Ginzburg-Landau equation

With the fomi of free energy fiinctional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3.48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise temi is ignored) the model B equation is often referred to as the Calm-Flilliard-Cook equation, and as the Calm-Flilliard equation in the absence of the noise temi. 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

Figure C3.6.10 Defect-mediated turbulence in tire complex Ginzburg-Landau equation, (a) The phase, arg( ), as grey shades, (b) The amplitude [A], witli a similar color coding. In tire left panel topological defects can be identified as points around which one finds all shades of grey. Note tire apparently random spatial pattern of amplitudes.

In the previous sections, we briefly introduced a number of different specific models for crystal growth. In this section we will make some further simplifications to treat some generic behavior of growth problems in the simplest possible form. This usually leads to some nonlinear partial differential equations, known under names like Burgers, Kardar-Parisi-Zhang (KPZ), Kuramoto-Sivashinsky, Edwards-Anderson, complex Ginzburg-Landau equation and others. 

If we allow for exchange with the gas phase, then h x) does not obey a conservation condition and we have the simple Ginzburg-Landau equation ... 

For the explanation of macroscopic phenomena, the thickness of the phase boundary (interface) often plays no important role. As an example, we describe the movement of a phase boundary in two dimensions or the movement of a step edge on a crystal surface. We start with a Ginzburg-Landau equation [69]... 

We start from the time-dependent Ginzburg-Landau equation for a non-conserved order parameter 0... 

In this paper we study the distribution of the magnehc held of a neutron star with superconducting CFL quark matter core in the framework of the Ginzburg-Landau theory. We solve the Ginzburg-Landau equations with proper boundary conditions. 

It should be emphasized that this way of including fluctuations has no other justification than that it is convenient and bypasses a description of the noise sources, compare IX.4. It may provide some qualitative insight into the effect of noise, but does not describe its actual mechanism. For instance, fluctuations in the pumping should give rise to randomness in the coefficient a, rather than to an additive term. Yet the equation (7.6) has been the subject of extensive study and it is famous in statistical mechanics under the name of generalized Ginzburg-Landau equation. It may well serve us as an illustration for a stochastic process.510... 

Fig. 2.47 Pseudostable perforated layer structure, observed following a quench from the lam to hex phase using a multimode analysis of the time-dependent Ginzburg-Landau equation, within the single-wavenumber approximation (Qi and Wang 1997). This structure results from the superposition of six BCC-type wavevectors.

L. P. Gor kov, Macroscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity. Sov. Physics JETP 36, 1364-1367 (1959). 

M.M. Doria, J.E. Gubernatis, D. Rainer, Solving the Ginzburg-Landau equations by simulated annealing. Physical Review B Condensed Matter, 41 (1990) 6335. 

H. Chate. Spatiotemporal intermittency regimes of the one-dimensional complex ginzburg-landau equation. Nonlinearity, 7 185— 204, 1994. 

H. Chate and P. Manneville. Phase diagram of the two-dimensional complex Ginzburg-Landau equation. Physica A, 224 348-368, 1996. 

B.I. Shraiman, A. Pumir, W. van Saarloos, P.C. Hohenberg, H. Chate, and M. Holen. Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation. Physica D, 57 241-248, 1992. 

The universal description of reaction-diffusion systems near a supercritical Hopf bifurcation is provided by the complex Ginzburg-Landau equation [11]. Action of global periodic forcing on the systems described by this... 

Under global resonant n l forcing, the complex Ginzburg-Landau equation (CGLE) for the slow complex oscillation amplitude rj is [12]... 

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