Amplitude Equations for Patterns

Abstract

A typical dispersion relation λ(κ) in a spatially extended system just beyond a bifurcation point may have one of the forms shown in Fig. 4.1. In both cases, a narrow band of wavenumbers adjacent to the maximum of the dispersion relation (which may be reached either at κ = 0 or at a finite κ = κ₀) becomes unstable. Since the dispersion relation should be, generically, parabolic near the maximum, and the leading eigenvalue can be assumed to depend linearly on a chosen bifurcation parameter, say, μ, the width of the excited band scales as the square root of the deviation from the bifurcation point μ–μ₀. A spectral band of a finite width can be modeled by allowing the amplitude to change on an extended spatial scale, large compared to either κ^–1 ₀ or any “natural” length scale characteristic to the underlying system.