The Swift-Hohenberg amplitude equation neglects a slaved third harmonic of amplitude…

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Pattern formation: instabilities that choose a wavelength

Problem

The Swift-Hohenberg amplitude equation neglects a slaved third harmonic of amplitude approximately A3/[4(64−r)]A^3/[4(64-r)], generated by the cubic term acting on Acos⁡xA\cos x. At r=0.1r = 0.1 this third-harmonic amplitude is about 2×10−42\times10^{-4}, negligible next to the fundamental A∗=0.365A^* = 0.365.Estimate the fractional size of the third harmonic relative to the fundamental at r=0.3r = 0.3, and comment on whether the weakly nonlinear (small-AA) approximation used to derive the amplitude equation is still trustworthy there.

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