# 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  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
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##### References:
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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