Ginzburg-Landau equation

A reaction-diffusion (RD) system,

where the concentration c depends on the spatial position vector r and time t, and D is a diffusion matrix.

Close to the onset of a supercritical Hopf bifurcation of a homogeneous solution, the RD system can be described by the complex Ginzburg-Landau equations (CGLE). The cubic form is:

The cubic complex Ginzburg-Landau equation is one of the most-studied nonlinear equations in the physics community. It describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose-Einstein condensation to liquid crystals and strings in field theory. The CGLE is a minimal, universal model that cannot be further simplified.
The CGLE admits plane wave solutions of the form, the amplitude, and frequency determined by the dispersion relation.

Forced Ginzburg-Landau equation: (n:1 forcing)
All parameters are real:

µ represents the distance from the Hopf bifurcation, which gives the exponential growth rate of homogeneous perturbations from the A=0 state;

nu=Omega-omega_f/n is a detuning, a frequency shift;

alpha represents nonlinear frequency correction;

beta represents dispersion, and

gamma is the forcing amplitude.

This equation describes the dynamics of the complex amplitude field in a stroboscopic representation determined by multiples of the driving period 2pi.n/omega_f. [about FCGLE:J.M. Gambaudo, J. Diff. Eqns. 57 (1985) 172; C. Elphick, G. Iooss, E. Tirapegui, Phys. Lett. A 120 (1987) 459; P. Coullet and K. Emilsson, Physica (Amsterdam) 61D,119 (1992).]

For 2:1 resonance, the dispersion relation:

Hopf bifurcation:

Turing bifurcation:

and the critical wavenumber locates:

The codimension-2 Turing-Hopf point locates:

If the linear term is positive (above Hopf bifurcation), the cubic term serves as a saturation to stop the growth of amplitude of oscillation. If the linear term is negative (subcritical), and the cubic provides an excitation, then, the quintic term is needed to play the saturation role. The quintic CGLE:

The subcritical Hopf bifurcation is at mr=0.

There are three possible modulus solutions, |A|=0, and

The 1st one is stable (unstable) for all negative (positive) real part of mu; The middle one (always unstable) exsists between a^2/(4h)a^2/(4h). and their stabilities (solid for stable, dash for unstable):

For wave solution:

Regarding the plane wave solution (not the steady state):
Ipsen and Soerensen [M. Ipsen, P.G. Soerensen, Finite wavelength instabilities in a slow mode coupled complex Ginzburg–landau equation, Phys. Rev. Lett. 84 (2000) 2389–2392.] investigated the effect of a real, slow mode in reaction–diffusion systems near a supercritical Hopf bifurcation. They found the slow mode leads to new finite-wavenumber instabilities which alter traditional Eckhaus and Benjamin–Feir stability criteria for periodic waves.

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Multiple scale analysis

Derive the amplitude equations from a reaction-diffusion model. By doing this, we can build a real connection to the real system, and get guidance for pattern formation.