Sources and sinks in traveling wave systems
In many pattern forming systems - Rayleigh Bénard convection in binary fluids, coupled chemical reactions, convection driven by horizontal temperature gradients - the first basic instability is one to traveling waves. Sometimes, these instabilities are refered to as a Hopf instability: this name refers to an instability to an oscillatory mode, and if one sits at a fixed position, a traveling wave gives rise to an oscillation of the basic field.Over the last two decades, systems close to a stationary instability (i.e., systems where the basic pattern is stationary, not a traveling wave pattern), 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 wavelenght pattern. The predictions of these amplitude approaches have been extensively tested for these systems.
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. Although experimental work has often the driving force behind much of the theoretical work on the Complex Ginzburg Landau equation, the situation regarding the comparison of theory and experiment in these systems is much less satisfactory than for systems exhibiting a stationary bifurcation. This is largely due to the fact that the various coefficients in the Complex Ginzburg Landau equations are often not too well known and difficult to extract from experiments. A few years ago, however, Martin van Hecke (then a graduate student at Leiden) and I were motivated by experiments of Alvarez (then at Drexel University) on traveling waves near a heated wire, to think about sources and sinks in traveling wave systems. Sources and sinks are examples of so-called coherent structures, structures with a well-defined shape and size that separate domains of the two different traveling wave states. Basically, a source sends out traveling waves to both sides, whereas a sink is sandwiched between two incoming traveling wave states.
We found that sources and sinks have a number of properties that makes them very suitable for confronting theory and experiment in traveling wave systems. This is because many of their properties (in particular their multiplity, i.e., roughly speaking, how many of them there are) are independent of the precise coefficients in the coupled amplitude equations for such systems.
In the last two years, we have done an extensive study, together with Kees Storm, who presently is a graduate student at the Instituut-Lorentz, of sources and sinks and their dynamical implications for the one-dimensional coupled Complex Ginzburg Landau equations. Since there typically is a unique source solutions for fixed parameters in the equations, this source organizes much of the dynamics, in much the same way as spirals organize much of the dynamics of the coupled equations in two dimensions. It turns out that there are various dynamical regimes that are mediated by this source behavior: (i) Sources can exhibit a core-instability, associated with the transition from convective to absolute instability in the core; (ii) The unique traveling wave state that a source selects can become absolutely or convectively unstable; (iii) When the coupling parameter of the two modes is not too strong, there can be a bimodal instability: even though the single mode selected by the source is itself stable to finite wavelength perturbations, the second mode is insufficiently suppressed and grows. (iv) Finally, there can be mixed instabilities, i.e., combinations of the above instabilities.
As a result of all these instabilities, there is a very rich phase diagram with a plethora of dynamical mechanisms and chaotic behaviors. Some of these can be reasonably well understood in terms of the behavior of sources and sinks, others remain to be explored.
It worth mentioning that we have regular contacts with Mark Westra and Willem van de Water of the Technical University of Eindhoven, who have built a large scale heated wire experiment of the type used by Alvarez, that allows them to explore the dynamical regimes in this experiment in detail.
References (with entries to the literature)
R. Alvarez, M. van Hecke and W. van Saarloos, Sources and sinks separating domains of left- and right-traveling waves: experiment versus amplitude equations, Phys. Rev. E 56, R1306 (1997).
M. van Hecke, C. Storm and W. van Saarloos, Sources, sinks and wavenumber selection in coupled CGL equations and implications for counterpropagating wave systems, to appear in Physica D.July 12, 1999
[Pattern formation] [Wim van Saarloos] [Instituut-Lorentz]