The Ginzburg-Landau Equations for Superconductivity with Random Fluctuations

2008 ◽  
pp. 25-133
Author(s):  
Andrei Fursikov ◽  
Max Gunzburger ◽  
Janet Peterson
Keyword(s):  
Ginzburg Landau ◽  
Ginzburg Landau Equations ◽  
Random Fluctuations

Virial theorem and Abrikosov’s solution of the Ginzburg-Landau equations

1991 ◽  
Vol 44 (14) ◽  
pp. 7704-7707 ◽  
Author(s):  
U. Klein ◽  
B. Pöttinger
Keyword(s):  
Virial Theorem ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equations

PATTERNS, DEFECTS, AND EVOLUTION OF BÉNARD–MARANGONI CELLS

1996 ◽  
Vol 06 (09) ◽  
pp. 1665-1671 ◽  
Author(s):  
J. BRAGARD ◽  
J. PONTES ◽  
M.G. VELARDE
Keyword(s):  
Fluid Layer ◽  
Ambient Air ◽  
Finite Geometry ◽  
Amplitude Equations ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Numerical Exploration ◽  
Ginzburg Landau Equations ◽  
Rigid Plane ◽  
Selection Of

We consider a thin fluid layer of infinite horizontal extent, confined below by a rigid plane and open above to the ambient air, with surface tension linearly depending on the temperature. The fluid is heated from below. First we obtain the weakly nonlinear amplitude equations in specific spatial directions. The procedure yields a set of generalized Ginzburg–Landau equations. Then we proceed to the numerical exploration of the solutions of these equations in finite geometry, hence to the selection of cells as a result of competition between the possible different modes of convection.

Geometric effect on the vortex configuration and motion in mesoscopic superconducting cylinders

2015 ◽  
Vol 29 (03) ◽  
pp. 1550009 ◽  
Author(s):  
Shan-Shan Wang ◽  
Guo-Qiao Zha
Keyword(s):  
Phase Transitions ◽  
Vortex Dynamics ◽  
Axial Symmetry ◽  
Time Dependent ◽  
Vortex Matter ◽  
Ginzburg Landau ◽  
Geometric Effect ◽  
Multiply Connected ◽  
Vortex States ◽  
Ginzburg Landau Equations

Based on the time-dependent Ginzburg–Landau equations, we study numerically the vortex configuration and motion in mesoscopic superconducting cylinders. We find that the effects of the geometric symmetry of the system and the noncircular multiply-connected boundaries can significantly influence the steady vortex states and the vortex matter moving. For the square cylindrical loops, the vortices can enter the superconducting region in multiples of 2 and the vortex configuration exhibits the axial symmetry along the square diagonal. Moreover, the vortex dynamics behavior exhibits more complications due to the existed centered hole, which can lead to the vortex entering from different edges and exiting into the hole at the phase transitions.

In–Out Intermittency with Nested Subspaces in a System of Globally Coupled, Complex Ginzburg–Landau Equations

2021 ◽  
Vol 31 (01) ◽  
pp. 2130001
Author(s):  
Gerhard Dangelmayr ◽  
Iuliana Oprea
Keyword(s):  
Normal Form ◽  
Traveling Waves ◽  
Complex Dynamics ◽  
Invariant Subspaces ◽  
Chaotic Attractors ◽  
Governing Equations ◽  
Ginzburg Landau ◽  
Higher Dimensional ◽  
Ginzburg Landau Equations ◽  
Anisotropic Systems

Chaos and intermittency are studied for the system of globally coupled, complex Ginzburg–Landau equations governing the dynamics of extended, two-dimensional anisotropic systems near an oscillatory (Hopf) instability of a basic state with two pairs of counterpropagating, oblique traveling waves. Parameters are chosen such that the underlying normal form, which governs the dynamics of the spatially constant modes, has two symmetry-conjugated chaotic attractors. Two main states residing in nested invariant subspaces are identified, a state referred to as Spatial Intermittency ([Formula: see text]) and a state referred to as Spatial Persistence ([Formula: see text]). The [Formula: see text]-state consists of laminar phases where the dynamics is close to a normal form attractor, without spatial variation, and switching phases with spatiotemporal bursts during which the system switches from one normal form attractor to the conjugated normal form attractor. The [Formula: see text]-state also consists of two symmetry-conjugated states, with complex spatiotemporal dynamics, that reside in higher dimensional invariant subspaces whose intersection forms the 8D space of the spatially constant modes. We characterize the repeated appearance of these states as (generalized) in–out intermittency. The statistics of the lengths of the laminar phases is studied using an appropriate Poincaré map. Since the Ginzburg–Landau system studied in this paper can be derived from the governing equations for electroconvection in nematic liquid crystals, the occurrence of in–out intermittency may be of interest in understanding spatiotemporally complex dynamics in nematic electroconvection.

scholarly journals Justification of mean-field coupled modulation equations

1997 ◽  
Vol 127 (3) ◽  
pp. 639-650 ◽  
Author(s):  
Guido Schneider
Keyword(s):  
Ground State ◽  
Model Problem ◽  
Mean Field ◽  
Periodic Pattern ◽  
Scaling Analysis ◽  
Ginzburg Landau ◽  
Modulation Equations ◽  
Evolution Problems ◽  
Ginzburg Landau Equations ◽  
Infinite Line

We are interested in reflection symmetric (x↦–x) evolution problems on the infinite line. In the systems which we have in mind, a trivial ground state loses stability and bifurcates into a temporally oscillating, spatial periodic pattern. A famous example of such a system is the Taylor-Couette problem in the case of strongly counter-rotating cylinders. In this paper, we consider a system of coupled Kuramoto–Shivashinsky equations as a model problem for such a system. We are interested in solutions which are slow modulations in time and in space of the bifurcating pattern. Multiple scaling analysis is used in the existing literature to derive mean-field coupled Ginzburg–Landau equations as approximation equations for the problem. The aim of this paper is to give exact estimates between the solutions of the coupled Kuramoto–Shivashinsky equations and the associated approximations.

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