Influence of lateral boundaries on the Eckhaus instability in anisotropic pattern-forming systems
This paper examines the influence of lateral boundaries upon the existence and stability of roll type instabilities in two-dimensional systems having axial symmetry, the lateral boundaries being perpendicular to the axis of the rolls. A weakly nonlinear analysis, incorporating slow space and time variables, is used to investigate stationary solutions of the amplitude equation which arises from a solvability condition. Solutions are shown to exist only when \(\epsilon \equiv (R-R_ c)/R_ c\) exceeds a certain critical value \(\epsilon_ c\), where \(R_ c\) is the linear threshold value. Both analytical and numerical methods are employed in performing a linear stability analysis of the nonlinear solution. It is found that deviations from the universal Eckhaus ratio of 1/\(\sqrt{3}\), due to the finite width of the system, are small. However the presence of lateral boundaries gives rise to an increase in the modulation wave number of the fastest growing linear mode and this effect can be considerable as the ratio \(\epsilon_ c/\epsilon\) becomes large. In concluding, a quantitative comparison is made with results concerning electrohydrodynamic instabilities in nematic liquid crystals. This paper should be of interest to researchers concerned with nonlinear convective instabilities in anisotropic systems.
scientific article; zbMATH DE number 4055241