The Complex Ginzburg Landau equation: coherent structures and spatiotemporal chaos
Over the last two decades, systems just above the onset of a finite-wavelength instability have been analyzed in detail using so-called amplitude equations. These amplitude equations describe the slow modulation in space and time of the envelope of the finite wavelength pattern. Especially near stationary bifurcations, the predictions of these amplitude approaches have been compared in detail with experiments. In fact, often experimental observations provided much of the impetus for theoretical studies - an extensive account of both the theory and the experiments as well as the confrontation between the two can be found in the review by Cross and Hohenberg cited below.The appropriate amplitude equations for traveling wave systems near threshold are extensions of the amplitude equations for stationary systems, in which the coefficients of the various terms are complex (the imaginary parts of these coefficients are associated with the change of the frequency of the waves with the wavenumber, and with the amplitude of mode). These equations are often refered to as the Complex Ginzburg Landau equations.
In comparison with the amplitude equation valid near a stationary bifurcation, which is sometimes refered to as the real Ginzburg Landau equation, even the single complex Ginzburg Landau equation, which describes the pattern envelope if just one of the two traveling wave modes is present, is extremely rich. This is basically due to the following. The real Ginzburg Landau equation can be derived from a Lyapunov functional or "free energy". This Lyapunov functional decreases under the dynamics of the equation. In other words: the dynamics of this equation is essentially relaxational. The Complex Ginzburg Landau equation, on the other hand, is not relaxational. It is, of course, in the limit in which it reduces to the real equation (i.e., the limit when all the complex coefficients become real), but in general it is not. In fact, in the opposite limit in which the real parts of the coefficients vanish, the cubic form of the equation reduces to the so-called Nonlinear Schroedinger equation, an equation that is Hamiltonian, has infinitely many conservations laws, and is integrable (it has true soliton solutions). Since the Complex Ginzburg Landau equation has such different behavior in the two extreme limits, the dynamics in the region in between is extremely complex. Not only are there new types of coherent structure, like stable pulse solutions, sources, sinks, homoclinic orbit solutions, etc., the equation has regimes where the behavior is intrinsically chaotic. The latter type of behavior is the reason that the Complex Ginzburg Landau dynamics is often studied as a prototype equation for spatiotemporal chaos.
In the early nineties, Pierre Hohenberg (then at AT&T Bell Labs, now at Yale) and I made an extensive study of coherent stuctures in the single Complex Ginzburg Landau equation, both with cubic and with quintic nonlinearities. These coherent structures are often the building blocks of more complicated dynamical behavior. One of the surprises we found was that for the quintic equation, one can find the exact nonlinear front solution that organizes much of the dynamics both in the subcritical and supercritical regime (in the more modern language of front propagation, this is the pushed front solution). E.g., the range of existence of stable pulse solutions (which were first studied by Thual and Fauve, and by Hakim and Pomeau in France) is governed by the properties of this solution, while in the parameter ranges where this solution does not exist, the front dynamics is usually incoherent in the supercritical range.
In another study with Chate (now at Saclay), Hohenberg, Shraiman (Bell Labs), Pumir (now at the INLN in Nice) and Holen (a summer student at Bell Labs) we identified various phases of spatio-temporal chaos in the cubic Complex Ginzburg Landau Equation. These phases are easily identified, e.g., by considering whether or not there are spatio-temporal defects. Nevertheless, whether there are really sharp phase transitions between these "phases" has remained a matter of debate, as it is an issue which is difficult to settle numerically.
Finally, there is an interesting twist to the story: shortly after leaving Leiden, where he did his PhD with me, Martin van Hecke discovered that Hohenberg and I overlooked an important coherent structure, a uniformly traveling solution which corresponds to a homoclinic orbit in phase space. Moreover, he showed that this solution is an important building block of the spatiotemporally chaotic regime of the cubic Complex Ginzburg Landau equation!
References
W. van Saarloos and P. H. Hohenberg, Fronts, Pulses, Sources and Sinks in generalized complex Ginzburg-Landau equations, Physica D 56 303-367 (1992).
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).Other references
H. Chate, Spatiotemporal intermitttency regimes of the one-dimensional complex Ginzburg-Landau equation, Nonlinearity 7 185 (1994).
M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 851 (1993). In section VI.B the above-mentioned work on coherent structures is reviewed.
M. L. van Hecke, E. de Wit, and W. van Saarloos, Coherent and incoherent drifting pulse dynamics in a complex Ginzburg-Landau equation, Phys. Rev. Lett. 75, 3830 (1995).
M. van Hecke, Building Blocks of Spatiotemporal Intermittency, Phys. Rev. Lett. 80 1896 (1998).July 13, 1999
[Pattern formation] [Wim van Saarloos] [Instituut-Lorentz]