scholarly journals Stable Solutions with Zeros to the Ginzburg–Landau Equation with Neumann Boundary Condition

1996 ◽  
Vol 128 (2) ◽  
pp. 596-613 ◽  
Author(s):  
Shuichi Jimbo ◽  
Yoshihisa Morita
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Landau Equation ◽  
Neumann Boundary ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Stable Solutions

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

2013 ◽  
Vol 14 (3) ◽  
pp. 819-850 ◽  
Author(s):  
Weizhu Bao ◽  
Qinglin Tang
Keyword(s):  
Boundary Condition ◽  
Numerical Methods ◽  
Neumann Boundary Condition ◽  
Landau Equation ◽  
Neumann Boundary ◽  
Quantized Vortex ◽  
Bounded Domains ◽  
Vortex Interaction ◽  
Ginzburg Landau ◽  
Homogeneous Neumann Boundary Condition

AbstractIn this paper, we study numerically quantized vortex dynamics and their interaction in the two-dimensional (2D) Ginzburg-Landau equation (GLE) with a dimensionless parameter ε > 0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a review of the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.

scholarly journals Non-Constant Stable Solutions of the Ginzburg-Landau Equation in Finite Domains

1992 ◽  
Vol 87 (2) ◽  
pp. 507-511
Author(s):  
M. Hirayama ◽  
H. Hirayama ◽  
M. Igai ◽  
J. Ishida
Keyword(s):  
Landau Equation ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Stable Solutions ◽  
Finite Domains

Highly chirped dissipative solitons as a one-parameter family of stable solutions of the cubic–quintic Ginzburg–Landau equation

2011 ◽  
Vol 28 (10) ◽  
pp. 2314 ◽  
Author(s):  
Denis S. Kharenko ◽  
Olga V. Shtyrina ◽  
Irina A. Yarutkina ◽  
Evgenii V. Podivilov ◽  
Mikhail P. Fedoruk ◽  
...  
Keyword(s):  
Landau Equation ◽  
Parameter Family ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Dissipative Solitons ◽  
Stable Solutions

scholarly journals VORTEX LIQUIDS AND THE GINZBURG–LANDAU EQUATION

2014 ◽  
Vol 2 ◽  
Author(s):  
MATTHIAS KURZKE ◽  
DANIEL SPIRN
Keyword(s):  
Vortex Dynamics ◽  
Hydrodynamic Limit ◽  
Convergence Rates ◽  
Landau Equation ◽  
Main Tool ◽  
Neumann Boundary ◽  
Time Dependent ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Large Numbers

AbstractWe establish vortex dynamics for the time-dependent Ginzburg–Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.

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