The general goal of this project is to contribute to the understanding of spatio-temporal structures that arise spontaneously in many driven dissipative systems. Mathematically, these systems represent dynamical systems with many degrees of freedom. The PI will focus on waves in two-dimensional systems with axial anisotropy. The immediate motivation is given by recent experiments by G. Ahlers et al. on electroconvection in nematic liquid crystals, where three interesting regimes have been observed: 1) extended waves exhibiting spatio-temporally chaotic dynamics, 2) waves localized into narrow, long patches (`worms'), and 3) temporally irregular, spatially localized bursting of the wave amplitude. These phenomena are found immediately above onset at small amplitudes of the waves. They therefore afford an excellent opportunity to develop and test mathematical ideas that will be relevant quite generally in weakly nonlinear theories of dissipative structures. Furthermore, due to the anisotropy this two-dimensional system can be described systematically by Ginzburg-Landau-type equations. No recourse to phenomenological equations of the Swift-Hohenberg-type is needed. The main part of the project is devoted to the worms. They cannot be described by the Ginzburg-Landau equations derived by straightforward asymptotic analysis. The analysis has to be extended to include certain additional modes that are strictly speaking not on the center manifold but become relevant already very close to threshold. The main questions to be addressed are: the localization mechanism of the worms and suitable one- dimensional reductions (standing- and traveling-wave pulses), the stability of long worms, and the nucleation of worms from small noise. The project will demonstrate how in cases like the one described here the qualitative and quantitative relevance of the weakly nonlinear theory can be drastically increased by including a single additional mode. This aspect carries over to a wide range of other systems. On a more specific level, the advection of a slow mode will be important in other wave systems as well. At the same time, for the electroconvection system detailed qualitative and quantitative comparisons with ongoing experiments will be achieved through close cooperation with the experimentalists.
In wide range of systems spatial structures appear spontaneously. These patterns can be time-independent or may have the form of traveling or standing waves. Just a few examples are animal coat markings, waves on the surface of vibrated fluids, cloud streets, waves in chemical reactions and in nerve conduction. If the properties of a system are the same over its whole extent one would usually expect that the patterns also cover the whole system. However, in quite a number of cases a spontaneous localization of the pattern in patches or even single wave crests has been observed. In repeated runs of the same experiment the patches occur in randomly varying locations. This indicates that their localization is not due to special properties of the system at that location. Such localized waves have been observed, e.g., in vibrated granular material, fluid convection in mixtures, and in nematic liquid crystals. The funded research will contribute to the understanding of the causes that lead to localization focussing on the `worms' observed in liquid crystals. It will elucidate the mechanism in detail and investigate the chaotic bursting of such structures and how they are generated (nucleated) from weak noise that is due to the thermal motion in the fluid. Using advanced mathematical methods, the research will bring out the central features that are relevant more generally in other systems as well.
