Eckhaus instability

One can see that the roll solutions within the interval 1/3 < K < 1 are unstable with respect to disturbances with k / 0, ky = 0, i.e. to longitudinal modulations. This kind of instability in nonlinear dissipative systems was discovered by Eckhaus [48] and is called the Eckhaus instability. 

Figure 17. Function kB k) defined by (134) and intervals of k corresponding to a stable pattern ks < k < kEs), Eckhaus instability (kL < k < kEL) and zigzag instability (fc < k < ks).

Tuning of the coefficients allows us to reproduce details of the numerically obtained stability diagrams. Specifically, in addition to the Eckhaus instability (the disturbance wavevector is parallel to that of the roll) and zigzag instability (the disturbance wavevector is orthogonal to that of the roll), we can predict a skewed-varicose instability characterized by a disturbance wavevector inclined with respect to the wavevector of the roll. 

In a next step the Eckhaus instability boundaries (curves E in Fig. 13.7), which probe non-linear aspects of the system, could be identified by observing the destabilization of the pattern via longitudinal modulations after a frequency jump [29, 36, 37] . Subsequently the... 

The mean flow also has the effect of transforming the Eckhaus instability on the large q side into a SV instability. This effect becomes noticeable only in the upper part as the ZZ line is approached because at smaller e the ratio Syls is very small, see the broken line in Fig. 13.8 a. In fact the SV instability then turns around and joins the ZZ line smoothly. This effect indicates that the SV instability may become relevant and could be a clue to the observations quoted above [29, 37, 108]. 

All the wavelengths which are amplified for b > b do not correspond to physically realisable patterns. The curve D.. (q, e ) = 0 and Dj (q.> e ) = 0 delineate the region where the structure (rolls) is stable against inhomogeneous fluctuations. From Eq. (111.6,7)> the system becomes unstable with respect to compression and dilatation of the rolls for > e / 3 (Eckhaus instability) whereas for Q < 0, it is unstable against wavy distor-... 

Fig. 4. Evolution of a spiral wave which results from the integration of Equation (1) with initial conditions similar to (8). (a) Breaking of the wave front emitted by the spiral due to the Eckhaus instability, (b) Proliferation of defects in the system. The latter are pushed out of the system by the front of the spiral. The parameters of Equation (1) are fi — -a = 0.8 and the size of the system is L = 170.

Our numerical simulations confirm thus the predictions of Aranson et al. These considerations are relevant for systems of large size. In the next section, we turn to systems having smaller sizes. In such systems we will see that the Eckhaus instability manifests itself in a different manner than in infinite and large systems. 

The looping motion reported in case (iv) is related to the Eckhaus instability. Indeed, as mentionned in the preceeding section, the value of /3 has been chosen such as the selected wavenumber qs P) of the spiral does not satisfy the Eckhaus condition (6). On the other hand we have performed other numerical simulations with (3 < (3c, i e., in the case where there is no Eckhaus instability. In this situation, no looping motion is observed. 

Fig. 18. Eckhaus instability with transient hexagonal state (a) Initial state (t = 0) (b) First transient state with local hexagonal phase (c) Second transient state with a pair of moving dislocations. Parameters (t = 100 000) a = 7.1, /3 = 5,7 = 1, d = 20, size 100 x 100.

When there is more than one mode in the initial solution, the analysis can be continued on the same lines. The case of hexagonal patterns close to onset has been discussed on the basis of amplitude equations [83, 84]. In this case, the zig-zag instability is masked by the Eckhaus instability. For more details on this particular case, see also [24] in this volume. 

The theory of pattern formation predicts other instabilities beyond the primary instability. One instability leads to the zigzag patterns discussed in Section 4. Another instability that has been well studied, particularly in convective systems, is the Eckhaus instability, which produces a long wavelength pattern. The Eckhaus instability has not been observed in reaction-diffusion systems but should exist for some parameter conditions. 

Eckhaus stability limit (E). The secondary and tertiary instabilities that limit the regions from above will be discussed further below. 

Eckhaus and zig-zag instabilities also play a predominant role in the phase stability of square patterns (M = 2) [55]. However, as this type of structure is not expected in chemical systems (because usually qd < Qnd) its phase instabilities will not be discussed further here. 

Fig. 5. Bifurcation diagrams in the control parameter (/x)/wavenumber (k) representation showing the results of the stability analysis in the sidebands of active modes (a) the Eckhaus and zig-zag instability reduce the band of stable stripes to the hatched tong (b) their own phase instabilities lead to similar effects for the patterns of hexagonal symmetry. The reduced stripes domain (cf. (a)) is also represented.

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