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Amplitude

# Amplitude - Theory of Amplitude Equations Previously we...

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Theory of Amplitude Equations Previously we have discussed the evolution of infinitesimal perturbations of a uniform state into saturated, stationary, spatially periodic solutions. By restricting attention to these simple solutions, it was straightforward to formulate the effects of the nonlinearities, using analytical methods near threshold and fairly simple numerical methods further from threshold. However, most realistic geometries do not permit spatially periodic solutions since these are usually not compatible with the boundary conditions at the lateral walls. Even if periodic solutions are consistent with some finite domain, they do not exhaust all the possibilities. More typically, patterns have an ideal form (stripes, hexagons, etc.) over small regions, and these ideal forms are distorted over long length scales, disrupted in localized regions by defects, and these distortions and defects are time-dependent. The amplitude equation formalism provides a method to study such effects. Amplitude equations capture three basic ingredients of pattern formation: the growth of the perturbation about the spatially uniform state, the saturation of the growth by nonlinearity, and what we will loosely call dispersion , namely the effect of spatial distortions. (In a Fourier decomposition, dispersion corresponds to a range of component modes with wave vectors centered about the critical wave vector q c .) The interplay of these three effects lies at the heart of pattern formation, and amplitude equations have yielded many useful quantitative insights. In addition, amplitude equations provide a natural extension of the classification based on the linear instability into the weakly nonlinear regime. We will see that the form of an amplitude equation is dictated by the linear classification (type-I s , type-III o , etc.) together with some simple assumptions about the effect of nonlinearity.

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Amplitude Equation for Stripe States Origin and Meaning of the Amplitude In analyzing the linear instability of a spatially uniform state we looked at the dynamics of monochromatic disturbances (described by a single Fourier mode): This ansatz may be appropriate for laterally finite systems with periodic boundary conditions close to onset of instability, but in the more general case (e.g., for infinite periodic boundary conditions) we have to consider a superposition of all unstable modes: The complex-valued amplitude represents slow modulation of the pattern corresponding to the critical onset mode Indeed, the integral is composed of spatial Fourier modes which vary slowly in space just above onset since where ε = ( p p c )/ p c is the reduced bifurcation parameter. Similarly, the amplitude varies very slowly in time as the growth rates for the unstable modes are small, just above onset.

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