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

• 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.

Mathematics Subject Classification: Primary: 34C60, 34D05; Secondary: 34A33, 34D30, 65L07.

 Citation: • • 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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