August  2018, 23(6): 2265-2297. doi: 10.3934/dcdsb.2018096

Quantized vortex dynamics and interaction patterns in superconductivity based on the reduced dynamical law

1. 

School of Mathematics, Jilin University, Changchun 130012, China

2. 

Department of Mathematics, National University of Singapore, 119076, Singapore

Received  December 2016 Revised  October 2017 Published  March 2018

We study analytically and numerically stability and interaction patterns of quantized vortex lattices governed by the reduced dynamical lawa system of ordinary differential equations (ODEs) - in superconductivity. By deriving several non-autonomous first integrals of the ODEs, we obtain qualitatively dynamical properties of a cluster of quantized vortices, including global existence, finite time collision, equilibrium solution and invariant solution manifolds. For a vortex lattice with 3 vortices, we establish orbital stability when they have the same winding number and find different collision patterns when they have different winding numbers. In addition, under several special initial setups, we can obtain analytical solutions for the nonlinear ODEs.

Citation: Zhiguo Xu, Weizhu Bao, Shaoyun Shi. Quantized vortex dynamics and interaction patterns in superconductivity based on the reduced dynamical law. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2265-2297. doi: 10.3934/dcdsb.2018096
References:
[1]

W. Bao, Numerical methods for the nonlinear Schrödinger equation with nonzero far-field conditions, Methods Appl. Anal., 11 (2004), 367-387.  doi: 10.4310/MAA.2004.v11.n3.a8.  Google Scholar

[2]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Mod., 6 (2013), 1-135.   Google Scholar

[3]

W. Bao and Q. Tang, Numerical study of quantized vortex interaction in the nonlinear Schroedinger equation on bounded domains, Multiscale Model. Simul., 12 (2014), 411-439.  doi: 10.1137/130906489.  Google Scholar

[4]

W. Bao and Q. Tang, Numerical study of quantized vortex interaction in the Ginzburg-Landau equation on bounded domains, Commun. Comput. Phys., 14 (2013), 819-850.  doi: 10.4208/cicp.250112.061212a.  Google Scholar

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W. BaoR. Zeng and Y. Zhang, Quantized vortex stability and interaction in the nonlinear wave equation, Phys. D, 237 (2008), 2391-2410.  doi: 10.1016/j.physd.2008.03.026.  Google Scholar

[6]

P. BaumanC. ChenD. Phillips and P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems, European J. Appl. Math., 6 (1995), 115-126.   Google Scholar

[7]

F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser, Boston, 1994.  Google Scholar

[8]

S. J. Chapman and G. Richardson, Motion of vortices in type Ⅱ superconductors, SIAM J. Appl. Math., 55 (1995), 1275-1296.  doi: 10.1137/S0036139994263872.  Google Scholar

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J. E. Colliander and R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau-Schrödinger equation, Internat. Math. Res. Notices, 7 (1998), 333-358.   Google Scholar

[10]

Q. Du, Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity, Comput. Math. Appl., 27 (1994), 119-133.  doi: 10.1016/0898-1221(94)90091-4.  Google Scholar

[11]

W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Phys. D, 77 (1994), 383-404.  doi: 10.1016/0167-2789(94)90298-4.  Google Scholar

[12]

R. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech., 142 (1998), 99-125.  doi: 10.1007/s002050050085.  Google Scholar

[13]

A. KleinD. JakschY. Zhang and W. Bao, Dynamics of vortices in weakly interacting BoseEinstein condensates, Phys. Rev. A, 76 (2007), 043602.  doi: 10.1103/PhysRevA.76.043602.  Google Scholar

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S. Kowalevski, Sur la probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232.  doi: 10.1007/BF02592182.  Google Scholar

[15] V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, SpringerVerlag, Berlin, 1996.   Google Scholar
[16]

O. Lange and B. Schroers, Unstable manifolds and Schrödinger dynamics of Ginzburg-Landau vortices, Nonlinearity, 15 (2002), 1471-1488.  doi: 10.1088/0951-7715/15/5/307.  Google Scholar

[17]

F. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323-360.  doi: 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E.  Google Scholar

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F. Lin, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds, Comm. Pure Appl. Math., 51 (1998), 385-441.  doi: 10.1002/(SICI)1097-0312(199804)51:4<385::AID-CPA3>3.0.CO;2-5.  Google Scholar

[19]

F. Lin and J. Xin, On the dynamical law of the Ginzburg-Landau vortices on the plane, Comm. Pure Appl. Math., 52 (1999), 1189-1212.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1189::AID-CPA1>3.0.CO;2-T.  Google Scholar

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P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), 334-344.  doi: 10.1006/jfan.1995.1073.  Google Scholar

[21]

P. K. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective, SIAM Rev., 51 (2009), 501-542.  doi: 10.1137/07068597X.  Google Scholar

[22]

J. Neu, Vortices in complex scalar fields, Phys. D, 43 (1990), 385-406.  doi: 10.1016/0167-2789(90)90143-D.  Google Scholar

[23]

J. Neu, Vortex dynamics of the nonlinear wave equation, Phys. D, 43 (1990), 407-420.  doi: 10.1016/0167-2789(90)90144-E.  Google Scholar

[24]

Y. Ovchinnikov and I. Sigal, Long-time behavior of Ginzburg-Landau vortices, Nonlinearity, 11 (1998), 1295-1309.  doi: 10.1088/0951-7715/11/5/007.  Google Scholar

[25]

Y. Ovchinnikov and I. Sigal, Asymptotic behavior of solutions of Ginzburg-Landau and relate equations, Rev. Math. Phys., 12 (2000), 287-299.  doi: 10.1142/S0129055X00000101.  Google Scholar

[26] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003.   Google Scholar
[27]

H. Poincaré, Sur l'intégrations des équations différentielles du premier order et du premier degré Ⅰ and Ⅱ, Rend. Circ. Mat. Palermo, 5 (1891), 161-191; 11 (1897), 193-239. Google Scholar

[28]

E. Sandier, The symmetry of minimizing harmonic maps from a two-dimernsional domain to the sphere, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 549-559.  doi: 10.1016/S0294-1449(16)30204-9.  Google Scholar

[29]

W. Shen and X. Zhao, Convergence in almost periodic cooperative systems with a first integral, Proc. Amer. Math. Soc., 133 (2005), 203-212.  doi: 10.1090/S0002-9939-04-07556-2.  Google Scholar

[30]

Y. ZhangW. Bao and Q. Du, The dynamics and interaction of quantized vortices in the Ginzburg-Landau-Schrödinger equation, SIAM J. Appl. Math., 67 (2007), 1740-1775.  doi: 10.1137/060671528.  Google Scholar

[31]

Y. ZhangW. Bao and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, European J. Appl. Math., 18 (2007), 607-630.   Google Scholar

show all references

References:
[1]

W. Bao, Numerical methods for the nonlinear Schrödinger equation with nonzero far-field conditions, Methods Appl. Anal., 11 (2004), 367-387.  doi: 10.4310/MAA.2004.v11.n3.a8.  Google Scholar

[2]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Mod., 6 (2013), 1-135.   Google Scholar

[3]

W. Bao and Q. Tang, Numerical study of quantized vortex interaction in the nonlinear Schroedinger equation on bounded domains, Multiscale Model. Simul., 12 (2014), 411-439.  doi: 10.1137/130906489.  Google Scholar

[4]

W. Bao and Q. Tang, Numerical study of quantized vortex interaction in the Ginzburg-Landau equation on bounded domains, Commun. Comput. Phys., 14 (2013), 819-850.  doi: 10.4208/cicp.250112.061212a.  Google Scholar

[5]

W. BaoR. Zeng and Y. Zhang, Quantized vortex stability and interaction in the nonlinear wave equation, Phys. D, 237 (2008), 2391-2410.  doi: 10.1016/j.physd.2008.03.026.  Google Scholar

[6]

P. BaumanC. ChenD. Phillips and P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems, European J. Appl. Math., 6 (1995), 115-126.   Google Scholar

[7]

F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser, Boston, 1994.  Google Scholar

[8]

S. J. Chapman and G. Richardson, Motion of vortices in type Ⅱ superconductors, SIAM J. Appl. Math., 55 (1995), 1275-1296.  doi: 10.1137/S0036139994263872.  Google Scholar

[9]

J. E. Colliander and R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau-Schrödinger equation, Internat. Math. Res. Notices, 7 (1998), 333-358.   Google Scholar

[10]

Q. Du, Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity, Comput. Math. Appl., 27 (1994), 119-133.  doi: 10.1016/0898-1221(94)90091-4.  Google Scholar

[11]

W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Phys. D, 77 (1994), 383-404.  doi: 10.1016/0167-2789(94)90298-4.  Google Scholar

[12]

R. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech., 142 (1998), 99-125.  doi: 10.1007/s002050050085.  Google Scholar

[13]

A. KleinD. JakschY. Zhang and W. Bao, Dynamics of vortices in weakly interacting BoseEinstein condensates, Phys. Rev. A, 76 (2007), 043602.  doi: 10.1103/PhysRevA.76.043602.  Google Scholar

[14]

S. Kowalevski, Sur la probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232.  doi: 10.1007/BF02592182.  Google Scholar

[15] V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, SpringerVerlag, Berlin, 1996.   Google Scholar
[16]

O. Lange and B. Schroers, Unstable manifolds and Schrödinger dynamics of Ginzburg-Landau vortices, Nonlinearity, 15 (2002), 1471-1488.  doi: 10.1088/0951-7715/15/5/307.  Google Scholar

[17]

F. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323-360.  doi: 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E.  Google Scholar

[18]

F. Lin, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds, Comm. Pure Appl. Math., 51 (1998), 385-441.  doi: 10.1002/(SICI)1097-0312(199804)51:4<385::AID-CPA3>3.0.CO;2-5.  Google Scholar

[19]

F. Lin and J. Xin, On the dynamical law of the Ginzburg-Landau vortices on the plane, Comm. Pure Appl. Math., 52 (1999), 1189-1212.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1189::AID-CPA1>3.0.CO;2-T.  Google Scholar

[20]

P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), 334-344.  doi: 10.1006/jfan.1995.1073.  Google Scholar

[21]

P. K. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective, SIAM Rev., 51 (2009), 501-542.  doi: 10.1137/07068597X.  Google Scholar

[22]

J. Neu, Vortices in complex scalar fields, Phys. D, 43 (1990), 385-406.  doi: 10.1016/0167-2789(90)90143-D.  Google Scholar

[23]

J. Neu, Vortex dynamics of the nonlinear wave equation, Phys. D, 43 (1990), 407-420.  doi: 10.1016/0167-2789(90)90144-E.  Google Scholar

[24]

Y. Ovchinnikov and I. Sigal, Long-time behavior of Ginzburg-Landau vortices, Nonlinearity, 11 (1998), 1295-1309.  doi: 10.1088/0951-7715/11/5/007.  Google Scholar

[25]

Y. Ovchinnikov and I. Sigal, Asymptotic behavior of solutions of Ginzburg-Landau and relate equations, Rev. Math. Phys., 12 (2000), 287-299.  doi: 10.1142/S0129055X00000101.  Google Scholar

[26] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003.   Google Scholar
[27]

H. Poincaré, Sur l'intégrations des équations différentielles du premier order et du premier degré Ⅰ and Ⅱ, Rend. Circ. Mat. Palermo, 5 (1891), 161-191; 11 (1897), 193-239. Google Scholar

[28]

E. Sandier, The symmetry of minimizing harmonic maps from a two-dimernsional domain to the sphere, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 549-559.  doi: 10.1016/S0294-1449(16)30204-9.  Google Scholar

[29]

W. Shen and X. Zhao, Convergence in almost periodic cooperative systems with a first integral, Proc. Amer. Math. Soc., 133 (2005), 203-212.  doi: 10.1090/S0002-9939-04-07556-2.  Google Scholar

[30]

Y. ZhangW. Bao and Q. Du, The dynamics and interaction of quantized vortices in the Ginzburg-Landau-Schrödinger equation, SIAM J. Appl. Math., 67 (2007), 1740-1775.  doi: 10.1137/060671528.  Google Scholar

[31]

Y. ZhangW. Bao and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, European J. Appl. Math., 18 (2007), 607-630.   Google Scholar

Figure 2.1.  Illustrations of a finite time collision of a vortex dipole in a vortex cluster with 3 vortices (a) and a (finite time) collision cluster with 3 vortices in a vortex cluster with 5 vortices (b). Here and in the following figures, '+' and '$-$' denote the initial vortex centers with winding numbers $m = +1$ and $m = -1$, respectively; and 'o' denotes the finite time collision position
Figure 3.1.  Interaction of $3$ vortices with the same winding number (a and b) and opposite winding numbers (c)
Figure 4.1.  Time evolution of $\rho_1(t)$ (left) and $\rho_2(t)$ (right) of (4.12) with $\rho_1^0 = 1$ and $\rho_2^0 = 4$ for different $n\ge2$
Figure 4.2.  Time evolution of $\rho_1(t)$ (left) and $\rho_2(t)$ (right) of (4.20) with $\rho_1^0 = 1$ and $\rho_2^0 = 4$ for different $n\ge2$
Figure 4.3.  Time evolution of $\rho_1(t)$ (left) and $\rho_2(t)$ (right) of (4.26) with $\rho_1^0 = 1$ and $\rho_2^0 = 4$ for different $n\ge2$
Figure 4.4.  Time evolution of $\rho_1(t)$ (left) and $\rho_2(t)$ (right) of (4.32) with $\rho_1^0 = 1$ and $\rho_2^0 = 4$ for different $n\ge2$
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