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

Abstract

The rich dynamics of quantized vortices governed by the Ginzburg-Landau-Schrödinger equation (GLSE) is an interesting problem studied in many application fields. Although recent mathematical analysis and numerical simulations have led to a much better understanding of such dynamics, many important questions remain open. In this article, we consider numerical simulations of the GLSE in two dimensions with non-zero far-field conditions. Using twodimensional polar coordinates, transversely highly oscillating far-field conditions can be efficiently resolved in the phase space, thus giving rise to an unconditionally stable, efficient and accurate time-splitting method for the problem under consideration. This method is also time reversible for the case of the non-linear Schrödinger equation. By applying this numerical method to the GLSE, we obtain some conclusive experimental findings on issues such as the stability of quantized vortex, interaction of two vortices, dynamics of the quantized vortex lattice and the motion of vortex with an inhomogeneous external potential. Discussions on these simulation results and the recent theoretical studies are made to provide further understanding of the vortex stability and vortex dynamics described by the GLSE.

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.…”

“…In this case, it is natural to adopt the polar coordinate (r,θ) in our numerical discretization. In order to discretize (3.2) with either (1.3) or (1.4), we apply the standard Fourier pseudospectral method in θ-direction [34], finite element method in r-direction, and Crank-Nicolson method in time [4,5,37]. Plugging the following truncated Fourier expansion for ψ ε ψ ε (r,θ,t) =…”

“…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].…”

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

“…One of the most well-studied equations in nonlinear science is the Ginzburg-Landau-Schrödinger equation (GLSE) of the form [36] (α − iβ)∂ t ψ(x, t) = ∇ 2 ψ + 1 ε 2 V (x) − |ψ| 2 ψ, x ∈ R 2 , t > 0, (1.1)…”

A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and Its Application to the Nonlinear Schrödinger Equation

“…As an additional example to test the MSD boundary condition, we simulate two equal-charge vortices whose interaction is known to produce a rotating circular motion of the two vortices orbiting each other [11,33,44,45]. Using a fixed grid size of 171 × 171, the simulations are run for long times using the L0 and MSD boundary conditions.…”