The Dynamics and Interaction of Quantized Vortices in the Ginzburg–Landau–Schrödinger Equation

Abstract

Abstract. The dynamic laws of quantized vortex interactions in the Ginzburg-Landau-Schrödinger equation (GLSE) are analytically and numerically studied. A review of the reduced dynamic laws governing the motion of vortex centers in the GLSE is provided. The reduced dynamic laws are solved analytically for some special initial data. By directly simulating the GLSE with an efficient and accurate numerical method proposed recently in [Y. Zhang, W. Bao, and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, European J. Appl. Math., to appear], we can qualitatively and quantitatively compare quantized vortex interaction patterns of the GLSE with those from the reduced dynamic laws. Some conclusive findings are obtained, and discussions on numerical and theoretical results are made to provide further understanding of vortex interactions in the GLSE. Finally, the vortex motion under an inhomogeneous potential in the GLSE is also studied.

Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains

“…Different steady state patterns of vortex lattices under the GLE dynamics were obtained numerically. From our numerical results, we observed that boundary conditions and domain geometry affect significantly on vortex dynamics and interaction, which showed different interaction patterns compared to those in the whole space case [37,38]. tion of Singapore grant R-146-000-120-112.…”

“…Based on the reduced dynamical laws which are sets of ordinary differential equations (ODEs) for the vortex centers, one can obtain that two vortices with opposite winding number attract each other, while the ones with the same winding number repel. Recently, by proposing efficient and accurate numerical methods for discretizing the GLE in the whole space, Zhang et al [37,38] compared the dynamics of quantized vortices from the reduced dynamical laws obtained by Neu with those from the direct numerical simulation results from GLE under different parameter and/or initial setups. They identified numerically the parameter regimes for quantized vortex dynamics when the reduced dynamical laws agree qualitatively and/or quantitatively and fail to agree with those from GLE dynamics [37,38].…”

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

“…Furthermore, we also carried out a detailed simulation of the NLSE dynamics of vortex lattice with 25 vortices, which is extremely challenging computationally and is the first one available in the literature. Finally, the efficient, unconditionally stable and accurate numerical method can be applied to study quantized vortex interactions in the GLSE with different initial set-ups, and comparisons to the solutions of the reduced dynamics [26,35,36,39] of the GLSE are reported [49]. In addition, we point out that the numerical method discussed here can be extended to study the dynamics and the interaction of vortex line states in three dimensions as well as in bounded domains for the GLSE.…”

Quantized vortex stability and interaction in the nonlinear wave equation

“…It is well known that there exist stationary vortex solutions with the single winding number (or index) m ∈ Z of the NLWE (1.1) with ε = 1 and V (x) ≡ 1 [25,22,14], which take the form φ m (x) = f m (r ) e imθ , x = (r cos θ, r sin θ ) Numerical solutions of the modulus for different winding numbers m were reported in the literature [25,34,35] by solving the boundary value problem (1.5)-(1.6) numerically. In addition, the core size r 0 m of a vortex state with winding number m is defined by the condition f m (r 0 m ) = 0.755, and then when m = ±1, the core size r 0 1 ≈ 1.75 [34,35].…”