Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T01:09:17.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation

Published online by Cambridge University Press:  01 October 2007

YANZHI ZHANG ,
WEIZHU BAO  and
QIANG DU
YANZHI ZHANG
Affiliation:
Department of Mathematics, National University of Singapore, Singapore117543 email: [email protected]
WEIZHU BAO
Affiliation:
Department of Mathematics and Center for Computational Science and Engineering National University of Singapore, Singapore117543 email: [email protected], URL: http://www.math.nus.edu.sg/~bao/
QIANG DU
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA email: [email protected], URL: http://www.math.psu.edu/qdu
Get access Rights & Permissions [Opens in a new window]

Abstract

The rich dynamics of quantized vortices governed by the Ginzburg-Landau-Schrödinger equation (GLSE) is an interesting problem studied in many application fields. Although recent mathematical analysis and numerical simulations have led to a much better understanding of such dynamics, many important questions remain open. In this article, we consider numerical simulations of the GLSE in two dimensions with non-zero far-field conditions. Using two-dimensional polar coordinates, transversely highly oscillating far-field conditions can be efficiently resolved in the phase space, thus giving rise to an unconditionally stable, efficient and accurate time-splitting method for the problem under consideration. This method is also time reversible for the case of the non-linear Schrödinger equation. By applying this numerical method to the GLSE, we obtain some conclusive experimental findings on issues such as the stability of quantized vortex, interaction of two vortices, dynamics of the quantized vortex lattice and the motion of vortex with an inhomogeneous external potential. Discussions on these simulation results and the recent theoretical studies are made to provide further understanding of the vortex stability and vortex dynamics described by the GLSE.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 5 , October 2007 , pp. 607 - 630
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adler, S. & Piran, T. (1984) Relaxation methods for gauge field equilibrium equations. Rev. Mod. Phys., 56, 1–40.Google Scholar
[2]Aranson, I. & Kramer, L. (2002) The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99–133.Google Scholar
[3]Bao, W. (2004) Numerical methods for the nonlinear Schrödinger equation with nonzero far-field conditions. Methods Appl. Anal., 11, 367–388.CrossRefGoogle Scholar
[4]Bao, W., Du, Q. & Zhang, Y. (2006) Dynamics of rotating Bose-Einstein condensates and their efficient and accurate numerical computation. SIAM J. Appl. Math., 66, 758–786.Google Scholar
[5]Bao, W. & Jaksch, D. (2003) An explicit unconditionally stable numerical methods for solving damped nonlinear Schrodinger equations with a focusing nonlinearity. SIAM J. Numer. Anal., 41, 1406–1426.CrossRefGoogle Scholar
[6]Bao, W., Jaksch, D. & Markowich, P. A. (2003) Numerical solution of the Gross-Pitaevskii Equation for Bose-Einstein condensation. J. Comput. Phys., 187, 318–342.Google Scholar
[7]Bao, W., Jin, S. & Markowich, P. A. (2002) On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175, 487–524.Google Scholar
[8]Bao, W., Jin, S. & Markowich, P. A. (2003) Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semi-classical regimes. SIAM J. Sci. Comp. 25, 27–64.Google Scholar
[9]Bao, W. & Zhang, Y. (2005) Dynamics of the ground state and central vortex states in Bose-Einstein condensation. Math. Mod. Meth. Appl. Sci. 15, 1863–1896.CrossRefGoogle Scholar
[10]Besse, C., Bidegaray, B. & Descombes, S. (2002) Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40, 26–40.CrossRefGoogle Scholar
[11]Chapman, J. & Richardson, G. (1995) Motion of vortices in type-II superconductors. SIAM J. Appl. Math. 55, 1275–1296.Google Scholar
[12]Chen, X. F., Elliott, C. M. & Tang, Q. (1994) Shooting method for vortex solutions of a complex-valued Ginzburg-Landau equation. Proc. R. Soc. Edinb. A—Math. 124, 1075–1088.Google Scholar
[13]Chen, Z. & Dai, S. (2001) Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity. SIAM J. Numer. Anal. 38, 1961–1985.Google Scholar
[14]Colliander, J. E. & Jerrard, R. L. (1998) Vortex dynamics for the Ginzburg-Landau-Schrödinger equation. IMRN Int. Math. Res. Not. 7, 333–358.CrossRefGoogle Scholar
[15]Deang, J., Du, Q. & Gunzburger, M. (2001) Stochastic dynamics of Ginzburg-Landau vortices in superconductors. Phys. Rev. B 64, 52506.CrossRefGoogle Scholar
[16]Deang, J., Du, Q., Gunzburger, M. & Peterson, J. (1997) Vortices in superconductors: Modeling and computer simulations. Philos. Trans. R. Soc. Lond. Ser. A 355, 1957–1968.CrossRefGoogle Scholar
[17]Du, Q. (1994) Finite element methods for the time dependent Ginzburg-Landau model of superconductivity. Comp. Math. Appl. 27, 119–133.Google Scholar
[18]Du, Q. (2003) Diverse vortex dynamics in superfluids. Contemp. Math. 329 105–117.CrossRefGoogle Scholar
[19]Du, Q., Gunzburger, M. & Peterson, J. (1992) Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34, 54–81.Google Scholar
[20]Du, Q., Gunzburger, M. & Peterson, J. (1995) Computational simulation of type-II superconductivity including pinning phenomena. Phys. Rev. B 51, 16194–16203.Google Scholar
[21]Du, Q. & Zhu, W. (2004) Stability analysis and applications of the exponential time differencing schemes. J. Comput. Math. 22, 200–209.Google Scholar
[22]E, W. (1994) Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity. Physica D 77, 383–404.Google Scholar
[23]Glowinski, R. & Tallec, P. L.Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, PA, 1989.CrossRefGoogle Scholar
[24]Jerrard, R. L. & Soner, H. M. (1998) Dynamics of Ginzburg-Landau vortices. Arch. Rational Mech. Anal. 142, 99–125.Google Scholar
[25]Jian, H. Y. (2000) The dynamical law of Ginzburg-Landau vortices with a pinning effect. Appl. Math. Lett. 13, 91–94.Google Scholar
[26]Jian, H. Y. & Song, B. H. (2001) Vortex dynamics of Ginzburg-Landau equations in inhomogeneous superconductors. J. Differential Equations 170, 123–141.CrossRefGoogle Scholar
[27]Jian, H. Y. & Wang, Y. D. (2002) Ginzburg-Landau vortices in inhomogeneous superconductors. J. Partial Differential Equations 15, 45–60.Google Scholar
[28]Karakashian, O. & Makridakis, C. (1998) A space-time finite element method for the nonlinear Schrödinger equation: The discontinuous Galerkin method. Math. Comp., 67, 479–499.Google Scholar
[29]Kwong, M. (1995) On the one-dimensional Ginzburg-Landau BVPs. Differential Int. Equations 8, 1395–1405.Google Scholar
[30]Lai, M.-C. & Wang, W.-C. (2002) Fast direct solvers for Poisson equation on 2D polar and spherical geometries. Numer. Methods Partial Differential Equation 18, 56–58.Google Scholar
[31]Lange, O. & Schroers, B. J. (2002) Unstable manifolds and Schrödinger dynamics of Ginzburg-Landau vortices. Nonlinearity 15, 1471–1488.Google Scholar
[32]Lin, F.-H. (1996) Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure. Appl. Math. XLIX, 323–359.Google Scholar
[33]Lin, F.-H. (1998) Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds. Comm. Pure Appl. Math. LI, 0385–0441.3.0.CO;2-5>CrossRefGoogle Scholar
[34]Lin, F.-H. & Du, Q. (1997) Ginzburg-Landau vortices: Dynamics, pinning, and hysteresis. SIAM J. Math. Anal. 28, 1265–1293.Google Scholar
[35]Lin, F.-H. & Xin, J. X. (1999) On the dynamical law of the Ginzburg-Landau vortices on the plane. Comm. Pure Appl. Math. LII, 1189–1212.Google Scholar
[36]Neu, J. C. (1990a) Vortices in complex scalar fields. Physica D 43, 385–406.Google Scholar
[37]Neu, J. C. (1990b) Vortex dynamics of the nonlinear wave equation. Physica D 43, 407– 420.Google Scholar
[38]Ovchinnikov, Y. N. & Sigal, I. M. (1997) The Ginzburg-Landau equation I. Static vortices. Partial Differential Equations Appl. CRM Pro. 12, 199–220.Google Scholar
[39]Ovchinnikov, Y. N. & Sigal, I. M. (1998a) Long-time behaviour of Ginzburg-Landau vortices. Nonlinearity 11, 1295–1309.Google Scholar
[40]Ovchinnikov, Y. N. & Sigal, I. M. (1998b) The Ginzburg-Landau equation III. Vortex dynamics. Nonlinearity 11, 1277–1294.Google Scholar
[41]Ovchinnikov, Y. N. & Sigal, I. M. (2000) Asymptotic behaviour of solutions of Ginzburg-Landau and related equations. Rev. Math. Phys. 12, 287–299.Google Scholar
[42]Ovchinnikov, Y. N. & Sigal, I. M. (2004) Symmetric breaking solutions to the Ginzburg-Landau equation. J. Exp. Theor. Phys. 99, 1090–1107.Google Scholar
[43]Peres, L. & Rubinstein, J. (1993) Vortex dynamics for U(1)-Ginzburg-Landau models. Physica D 64, 299–309.Google Scholar
[44]Pismen, L. & Rodriguez, J. D. (1990) Mobilities of singularities in dissipative Ginzburg-Landau equations. Phys. Rev. A 42, 2471–2474.Google Scholar
[45]Strang, G. (1968) On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 505–517.Google Scholar
[46]Taha, T. R. & Ablowitz, M. J. (1984) Analytical and numerical aspects of certain nonlinear evolution equations II: Numerical, nonlinear Schrödinger equation. J. Comput. Phys. 55, 203–230.Google Scholar
[47]Weideman, J. & Herbst, B. (1986) Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 23, 485–507.Google Scholar
[48]Weinstein, M. I. & Xin, J. (1996) Dynamics stability of vortex solutions of Ginzburg-Landau and nonlinear Schrödinger equations, Comm. Math. Phys. 180, 389–428.CrossRefGoogle Scholar
[49]Zhang, Y., Bao, W. & Du, Q. (to appear) The dynamics and interaction of quantized vortices in Ginzburg-Landau-Schrödinger equation. SIAM J. Appl. Math.Google Scholar