# American Institute of Mathematical Sciences

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  August 2018 Early access  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 and 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. [2] W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Mod., 6 (2013), 1-135. [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. [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. [5] W. Bao, R. 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. [6] P. Bauman, C. Chen, D. Phillips and P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems, European J. Appl. Math., 6 (1995), 115-126. [7] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser, Boston, 1994. [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. [9] J. E. Colliander and R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau-Schrödinger equation, Internat. Math. Res. Notices, 7 (1998), 333-358. [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. [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. [12] R. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech., 142 (1998), 99-125.  doi: 10.1007/s002050050085. [13] A. Klein, D. Jaksch, Y. Zhang and W. Bao, Dynamics of vortices in weakly interacting BoseEinstein condensates, Phys. Rev. A, 76 (2007), 043602.  doi: 10.1103/PhysRevA.76.043602. [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. [15] V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, SpringerVerlag, Berlin, 1996. [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. [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. [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. [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. [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. [21] P. K. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective, SIAM Rev., 51 (2009), 501-542.  doi: 10.1137/07068597X. [22] J. Neu, Vortices in complex scalar fields, Phys. D, 43 (1990), 385-406.  doi: 10.1016/0167-2789(90)90143-D. [23] J. Neu, Vortex dynamics of the nonlinear wave equation, Phys. D, 43 (1990), 407-420.  doi: 10.1016/0167-2789(90)90144-E. [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. [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. [26] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003. [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. [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. [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. [30] Y. Zhang, W. 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. [31] Y. Zhang, W. Bao and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, European J. Appl. Math., 18 (2007), 607-630.

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##### 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. [2] W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Mod., 6 (2013), 1-135. [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. [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. [5] W. Bao, R. 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. [6] P. Bauman, C. Chen, D. Phillips and P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems, European J. Appl. Math., 6 (1995), 115-126. [7] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser, Boston, 1994. [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. [9] J. E. Colliander and R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau-Schrödinger equation, Internat. Math. Res. Notices, 7 (1998), 333-358. [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. [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. [12] R. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech., 142 (1998), 99-125.  doi: 10.1007/s002050050085. [13] A. Klein, D. Jaksch, Y. Zhang and W. Bao, Dynamics of vortices in weakly interacting BoseEinstein condensates, Phys. Rev. A, 76 (2007), 043602.  doi: 10.1103/PhysRevA.76.043602. [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. [15] V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, SpringerVerlag, Berlin, 1996. [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. [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. [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. [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. [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. [21] P. K. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective, SIAM Rev., 51 (2009), 501-542.  doi: 10.1137/07068597X. [22] J. Neu, Vortices in complex scalar fields, Phys. D, 43 (1990), 385-406.  doi: 10.1016/0167-2789(90)90143-D. [23] J. Neu, Vortex dynamics of the nonlinear wave equation, Phys. D, 43 (1990), 407-420.  doi: 10.1016/0167-2789(90)90144-E. [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. [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. [26] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003. [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. [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. [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. [30] Y. Zhang, W. 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. [31] Y. Zhang, W. Bao and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, European J. Appl. Math., 18 (2007), 607-630.
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
Interaction of $3$ vortices with the same winding number (a and b) and opposite winding numbers (c)
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 $n\ge2$
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 $n\ge2$
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 $n\ge2$
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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