Vortex solutions of the evolutionary Ginzburg-Landau type equations

TL;DR

This paper investigates vortex solutions in 2D for two types of time-dependent Ginzburg-Landau equations, deriving vortex evolution equations, analyzing stationary points, and illustrating vortex trajectories in different physical scenarios.

Contribution

It introduces a unified ODE system describing vortex dynamics for both heat-flow and Schrödinger Ginzburg-Landau equations, including analysis of stationary states and vortex trajectories.

Findings

Existence of stationary vortex solutions.

Derived ODE system for vortex evolution.

Examples of vortex trajectories in different regimes.

Abstract

We consider two types of the time-dependent Ginzburg-Landau equation in 2D bounded domains: the heat-flow equation and the Schroedinger equation. The system of ordinary differential equations is obtained that describes the evolution of the vortices. It is shown that there exist the stationary points for the both types of the equations. The motion of the particles is studied and the examples of the trajectories are presented in the cases when the particles move like electric charges and hydrodynamic vortices.