Finite amplitude perturbations

In an attempt to shed some light on the wavelength selection Datye and Langer [139] considered finite amplitude perturbations of the local wavelength. This type of approach was used in a somewhat refined version by... 

Figure 19. Enlargement of the region of suflicient conditions for the existence of DS when finite amplitude perturbations are taken into account k = 0.2, e = 2-10 p = 0.011, = var.

It provides a simple model case that illustrates formation of spatio-temporal patterns due to such finite-amplitude perturbations. 

Thus, consider the disturbance of the wavelength A0 located not far from the channel axis at y = y0 and determine how it would grow up until the moment of contact with the wall. Let A0 be an initial amplitude of the disturbance, and Aw the finite amplitude near the wall. The shortwave perturbations are assumed to be small in order to use the linear approach. Since in accordance with Landau theory... 

For positive values of the control parameter , stationary, spatially periodic solutions y/s(x) = y/x(x I 2n/q) of (53) may be found with and without forcing. However, in the case of a vanishing forcing amplitude (a = 0) in (53), this equation has a i//-symmetry and one has a pitchfork bifurcation from the trivial solution l/r = 0 to finite amplitude periodic solutions as indicated in Fig. 19. In the unforced case, however, periodic solutions of (53) are unstable for any wave number q against infinitesimal perturbations that induce coarsening processes [114, 121],... 

Figure 2.43 Finite amplitude, oscillatory perturbation amplitudes 0.01, 0.001 and 0.0001...

VI. On the Formation of DS in Response to Nonuniform Perturbations of Finite Amplitude... 

VI. ON THE FORMATION OF DS IN RESPONSE TO NONUNIFORM PERTURBATIONS OF FINITE AMPLITUDE... 

We find that the pulsating planar solution is stable for small R, but becomes unstable to nonplanar perturbations for larger values of R. Our results show that the transition from pulsating planar, i.e., one dimensional behavior, to spot behavior is a jump transition, i.e., the spots enter with finite amplitude as R increases, consistent with the phenomenological description in [2]. This is not surprising since instability of the uniformly propagating solution is also expected to lead to instability of spinning solutions which differ infinitesimally from it. 

The above equation is based on a linear analysis that applies only for infinitesimal paturbations. Perturbations of finite amplitude are required, however, for a practical experiment. An early papa- of Niels Bohr (1909) dealt with the necessary extension. He also included the effect of liquid viscosity. A corrected equation suitable for perturbations whose amplitude b is finite but still smaller than the mean radius R is... 

Weakly perturbed or weakly interacting finite-amplitude oscillations form a particular class of systems whose dynamics finds an extremely simplified description through the method presented here. 

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