For the Swift-Hohenberg growth rate

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Fourier analysis and integral transforms

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

Problem

For the Swift-Hohenberg growth rate σ(q)=r−(1−q2)2\sigma(q) = r-(1-q^2)^2, write q=1+δq=1+\delta with δ\delta small. (a) Expand 1−q21-q^2 to first order in δ\delta, square the result, and hence derive an approximate band width Δq\Delta q as a function of rr (i.e.\ show that it takes the generic finite-wavenumber form Δq∝r1/2\Delta q \propto r^{1/2}). (b) Evaluate the approximate band width at r=0.09r=0.09 and compare it with the exact band width obtained by solving q2=1±rq^2=1\pm\sqrt r directly.

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