Projecting the Swift-Hohenberg equation onto a single mode gives the amplitude equation

PHYSICS · Multi-part · 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

Projecting the Swift-Hohenberg equation onto a single mode u=A(t)cos⁡xu = A(t)\cos x gives the amplitude equation A˙=rA−34A3\dot A = rA - \tfrac34 A^3, with the neglected third-harmonic amplitude estimated as A3/[4(64−r)]A^3/[4(64-r)]. ((a)) Find the saturated amplitude A∗A^* for r=0.09r = 0.09. ((b)) Estimate the size of the neglected third harmonic at this rr, and comment on whether dropping it was justified.

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