginzburg-Landau and nonlinear Schrödinger equations, Comm. Math. Phys., 180 (1996), 389–428.Google Scholar
[37]Zhang, Y., Bao, W. and Du, Q., Numerical simulation of vortex dynamics in Ginzburg-Landau-Schroädinger equation, Euro. J. Appl. Math., 18 (2007), 607–630.Google Scholar
[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), 1740–1775.Google Scholar