The defects of a system where hexagons and rolls are both stable solutions are considered. On the basis of topological arguments we show that the unstable phase is present in the core of the defects. This means that a roll is present in the penta-hepta defect of hexagons and that a hexagon is found in the core of a grain boundary connecting rolls with different orientations. These results are verified in an experiment of thermal convection under non-Boussinesq conditions. PACS numbers: 47.20. Bp, 47.25.Qv Defects play an important role in the dynamics of pattern-forming systems. Specifically, dislocations and grain boundaries in convective patterns of rolls, and spirals and centered defects in chemical reactions, have been the object of several studies. ! However, the structure of defects has not been carefully analyzed in systems where two different symmetries coexist. This is a very important case that appears very often in nature, a typical example being the transition between hexagons and rolls in thermal convection. The competition between patterns associated with different symmetries has recently been discussed on the basis of general arguments. 2 The purpose of this Letter is to study defect properties when hexagons and rolls are stable solutions in a nonequilibrium pattern-forming system.The competition between hexagons and rolls can be described by means of three coupled Ginzburg-Landau equations (GLH), which determine the behavior of the three complex amplitudes A t of the sets of rolls describing the hexagonal structure. Each of them makes an angle of 2;r/3 with each of the others. A qualitative description of the nature of the cores of the various defects which may be observed in this problem can be deduced 3 from an elementary study of the following sixdimensional dynamical system, obtained from GLH, in the limit of homogeneous patterns: 4 d

The selection and competition of Turing patterns in the Brusselator model are reviewed. The stability of stripes and hexagons towards spatial perturbations is studied using the amplitude equation formalism. For hexagonal patterns these equations include both linear and nonpotential spatial terms enabling distorted solutions. The latter modify substantially the stability diagrams and select patterns with wave numbers quite different from the critical value. The analytical results from the amplitude formalism agree with direct simulations of the model. Moreover, we show that slightly squeezed hexagons are locally stable in a full range of distortion angles. The stability regions resulting from the phase equation are similar to those obtained numerically by other authors and to those observed in experiments.

Pinning effects in domain walls separating different orientations in patterns in nonequilibrium systems are studied. Usually, theoretical studies consider perfect structures, but in experiments, point defects, grain boundaries, etc., always appear. The aim of this paper is to perform an analysis of the stability of fronts between hexagons and squares in a generalized Swift-Hohenberg model equation. We focus the analysis on pinned fronts between domains with different symmetries by using amplitude equations and by considering the small-scale structure in the pattern. The conditions for pinning effects and stable fronts are determined. This study is completed with direct simulations of the generalized Swift-Hohenberg equation. The results agree qualitatively with recent observations in convection and in ferrofluid instabilities. ͓S1063-651X͑96͒10707-8͔

The onset of convection in a viscoelastic fluid that obeys the Jeffreys model is investigated. Two boundary conditions have been considered separately: free-free and rigid-rigid. The role played by the retardation time, characteristic of the Jeffreys model, is emphasised. The threshold values of the parameters (critical Rayleigh number, critical wavenumber, onset frequency, etc.) for stationary and oscillatory convection are obtained. The frontier between oscillatory and stationary convection is calculated and the possibility to obtain a codimension-two point is discussed.

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Linear stability analysis of the Bénard–Marangoni problem in a layer of fluid with a deformable free surface is considered. The analysis is restricted to fixed values of the Prandtl and the Biot numbers in order to determine the role of the Crispation number on convection. For a deformable upper interface both stationary and oscillatory instabilities are obtained. These two kinds of instabilities have been studied separately and the corresponding critical wave numbers kc and critical Rayleigh numbers Rc have been obtained numerically. The conditions under which two stationary states, an overstable mode and stationary mode, or two overstable modes can coexist simultaneously are determined. In the last case the possibility to obtain a strong resonance between two overstable modes is also discussed.

The general form of the amplitude equations for a hexagonal pattern including spatial terms is discussed. At the lowest order we obtain the phase equation for such patterns. The general expression of the diffusion coefficients is given and the contributions of the new spatial terms are analysed in this paper. From these coefficients the phase stability regions in a hexagonal pattern are determined. In the case of Benard-Marangoni instability our results agree qualitatively with numerical simulations performed recently.

The transition between hexagons and rolls in convective patterns has been studied. The transition thresholds and changes in the Nusselt number are discussed theoretically in terms of calculations made by Busse (1967a) and with amplitude equations. Experiments have been done in a shallow layer of pure water under non-Boussinesq conditions using complementary techniques: shadowgraph (qualitative), optical (based on the deflections of a laser beam) and calorimetric. The experimental values of the critical Rayleigh number Rc and the critical wavenumber kc are in agreement with theoretical ones. However, theory and experiments show some discrepancies in the slopes of the non-dimensional convective heat flow curve and in the thresholds of the hysteretic hexagons-rolls transition. These discrepancies are discussed in terms of lateral effects and of the presence of defects in the pattern.