A growth-rate curve is given by with (the Swift-Hohenberg form)
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
A growth-rate curve is given by σ(q)=r−(1−q2)2 with r=0.16 (the Swift-Hohenberg form). Someone reasons: "The uniform mode has σ(0)=0.16−(1−0)2=−0.84, which is negative, so the uniform state is stable. Stability of a uniform state is a single number, and it cannot depend on the wavelength of the disturbance one happens to choose to test it with. So the whole system is stable at r=0.16."
((a)) Is this reasoning correct? Justify your answer using σ(q).
((b)) If the system is in fact unstable, find the unstable band of wavenumbers and the fastest-growing mode with its growth rate.