Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 14
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Li, Leonard Z. Sun, Hai-Wei and Tam, Sik-Chung 2015. A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schrödinger equations. Computer Physics Communications, Vol. 187, p. 38.

    Yan, Yun Moxley, Frederick Ira and Dai, Weizhong 2015. A new compact finite difference scheme for solving the complex Ginzburg–Landau equation. Applied Mathematics and Computation, Vol. 260, p. 269.

    Bao, Weizhu and Tang, Qinglin 2014. Numerical Study of Quantized Vortex Interactions in the Nonlinear Schrödinger Equation on Bounded Domains. Multiscale Modeling & Simulation, Vol. 12, Issue. 2, p. 411.

    Caplan, R. M. and Carretero-González, R. 2014. A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and Its Application to the Nonlinear Schrödinger Equation. SIAM Journal on Scientific Computing, Vol. 36, Issue. 1, p. A1.

    Jiang, Wei and Tang, Qinglin 2013. Numerical study of quantized vortex interaction in complex Ginzburg–Landau equation on bounded domains. Applied Mathematics and Computation, Vol. 222, p. 210.

    Wang, Shanshan and Zhang, Luming 2013. An efficient split-step compact finite difference method for cubic–quintic complex Ginzburg–Landau equations. Computer Physics Communications, Vol. 184, Issue. 6, p. 1511.

    Zhang, Ya-nan Sun, Zhi-zhong and Wang, Ting-chun 2013. Convergence analysis of a linearized Crank-Nicolson scheme for the two-dimensional complex Ginzburg-Landau equation. Numerical Methods for Partial Differential Equations, Vol. 29, Issue. 5, p. 1487.

    Gelantalis, Michael and Sternberg, Peter 2012. Rotating 2N-vortex solutions to the Gross-Pitaevskii equation on S2. Journal of Mathematical Physics, Vol. 53, Issue. 8, p. 083701.

    Smirnov, L. A. and Mironov, V. A. 2012. Dynamics of two-dimensional dark quasisolitons in a smoothly inhomogeneous Bose-Einstein condensate. Physical Review A, Vol. 85, Issue. 5,

    Kong, Linghua Hong, Jialin Fu, Fangfang and Chen, Jing 2011. Symplectic structure-preserving integrators for the two-dimensional Gross–Pitaevskii equation for BEC. Journal of Computational and Applied Mathematics, Vol. 235, Issue. 17, p. 4937.

    Zhang, Yanzhi Peterson, Janet and Gunzburger, Max 2011. Maximizing critical currents in superconductors by optimization of normal inclusion properties. Physica D: Nonlinear Phenomena, Vol. 240, Issue. 21, p. 1701.

    Wang, Hanquan 2010. An efficient Chebyshev–Tau spectral method for Ginzburg–Landau–Schrödinger equations. Computer Physics Communications, Vol. 181, Issue. 2, p. 325.

    Bao, Weizhu Zeng, Rong and Zhang, Yanzhi 2008. Quantized vortex stability and interaction in the nonlinear wave equation. Physica D: Nonlinear Phenomena, Vol. 237, Issue. 19, p. 2391.

    Du, Qiang 2007. Quantized vortices in BEC and superconductors. PAMM, Vol. 7, Issue. 1, p. 1023901.

  • European Journal of Applied Mathematics, Volume 18, Issue 5
  • October 2007, pp. 607-630

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

  • YANZHI ZHANG (a1), WEIZHU BAO (a2) and QIANG DU (a3)
  • DOI:
  • Published online: 01 October 2007

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:
  • EISSN:
  • 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? *