Ginzburg-Landau equation complex

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. 

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]... 

Abstract Pattern formation is a widespread phenomenon observed in different physical, chemical and biological systems on varions spatial scales, including the nanometer scale. In this chapter discussed are the universal features of pattern formation pattern selection, modulational instabilities, structure and dynamics of domain walls, fronts and defects, as well as non-potential effects and wavy patterns. Principal mathematical models used for the description of patterns (Swift-Hohenberg equation, Newell-Whitehead-Segel equation, Cross-Newell equation, complex Ginzburg-Landau equation) are introduced and some asymptotic methods of their analysis are presented. 

The latter equation is called the complex Ginzburg-Landau equation. Using the notation A = Rexp(ib), one can rewrite (152) as a system of two real equations... 

A vast literature is devoted to the investigation of properties of the complex Ginzburg-Landau equation (see the review paper [65]). Here we discuss only a few of the most basic topics. 

D complex Ginzburg-Landau equation spiral wave... 

In the 2D case, the complex Ginzburg-Landau equation reads ... 

B. J. Matkowsky and V. Volpert, Coupled nonlocal complex Ginzburg-landau equations in gasless combustion, Physica D, 54 (1992), pp. 203-219. 

In [23] a weakly nonlinear analysis for the case of counterpropagating waves was considered. In this case coupled nonlocal complex Ginzburg-Landau equations for the amplitudes of the coimterpropagating waves were derived. We now summarize the derivation. Again, for simplicity, we consider the problem in Cartesian coordinates. Moreover, though the derivation in [23] was for the case when the effect of melting was accounted for, here we do not include melting effects. 

Equations (3.63) and (3.63) constitute a coupled set of complex Ginzburg-Landau equations. They differ from the usual complex Ginzburg-Landau equations by the appearance of the averaged terms, which may be interpreted as follows. The functions R and Si (together with their complex conjugates) are... 

Complex Ginzburg-Landau equations, 266 Contact line, 160, 163-164, 172, 179, 184 Convection, 61... 

Perhaps a sign of the growing importance of this approach is a recent review article in the journal Reviews of Modern Physics under the title The world of the complex Ginzburg-Landau equation (by I. Aranson and L. Kramer), just on solutions and properties of a particular type of order parameter model. 

It should be noted that the Ginzburg-Landau equation subject to the no-flux boundary conditions is invariant under the spatial inversion -x, which was also the case for the phase turbulence equation. Although this kind of symmetry property was not very important for the onset of phase turbulence, the same property is crucial to the understanding of the peculiar bifurcation structure in the present case. It is appropriate to make use of the system s symmetry by introducing a complex variable W(x, t) via... 

By taking the complex conjugate of this equation, and changing the sign before C2 at the same time, the equation remains invariant. This says that the only relevant parameter is the absolute value of C2. As we see below, sufficiently large c21 causes turbulence. Since C2 = Im /Re, where g is the nonlinear parameter in the original form of the Ginzburg-Landau equation (2.4.10), c2 oo as Re 0 (i.e., as the system approaches the borderline between supercritical and subcritical bifurcations). A number of kinetic models can have parameter values for which Re = 0, so that such systems should in principle exhibit chemical turbulence of the type discussed below. 

One is the time-dependent Ginzburg-Landau equation which is described by a complex order parameter and vector potential. The other is the Langevin-type stochastic equation of motion for magnetic vortices in two and three dimensions, which is described in terms of vortex position variables. 

Many features become more transparent when formulated in real (position) space in terms of ampbtude (envelope) or Ginzburg-Landau equations (GLE). Then one sees that the important information is really condensed in a few parameters and the universal aspects of the systems become apparent. By model calculations, which can often be performed analytically, stability boundaries and secondary bifurcation scenarios are traced out. The real space formulation is essential when it comes to the description of more complex spatio-temporal patterns with disorder and defects, which have been studied extensively in EHC slightly above threshold (Figs. 13.1b, 13.3b). One introduces a modulation ampbtude y4(x) defined as... 

The symbols su (space units) and tu (time units) designate here the spatial and temporal units of the complex Ginzburg- Landau equation [Equation (1)]. 

In some cases, one is interested in the structures of complex fluids only at the continuum level, and the detailed molecular structure is not important. For example, long polymer molecules, especially block copolymers, can form phases whose microstructure has length scales ranging from nanometers almost up to microns. Computer simulations of such structures at the level of atoms is not feasible. However, composition field equations can be written that account for the dynamics of some slow variable such as 0 (x), the concentration of one species in a binary polymer blend, or of one block of a diblock copolymer. If an expression for the free energy / of the mixture exists, then a Ginzburg-Landau type of equation can sometimes be written for the time evolution of the variable 0 with or without flow. An example of such an equation is (Ohta et al. 1990 Tanaka 1994 Kodama and Doi 1996)... 

MSI) that uses the same time-dependent Ginzburg Landau kinetic equation as CDS, but starts from (arbitrary) bead models for polymer chains. The methods have been summarized elsewhere. Examples of recent applications include LB simulations of viscoelastic effects in complex fluids under oscillatory shear,DPD simulations of microphase separation in block copoly-mers ° and mesophase formation in amphiphiles, and cell dynamics simulations applied to block copolymers under shear. - DPD is able to reproduce many features of analytical mean field theory but in addition it is possible to study effects such as hydrodynamic interactions. The use of cell dynamics simulations to model non-linear rheology (especially the effect of large amplitude oscillatory shear) in block copolymer miscrostructures is currently being investigated. ... 

A typical feature of a non-potential systems is the non-stationary oscillatory behavior that usually manifests itself in the propagation of waves. We have shown that the nonlinear evolution of waves near the instability threshold is described by the complex Ginzburg-Landau (CGL) equation. This equation is capable of describing various kinds of instabilities of wave patterns, like the Benjamin-Feir instability. In two dimensions, the CGL equation describes the formation of spiral waves that are observed in many biological and chemical systems characterized by the interplay of diffusion and chemical reactions at nano-scales. 

We first describe some analytical results and then describe results obtained from computations. We analytically derive coupled nonlocal complex Ginzburg-Landau type equations for the amplitudes of counterpropagating waves along the front as functions of slow temporal and spatial variables. The equations are... 

Consider a pair of Ginzburg-Landau oscillators which are interacting through a discretized version of diffusion. The evolution equations for the complex amplitudes W = X+ Y and W - X are then given by... 

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