Skip to main content
    • Aa
    • Aa

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

  • YANZHI ZHANG (a1), WEIZHU BAO (a2) and QIANG DU (a3)

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] S. Adler & T. Piran (1984) Relaxation methods for gauge field equilibrium equations. Rev. Mod. Phys., 56, 140.

[2] I. Aranson & L. Kramer (2002) The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99133.

[4] W. Bao , Q. Du & Y. Zhang (2006) Dynamics of rotating Bose-Einstein condensates and their efficient and accurate numerical computation. SIAM J. Appl. Math., 66, 758786.

[5] W. Bao & D. Jaksch (2003) An explicit unconditionally stable numerical methods for solving damped nonlinear Schrodinger equations with a focusing nonlinearity. SIAM J. Numer. Anal., 41, 14061426.

[6] W. Bao , D. Jaksch & P. A. Markowich (2003) Numerical solution of the Gross-Pitaevskii Equation for Bose-Einstein condensation. J. Comput. Phys., 187, 318342.

[7] W. Bao , S. Jin & P. A. Markowich (2002) On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175, 487524.

[8] W. Bao , S. Jin & P. A. Markowich (2003) Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semi-classical regimes. SIAM J. Sci. Comp. 25, 2764.

[9] W. Bao & Y. Zhang (2005) Dynamics of the ground state and central vortex states in Bose-Einstein condensation. Math. Mod. Meth. Appl. Sci. 15, 18631896.

[10] C. Besse , B. Bidegaray & S. Descombes (2002) Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40, 2640.

[11] J. Chapman & G. Richardson (1995) Motion of vortices in type-II superconductors. SIAM J. Appl. Math. 55, 12751296.

[13] Z. Chen & S. Dai (2001) Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity. SIAM J. Numer. Anal. 38, 19611985.

[15] J. Deang , Q. Du & M. Gunzburger (2001) Stochastic dynamics of Ginzburg-Landau vortices in superconductors. Phys. Rev. B 64, 52506.

[16] J. Deang , Q. Du , M Gunzburger . & J. Peterson (1997) Vortices in superconductors: Modeling and computer simulations. Philos. Trans. R. Soc. Lond. Ser. A 355, 19571968.

[17] Q. Du (1994) Finite element methods for the time dependent Ginzburg-Landau model of superconductivity. Comp. Math. Appl. 27, 119133.

[18] Q. Du (2003) Diverse vortex dynamics in superfluids. Contemp. Math. 329 105117.

[19] Q. Du , M. Gunzburger & J. Peterson (1992) Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34, 5481.

[20] Q. Du , M. Gunzburger & J. Peterson (1995) Computational simulation of type-II superconductivity including pinning phenomena. Phys. Rev. B 51, 1619416203.

[24] R. L. Jerrard & H. M. Soner (1998) Dynamics of Ginzburg-Landau vortices. Arch. Rational Mech. Anal. 142, 99125.

[25] H. Y. Jian (2000) The dynamical law of Ginzburg-Landau vortices with a pinning effect. Appl. Math. Lett. 13, 9194.

[26] H. Y. Jian & B. H. Song (2001) Vortex dynamics of Ginzburg-Landau equations in inhomogeneous superconductors. J. Differential Equations 170, 123141.

[28] O. Karakashian & C. Makridakis (1998) A space-time finite element method for the nonlinear Schrödinger equation: The discontinuous Galerkin method. Math. Comp., 67, 479499.

[30] M.-C. Lai & W.-C. Wang (2002) Fast direct solvers for Poisson equation on 2D polar and spherical geometries. Numer. Methods Partial Differential Equation 18, 5658.

[31] O. Lange & B. J. Schroers (2002) Unstable manifolds and Schrödinger dynamics of Ginzburg-Landau vortices. Nonlinearity 15, 14711488.

[34] F.-H. Lin & Q. Du (1997) Ginzburg-Landau vortices: Dynamics, pinning, and hysteresis. SIAM J. Math. Anal. 28, 12651293.

[36] J. C. Neu (1990a) Vortices in complex scalar fields. Physica D 43, 385406.

[37] J. C. Neu (1990b) Vortex dynamics of the nonlinear wave equation. Physica D 43, 407420.

[39] Y. N. Ovchinnikov & I. M. Sigal (1998a) Long-time behaviour of Ginzburg-Landau vortices. Nonlinearity 11, 12951309.

[40] Y. N. Ovchinnikov & I. M. Sigal (1998b) The Ginzburg-Landau equation III. Vortex dynamics. Nonlinearity 11, 12771294.

[41] Y. N. Ovchinnikov & I. M. Sigal (2000) Asymptotic behaviour of solutions of Ginzburg-Landau and related equations. Rev. Math. Phys. 12, 287299.

[42] Y. N. Ovchinnikov & I. M. Sigal (2004) Symmetric breaking solutions to the Ginzburg-Landau equation. J. Exp. Theor. Phys. 99, 10901107.

[43] L. Peres & J. Rubinstein (1993) Vortex dynamics for U(1)-Ginzburg-Landau models. Physica D 64, 299309.

[44] L. Pismen & J. D. Rodriguez (1990) Mobilities of singularities in dissipative Ginzburg-Landau equations. Phys. Rev. A 42, 24712474.

[45] G. Strang (1968) On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 505517.

[46] T. R. Taha & M. J. Ablowitz (1984) Analytical and numerical aspects of certain nonlinear evolution equations II: Numerical, nonlinear Schrödinger equation. J. Comput. Phys. 55, 203230.

[47] J. Weideman & B. Herbst (1986) Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 23, 485507.

[48] M. I. Weinstein & J. Xin (1996) Dynamics stability of vortex solutions of Ginzburg-Landau and nonlinear Schrödinger equations, Comm. Math. Phys. 180, 389428.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 20 *
Loading metrics...

Abstract views

Total abstract views: 184 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd September 2017. This data will be updated every 24 hours.